Symbolic integration aims to develop algorithms for evaluating integrals in finite terms. One of its classical topics is determining whether an integrand has an elementary integral, and computing such an integral if one exists. Fundamental results on this topic are collected and reviewed in~\cite{RaSi2022}. Algorithms for integrating transcendental functions are presented in~\cite{Bron2005}. Additive decompositions compute \lq\lq remainders\rq\rq\ and determine the so-called \lq\lq in-field integrability\rq\rq. More precisely, for a given function $f$ in a differential field~$F$, an additive decomposition computes $g, r \in F$ such that \[ f = g^\prime + r \] with the following two properties: \begin{itemize} \item[(i)] (minimality) $r$ is minimal in some technical sense, \item[(ii)] (in-field integrability) $f$ has an integral in $F$ if and only if $r = 0$. \end{itemize} The Hermite-Ostrogradsky reduction \cite{Herm1872,Ostr1845} developed in the 19th century is an additive decomposition for rational functions. %This reduction is considered as the beginning of symbolic integration. However, additive decompositions for more sophisticated functions had not yet been developed until the Hermite reduction for hyperexponential functions \cite{BCCLX2013} was introduced in 2013. We are going to review classical Risch's algorithm for integrating transcendental functions \cite{Risc1969b,Risc1970, Roth1976}, and present recent progress in additive decompositions \cite{CKK2016,CvHKK2018,CDL2018,DGLW2020,vdHo2021,DGGL2023,CDK2023} in this talk. Additionally, new results will be discussed on how to combine classical algorithms with additive decompositions so as to enhance and accelerate the integrators implemented in computer algebra systems such as {\sc Maple} and {\sc Mathematica}. This talk is dedicated to a special memorial of Professor Marko Petkov{\v{s}}ek. His paper \cite{AbPe2002b} coauthored with Sergei Abramov played a key role in bringing additive decompositions back to symbolic integration. %inspired Hermite reduction for hyperexponential functions. %played a key role to bring additive decompositions back to symbolic integration. {\small \begin{thebibliography}{10} \bibitem{AbPe2002b} Sergei~A.~Abramov and Marko Petkov{\v{s}}ek. \newblock Rational normal forms and minimal decompositions of hypergeometric terms. \newblock {\em Journal of Symbolic Computation}, 33(5):521--543, 2002. \bibitem{BCCLX2013} Alin Bostan, Shaoshi Chen, Fr{\'e}d{\'e}ric Chyzak, Ziming Li, and Guoce Xin. \newblock Hermite reduction and creative telescoping for hyperexponential functions. \newblock In {\em {P}roceedings of {ISSAC}'13}, pages 77--84. ACM, New York, 2013. \bibitem{Bron2005} Manuel Bronstein. \newblock {\em Symbolic {I}ntegration {I}: Transcendental Functions}, volume~1 of {\em Algorithms and Computation in Mathematics}. \newblock Springer-Verlag, Berlin, second edition, 2005. \bibitem{CDL2018} Shaoshi Chen, Hao Du, and Ziming Li. \newblock Additive decompositions in primitive extensions. \newblock In {\em Proceedings of {ISSAC}'18}, pages 135--142. ACM, New York, 2018. \bibitem{CDK2023} Shaoshi Chen, Lixin Du and Manuel Kauers. \newblock Hermite reduction for D-finite functions via integral bases. \newblock To appear in {\em Proceedings of {ISSAC}'23}. ACM, New York, 2023. \bibitem{CvHKK2018} Shaoshi Chen, Mark van Hoeij, Manuel Kauers, and Christoph Koutschan. \newblock Reduction-based creative telescoping for fuchsian {D}-finite functions. \newblock {\em Journal of Symbolic Computation.}, 85:108--127, 2018. %\bibitem{CHKL2015} %Shaoshi Chen, Hui Huang, Manuel Kauers, and Ziming Li. %\newblock A modified {A}bramov-{P}etkov\v sek reduction and creative % telescoping for hypergeometric terms. %\newblock In {\em Proceedings of {ISSAC}'15}, pages 117--124. ACM, New York, % 2015. \bibitem{CKK2016} Shaoshi Chen, Manuel Kauers, and Christoph Koutschan. \newblock Reduction-based creative telescoping for algebraic functions. \newblock In {\em Proceedings of {ISSAC}'16}, pages 175--182. ACM, New York, 2016. \bibitem{DGGL2023} Hao Du, Yiman Gao,Jing Guo, and Ziming Li. \newblock Computing logarithmic parts by evaluation homomorphisms. \newblock To appear in {\em Proceedings of {ISSAC}'23}. ACM, New York, 2023. \bibitem{DGLW2020} Hao Du, Jing Guo, Ziming Li, and Elaine Wong. \newblock An additive decomposition in logarithmic towers and beyond. \newblock In {\em Proceedings of {ISSAC}'20}, pages 146--153. ACM, New York, 2020. \bibitem{Herm1872} Charles Hermite. \newblock Sur l'int\'egration des fractions rationnelles. \newblock {\em Ann. Sci. \'Ecole Norm. Sup. (2)}, 1:215--218, 1872. \bibitem{vdHo2021} Joris van der Hoeven. \newblock Constructing reductions for creative telescoping: the general differentially finite case. \newblock {\em Appl. Algebra Engrg. Comm. Comput.}, 32(5):575--602, 2021. \bibitem{Ostr1845} Mikhail V.\ Ostrogradsky. \newblock De l'int{\'e}gration des fractions rationnelles. \newblock {\em Bull.\ de la classe physico-math{\'e}matique de l'Acad.\ Imp{\'e}riale des Sciences de Saint-P{\'e}tersbourg}, 4(145--167):286--300, 1845. \bibitem{RaSi2022} Clemens G.\ Raab and Michael F.\ Singer. \newblock {\em Integration in Finite Terms: Fundamental Sources}. \newblock Text \& Monographs in Symbolic Computation, Springer, 2022. \bibitem{Risc1969b} Robert~H. Risch. \newblock The problem of integration in finite terms. \newblock {\em Trans. Amer. Math. Soc.}, 139:167--189, 1969. \bibitem{Risc1970} Robert~H. Risch. \newblock The solution of the problem of integration in finite terms. \newblock {\em Bull. Amer. Math. Soc.}, 76:605--608, 1970. \bibitem{Roth1976} Michael Rothstein. \newblock {\em {Aspects of Symbolic Integration and Simplification of Exponential and Primitive Functions}}. \newblock PhD thesis, Univ.\ of Wisconsin, Madison, 1976. \end{thebibliography} }