\[ \int \frac {\cos (c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \]
Optimal antiderivative \[ \frac {2 \ln \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} \sin \left (d x +c \right )\right )}{9 a^{\frac {5}{3}} b^{\frac {1}{3}} d}-\frac {\ln \left (a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} \sin \left (d x +c \right )+b^{\frac {2}{3}} \left (\sin ^{2}\left (d x +c \right )\right )\right )}{9 a^{\frac {5}{3}} b^{\frac {1}{3}} d}+\frac {\sin \left (d x +c \right )}{3 a d \left (a +b \left (\sin ^{3}\left (d x +c \right )\right )\right )}-\frac {2 \arctan \left (\frac {\left (a^{\frac {1}{3}}-2 b^{\frac {1}{3}} \sin \left (d x +c \right )\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}}{9 a^{\frac {5}{3}} b^{\frac {1}{3}} d} \]
command
integrate(cos(d*x+c)/(a+b*sin(d*x+c)**3)**2,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \frac {\tilde {\infty } x \cos {\left (c \right )}}{\sin ^{6}{\left (c \right )}} & \text {for}\: a = 0 \wedge b = 0 \wedge d = 0 \\- \frac {1}{5 b^{2} d \sin ^{5}{\left (c + d x \right )}} & \text {for}\: a = 0 \\\frac {\tilde {\infty } \sin {\left (c + d x \right )}}{d} & \text {for}\: b = - \frac {a}{\sin ^{3}{\left (c + d x \right )}} \\\frac {x \cos {\left (c \right )}}{\left (a + b \sin ^{3}{\left (c \right )}\right )^{2}} & \text {for}\: d = 0 \\\frac {\sin {\left (c + d x \right )}}{a^{2} d} & \text {for}\: b = 0 \\- \frac {2 a \sqrt [3]{- \frac {a}{b}} \log {\left (- \sqrt [3]{- \frac {a}{b}} + \sin {\left (c + d x \right )} \right )}}{9 a^{3} d + 9 a^{2} b d \sin ^{3}{\left (c + d x \right )}} + \frac {a \sqrt [3]{- \frac {a}{b}} \log {\left (4 \left (- \frac {a}{b}\right )^{\frac {2}{3}} + 4 \sqrt [3]{- \frac {a}{b}} \sin {\left (c + d x \right )} + 4 \sin ^{2}{\left (c + d x \right )} \right )}}{9 a^{3} d + 9 a^{2} b d \sin ^{3}{\left (c + d x \right )}} + \frac {2 \sqrt {3} a \sqrt [3]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {3}}{3} + \frac {2 \sqrt {3} \sin {\left (c + d x \right )}}{3 \sqrt [3]{- \frac {a}{b}}} \right )}}{9 a^{3} d + 9 a^{2} b d \sin ^{3}{\left (c + d x \right )}} - \frac {2 a \sqrt [3]{- \frac {a}{b}} \log {\left (2 \right )}}{9 a^{3} d + 9 a^{2} b d \sin ^{3}{\left (c + d x \right )}} + \frac {3 a \sin {\left (c + d x \right )}}{9 a^{3} d + 9 a^{2} b d \sin ^{3}{\left (c + d x \right )}} - \frac {2 b \sqrt [3]{- \frac {a}{b}} \log {\left (- \sqrt [3]{- \frac {a}{b}} + \sin {\left (c + d x \right )} \right )} \sin ^{3}{\left (c + d x \right )}}{9 a^{3} d + 9 a^{2} b d \sin ^{3}{\left (c + d x \right )}} + \frac {b \sqrt [3]{- \frac {a}{b}} \log {\left (4 \left (- \frac {a}{b}\right )^{\frac {2}{3}} + 4 \sqrt [3]{- \frac {a}{b}} \sin {\left (c + d x \right )} + 4 \sin ^{2}{\left (c + d x \right )} \right )} \sin ^{3}{\left (c + d x \right )}}{9 a^{3} d + 9 a^{2} b d \sin ^{3}{\left (c + d x \right )}} + \frac {2 \sqrt {3} b \sqrt [3]{- \frac {a}{b}} \sin ^{3}{\left (c + d x \right )} \operatorname {atan}{\left (\frac {\sqrt {3}}{3} + \frac {2 \sqrt {3} \sin {\left (c + d x \right )}}{3 \sqrt [3]{- \frac {a}{b}}} \right )}}{9 a^{3} d + 9 a^{2} b d \sin ^{3}{\left (c + d x \right )}} - \frac {2 b \sqrt [3]{- \frac {a}{b}} \log {\left (2 \right )} \sin ^{3}{\left (c + d x \right )}}{9 a^{3} d + 9 a^{2} b d \sin ^{3}{\left (c + d x \right )}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________