Integrand size = 29, antiderivative size = 188 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {a \left (4 a^2-b^2\right ) x}{b^5}-\frac {2 a^2 \left (4 a^2-3 b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^5 \sqrt {a^2-b^2} d}+\frac {\left (12 a^2-b^2\right ) \cos (c+d x)}{3 b^4 d}-\frac {2 a \cos (c+d x) \sin (c+d x)}{b^3 d}+\frac {4 \cos (c+d x) \sin ^2(c+d x)}{3 b^2 d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{b d (a+b \sin (c+d x))} \]
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Time = 0.51 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2968, 3127, 3129, 3128, 3102, 2814, 2739, 632, 210} \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {2 a^2 \left (4 a^2-3 b^2\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^5 d \sqrt {a^2-b^2}}+\frac {a x \left (4 a^2-b^2\right )}{b^5}+\frac {\left (12 a^2-b^2\right ) \cos (c+d x)}{3 b^4 d}-\frac {2 a \sin (c+d x) \cos (c+d x)}{b^3 d}-\frac {\sin ^3(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}+\frac {4 \sin ^2(c+d x) \cos (c+d x)}{3 b^2 d} \]
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Rule 210
Rule 632
Rule 2739
Rule 2814
Rule 2968
Rule 3102
Rule 3127
Rule 3128
Rule 3129
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sin ^3(c+d x) \left (1-\sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx \\ & = -\frac {\cos (c+d x) \sin ^3(c+d x)}{b d (a+b \sin (c+d x))}-\frac {\int \frac {\sin ^2(c+d x) \left (-3 \left (a^2-b^2\right )+4 \left (a^2-b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{b \left (a^2-b^2\right )} \\ & = \frac {4 \cos (c+d x) \sin ^2(c+d x)}{3 b^2 d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{b d (a+b \sin (c+d x))}-\frac {\int \frac {\sin (c+d x) \left (8 a \left (a^2-b^2\right )-b \left (a^2-b^2\right ) \sin (c+d x)-12 a \left (a^2-b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{3 b^2 \left (a^2-b^2\right )} \\ & = -\frac {2 a \cos (c+d x) \sin (c+d x)}{b^3 d}+\frac {4 \cos (c+d x) \sin ^2(c+d x)}{3 b^2 d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{b d (a+b \sin (c+d x))}-\frac {\int \frac {-12 a^2 \left (a^2-b^2\right )+4 a b \left (a^2-b^2\right ) \sin (c+d x)+2 \left (a^2-b^2\right ) \left (12 a^2-b^2\right ) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx}{6 b^3 \left (a^2-b^2\right )} \\ & = \frac {\left (12 a^2-b^2\right ) \cos (c+d x)}{3 b^4 d}-\frac {2 a \cos (c+d x) \sin (c+d x)}{b^3 d}+\frac {4 \cos (c+d x) \sin ^2(c+d x)}{3 b^2 d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{b d (a+b \sin (c+d x))}-\frac {\int \frac {-12 a^2 b \left (a^2-b^2\right )-6 a \left (a^2-b^2\right ) \left (4 a^2-b^2\right ) \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{6 b^4 \left (a^2-b^2\right )} \\ & = \frac {a \left (4 a^2-b^2\right ) x}{b^5}+\frac {\left (12 a^2-b^2\right ) \cos (c+d x)}{3 b^4 d}-\frac {2 a \cos (c+d x) \sin (c+d x)}{b^3 d}+\frac {4 \cos (c+d x) \sin ^2(c+d x)}{3 b^2 d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{b d (a+b \sin (c+d x))}-\frac {\left (a^2 \left (4 a^2-3 b^2\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{b^5} \\ & = \frac {a \left (4 a^2-b^2\right ) x}{b^5}+\frac {\left (12 a^2-b^2\right ) \cos (c+d x)}{3 b^4 d}-\frac {2 a \cos (c+d x) \sin (c+d x)}{b^3 d}+\frac {4 \cos (c+d x) \sin ^2(c+d x)}{3 b^2 d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{b d (a+b \sin (c+d x))}-\frac {\left (2 a^2 \left (4 a^2-3 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^5 d} \\ & = \frac {a \left (4 a^2-b^2\right ) x}{b^5}+\frac {\left (12 a^2-b^2\right ) \cos (c+d x)}{3 b^4 d}-\frac {2 a \cos (c+d x) \sin (c+d x)}{b^3 d}+\frac {4 \cos (c+d x) \sin ^2(c+d x)}{3 b^2 d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{b d (a+b \sin (c+d x))}+\frac {\left (4 a^2 \left (4 a^2-3 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^5 d} \\ & = \frac {a \left (4 a^2-b^2\right ) x}{b^5}-\frac {2 a^2 \left (4 a^2-3 b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^5 \sqrt {a^2-b^2} d}+\frac {\left (12 a^2-b^2\right ) \cos (c+d x)}{3 b^4 d}-\frac {2 a \cos (c+d x) \sin (c+d x)}{b^3 d}+\frac {4 \cos (c+d x) \sin ^2(c+d x)}{3 b^2 d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{b d (a+b \sin (c+d x))} \\ \end{align*}
Time = 2.11 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.31 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {-\frac {48 a^2 \left (4 a^2-3 b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+\frac {96 a^4 c-24 a^2 b^2 c+96 a^4 d x-24 a^2 b^2 d x+12 a b \left (8 a^2-b^2\right ) \cos (c+d x)+4 a b^3 \cos (3 (c+d x))+96 a^3 b c \sin (c+d x)-24 a b^3 c \sin (c+d x)+96 a^3 b d x \sin (c+d x)-24 a b^3 d x \sin (c+d x)+24 a^2 b^2 \sin (2 (c+d x))-2 b^4 \sin (2 (c+d x))-b^4 \sin (4 (c+d x))}{a+b \sin (c+d x)}}{24 b^5 d} \]
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Time = 1.18 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.32
method | result | size |
derivativedivides | \(\frac {-\frac {4 a^{2} \left (\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2}}{2}-\frac {a b}{2}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {\left (4 a^{2}-3 b^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{5}}+\frac {\frac {4 \left (\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}}{2}+\left (\frac {3}{2} a^{2} b -\frac {1}{2} b^{3}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b -\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a \,b^{2}}{2}+\frac {3 a^{2} b}{2}-\frac {b^{3}}{6}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+2 a \left (4 a^{2}-b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{5}}}{d}\) | \(249\) |
default | \(\frac {-\frac {4 a^{2} \left (\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2}}{2}-\frac {a b}{2}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {\left (4 a^{2}-3 b^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{5}}+\frac {\frac {4 \left (\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}}{2}+\left (\frac {3}{2} a^{2} b -\frac {1}{2} b^{3}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b -\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a \,b^{2}}{2}+\frac {3 a^{2} b}{2}-\frac {b^{3}}{6}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+2 a \left (4 a^{2}-b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{5}}}{d}\) | \(249\) |
risch | \(\frac {4 a^{3} x}{b^{5}}-\frac {a x}{b^{3}}+\frac {i a \,{\mathrm e}^{2 i \left (d x +c \right )}}{4 b^{3} d}+\frac {3 \,{\mathrm e}^{i \left (d x +c \right )} a^{2}}{2 b^{4} d}-\frac {{\mathrm e}^{i \left (d x +c \right )}}{8 b^{2} d}+\frac {3 \,{\mathrm e}^{-i \left (d x +c \right )} a^{2}}{2 b^{4} d}-\frac {{\mathrm e}^{-i \left (d x +c \right )}}{8 b^{2} d}-\frac {i a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{4 b^{3} d}+\frac {2 i a^{3} \left (-i a \,{\mathrm e}^{i \left (d x +c \right )}+b \right )}{b^{5} d \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}-b +2 i a \,{\mathrm e}^{i \left (d x +c \right )}\right )}+\frac {4 i a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, d \,b^{5}}-\frac {3 i a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, d \,b^{3}}-\frac {4 i a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, d \,b^{5}}+\frac {3 i a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, d \,b^{3}}-\frac {\cos \left (3 d x +3 c \right )}{12 d \,b^{2}}\) | \(496\) |
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Time = 0.35 (sec) , antiderivative size = 643, normalized size of antiderivative = 3.42 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\left [\frac {4 \, {\left (a^{3} b^{3} - a b^{5}\right )} \cos \left (d x + c\right )^{3} + 6 \, {\left (4 \, a^{6} - 5 \, a^{4} b^{2} + a^{2} b^{4}\right )} d x + 3 \, {\left (4 \, a^{5} - 3 \, a^{3} b^{2} + {\left (4 \, a^{4} b - 3 \, a^{2} b^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} + 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) + 6 \, {\left (4 \, a^{5} b - 5 \, a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right ) - 2 \, {\left ({\left (a^{2} b^{4} - b^{6}\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left (4 \, a^{5} b - 5 \, a^{3} b^{3} + a b^{5}\right )} d x - 6 \, {\left (a^{4} b^{2} - a^{2} b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, {\left ({\left (a^{2} b^{6} - b^{8}\right )} d \sin \left (d x + c\right ) + {\left (a^{3} b^{5} - a b^{7}\right )} d\right )}}, \frac {2 \, {\left (a^{3} b^{3} - a b^{5}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (4 \, a^{6} - 5 \, a^{4} b^{2} + a^{2} b^{4}\right )} d x + 3 \, {\left (4 \, a^{5} - 3 \, a^{3} b^{2} + {\left (4 \, a^{4} b - 3 \, a^{2} b^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) + 3 \, {\left (4 \, a^{5} b - 5 \, a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right ) - {\left ({\left (a^{2} b^{4} - b^{6}\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left (4 \, a^{5} b - 5 \, a^{3} b^{3} + a b^{5}\right )} d x - 6 \, {\left (a^{4} b^{2} - a^{2} b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3 \, {\left ({\left (a^{2} b^{6} - b^{8}\right )} d \sin \left (d x + c\right ) + {\left (a^{3} b^{5} - a b^{7}\right )} d\right )}}\right ] \]
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Timed out. \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.42 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.39 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {\frac {3 \, {\left (4 \, a^{3} - a b^{2}\right )} {\left (d x + c\right )}}{b^{5}} - \frac {6 \, {\left (4 \, a^{4} - 3 \, a^{2} b^{2}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} b^{5}} + \frac {6 \, {\left (a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )} b^{4}} + \frac {2 \, {\left (3 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 3 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 18 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, a^{2} - b^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} b^{4}}}{3 \, d} \]
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Time = 13.28 (sec) , antiderivative size = 1688, normalized size of antiderivative = 8.98 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \]
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