Integrand size = 29, antiderivative size = 307 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {3 a \left (8 a^4-8 a^2 b^2+b^4\right ) x}{4 b^7}+\frac {6 a^2 \left (2 a^4-3 a^2 b^2+b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^7 \sqrt {a^2-b^2} d}-\frac {\left (30 a^4-25 a^2 b^2+b^4\right ) \cos (c+d x)}{5 b^6 d}+\frac {3 a \left (4 a^2-3 b^2\right ) \cos (c+d x) \sin (c+d x)}{4 b^5 d}-\frac {\left (10 a^2-7 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{5 b^4 d}+\frac {\left (3 a^2-2 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{2 a b^3 d}-\frac {\cos (c+d x) \sin ^4(c+d x)}{5 b^2 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{a b^2 d (a+b \sin (c+d x))} \]
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Time = 0.71 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2971, 3128, 3102, 2814, 2739, 632, 210} \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d (a+b \sin (c+d x))}+\frac {3 a \left (4 a^2-3 b^2\right ) \sin (c+d x) \cos (c+d x)}{4 b^5 d}-\frac {\left (10 a^2-7 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{5 b^4 d}+\frac {\left (3 a^2-2 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 a b^3 d}+\frac {6 a^2 \left (2 a^4-3 a^2 b^2+b^4\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^7 d \sqrt {a^2-b^2}}-\frac {3 a x \left (8 a^4-8 a^2 b^2+b^4\right )}{4 b^7}-\frac {\left (30 a^4-25 a^2 b^2+b^4\right ) \cos (c+d x)}{5 b^6 d}-\frac {\sin ^4(c+d x) \cos (c+d x)}{5 b^2 d} \]
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Rule 210
Rule 632
Rule 2739
Rule 2814
Rule 2971
Rule 3102
Rule 3128
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos (c+d x) \sin ^4(c+d x)}{5 b^2 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{a b^2 d (a+b \sin (c+d x))}+\frac {\int \frac {\sin ^3(c+d x) \left (3 \left (8 a^2-5 b^2\right )-a b \sin (c+d x)-10 \left (3 a^2-2 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{5 a b^2} \\ & = \frac {\left (3 a^2-2 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{2 a b^3 d}-\frac {\cos (c+d x) \sin ^4(c+d x)}{5 b^2 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{a b^2 d (a+b \sin (c+d x))}+\frac {\int \frac {\sin ^2(c+d x) \left (-30 a \left (3 a^2-2 b^2\right )+6 a^2 b \sin (c+d x)+12 a \left (10 a^2-7 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{20 a b^3} \\ & = -\frac {\left (10 a^2-7 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{5 b^4 d}+\frac {\left (3 a^2-2 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{2 a b^3 d}-\frac {\cos (c+d x) \sin ^4(c+d x)}{5 b^2 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{a b^2 d (a+b \sin (c+d x))}+\frac {\int \frac {\sin (c+d x) \left (24 a^2 \left (10 a^2-7 b^2\right )-6 a b \left (5 a^2-2 b^2\right ) \sin (c+d x)-90 a^2 \left (4 a^2-3 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{60 a b^4} \\ & = \frac {3 a \left (4 a^2-3 b^2\right ) \cos (c+d x) \sin (c+d x)}{4 b^5 d}-\frac {\left (10 a^2-7 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{5 b^4 d}+\frac {\left (3 a^2-2 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{2 a b^3 d}-\frac {\cos (c+d x) \sin ^4(c+d x)}{5 b^2 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{a b^2 d (a+b \sin (c+d x))}+\frac {\int \frac {-90 a^3 \left (4 a^2-3 b^2\right )+6 a^2 b \left (20 a^2-11 b^2\right ) \sin (c+d x)+24 a \left (30 a^4-25 a^2 b^2+b^4\right ) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx}{120 a b^5} \\ & = -\frac {\left (30 a^4-25 a^2 b^2+b^4\right ) \cos (c+d x)}{5 b^6 d}+\frac {3 a \left (4 a^2-3 b^2\right ) \cos (c+d x) \sin (c+d x)}{4 b^5 d}-\frac {\left (10 a^2-7 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{5 b^4 d}+\frac {\left (3 a^2-2 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{2 a b^3 d}-\frac {\cos (c+d x) \sin ^4(c+d x)}{5 b^2 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{a b^2 d (a+b \sin (c+d x))}+\frac {\int \frac {-90 a^3 b \left (4 a^2-3 b^2\right )-90 a^2 \left (8 a^4-8 a^2 b^2+b^4\right ) \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{120 a b^6} \\ & = -\frac {3 a \left (8 a^4-8 a^2 b^2+b^4\right ) x}{4 b^7}-\frac {\left (30 a^4-25 a^2 b^2+b^4\right ) \cos (c+d x)}{5 b^6 d}+\frac {3 a \left (4 a^2-3 b^2\right ) \cos (c+d x) \sin (c+d x)}{4 b^5 d}-\frac {\left (10 a^2-7 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{5 b^4 d}+\frac {\left (3 a^2-2 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{2 a b^3 d}-\frac {\cos (c+d x) \sin ^4(c+d x)}{5 b^2 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{a b^2 d (a+b \sin (c+d x))}+\frac {\left (3 a^2 \left (2 a^4-3 a^2 b^2+b^4\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{b^7} \\ & = -\frac {3 a \left (8 a^4-8 a^2 b^2+b^4\right ) x}{4 b^7}-\frac {\left (30 a^4-25 a^2 b^2+b^4\right ) \cos (c+d x)}{5 b^6 d}+\frac {3 a \left (4 a^2-3 b^2\right ) \cos (c+d x) \sin (c+d x)}{4 b^5 d}-\frac {\left (10 a^2-7 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{5 b^4 d}+\frac {\left (3 a^2-2 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{2 a b^3 d}-\frac {\cos (c+d x) \sin ^4(c+d x)}{5 b^2 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{a b^2 d (a+b \sin (c+d x))}+\frac {\left (6 a^2 \left (2 a^4-3 a^2 b^2+b^4\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^7 d} \\ & = -\frac {3 a \left (8 a^4-8 a^2 b^2+b^4\right ) x}{4 b^7}-\frac {\left (30 a^4-25 a^2 b^2+b^4\right ) \cos (c+d x)}{5 b^6 d}+\frac {3 a \left (4 a^2-3 b^2\right ) \cos (c+d x) \sin (c+d x)}{4 b^5 d}-\frac {\left (10 a^2-7 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{5 b^4 d}+\frac {\left (3 a^2-2 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{2 a b^3 d}-\frac {\cos (c+d x) \sin ^4(c+d x)}{5 b^2 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{a b^2 d (a+b \sin (c+d x))}-\frac {\left (12 a^2 \left (2 a^4-3 a^2 b^2+b^4\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^7 d} \\ & = -\frac {3 a \left (8 a^4-8 a^2 b^2+b^4\right ) x}{4 b^7}+\frac {6 a^2 \left (2 a^4-3 a^2 b^2+b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^7 \sqrt {a^2-b^2} d}-\frac {\left (30 a^4-25 a^2 b^2+b^4\right ) \cos (c+d x)}{5 b^6 d}+\frac {3 a \left (4 a^2-3 b^2\right ) \cos (c+d x) \sin (c+d x)}{4 b^5 d}-\frac {\left (10 a^2-7 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{5 b^4 d}+\frac {\left (3 a^2-2 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{2 a b^3 d}-\frac {\cos (c+d x) \sin ^4(c+d x)}{5 b^2 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{a b^2 d (a+b \sin (c+d x))} \\ \end{align*}
Time = 3.18 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.23 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {\frac {960 a^2 \left (2 a^4-3 a^2 b^2+b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-\frac {960 a^6 c-960 a^4 b^2 c+120 a^2 b^4 c+960 a^6 d x-960 a^4 b^2 d x+120 a^2 b^4 d x+60 a b \left (16 a^4-14 a^2 b^2+b^4\right ) \cos (c+d x)+5 \left (8 a^3 b^3-5 a b^5\right ) \cos (3 (c+d x))-3 a b^5 \cos (5 (c+d x))+960 a^5 b c \sin (c+d x)-960 a^3 b^3 c \sin (c+d x)+120 a b^5 c \sin (c+d x)+960 a^5 b d x \sin (c+d x)-960 a^3 b^3 d x \sin (c+d x)+120 a b^5 d x \sin (c+d x)+240 a^4 b^2 \sin (2 (c+d x))-200 a^2 b^4 \sin (2 (c+d x))+5 b^6 \sin (2 (c+d x))-10 a^2 b^4 \sin (4 (c+d x))+4 b^6 \sin (4 (c+d x))+b^6 \sin (6 (c+d x))}{a+b \sin (c+d x)}}{160 b^7 d} \]
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Time = 2.42 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.41
method | result | size |
derivativedivides | \(\frac {\frac {4 a^{2} \left (\frac {-\frac {b^{2} \left (a^{2}-b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {a^{3} b}{2}+\frac {a \,b^{3}}{2}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {3 \left (2 a^{4}-3 a^{2} b^{2}+b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{7}}-\frac {4 \left (\frac {\left (a^{3} b^{2}-\frac {5}{8} a \,b^{4}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {5}{2} a^{4} b -3 a^{2} b^{3}+\frac {1}{2} b^{5}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (2 a^{3} b^{2}-\frac {1}{4} a \,b^{4}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (10 a^{4} b -9 a^{2} b^{3}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (15 a^{4} b -11 a^{2} b^{3}+b^{5}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-2 a^{3} b^{2}+\frac {1}{4} a \,b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (10 a^{4} b -7 a^{2} b^{3}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-a^{3} b^{2}+\frac {5}{8} a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {5 a^{4} b}{2}-2 a^{2} b^{3}+\frac {b^{5}}{10}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {3 a \left (8 a^{4}-8 a^{2} b^{2}+b^{4}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}\right )}{b^{7}}}{d}\) | \(434\) |
default | \(\frac {\frac {4 a^{2} \left (\frac {-\frac {b^{2} \left (a^{2}-b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {a^{3} b}{2}+\frac {a \,b^{3}}{2}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {3 \left (2 a^{4}-3 a^{2} b^{2}+b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{7}}-\frac {4 \left (\frac {\left (a^{3} b^{2}-\frac {5}{8} a \,b^{4}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {5}{2} a^{4} b -3 a^{2} b^{3}+\frac {1}{2} b^{5}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (2 a^{3} b^{2}-\frac {1}{4} a \,b^{4}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (10 a^{4} b -9 a^{2} b^{3}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (15 a^{4} b -11 a^{2} b^{3}+b^{5}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-2 a^{3} b^{2}+\frac {1}{4} a \,b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (10 a^{4} b -7 a^{2} b^{3}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-a^{3} b^{2}+\frac {5}{8} a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {5 a^{4} b}{2}-2 a^{2} b^{3}+\frac {b^{5}}{10}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {3 a \left (8 a^{4}-8 a^{2} b^{2}+b^{4}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}\right )}{b^{7}}}{d}\) | \(434\) |
risch | \(-\frac {6 a^{5} x}{b^{7}}+\frac {6 a^{3} x}{b^{5}}-\frac {3 a x}{4 b^{3}}+\frac {i a \,{\mathrm e}^{2 i \left (d x +c \right )}}{4 b^{3} d}+\frac {i a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{2 b^{5} d}-\frac {5 \,{\mathrm e}^{i \left (d x +c \right )} a^{4}}{2 b^{6} d}+\frac {15 \,{\mathrm e}^{i \left (d x +c \right )} a^{2}}{8 b^{4} d}-\frac {{\mathrm e}^{i \left (d x +c \right )}}{16 b^{2} d}-\frac {5 \,{\mathrm e}^{-i \left (d x +c \right )} a^{4}}{2 b^{6} d}+\frac {15 \,{\mathrm e}^{-i \left (d x +c \right )} a^{2}}{8 b^{4} d}-\frac {{\mathrm e}^{-i \left (d x +c \right )}}{16 b^{2} d}-\frac {i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{2 b^{5} d}+\frac {2 i a^{3} \left (-a^{2}+b^{2}\right ) \left (-i a \,{\mathrm e}^{i \left (d x +c \right )}+b \right )}{b^{7} d \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}-b +2 i a \,{\mathrm e}^{i \left (d x +c \right )}\right )}-\frac {i a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{4 b^{3} d}+\frac {3 i \sqrt {a^{2}-b^{2}}\, a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,b^{5}}-\frac {6 i \sqrt {a^{2}-b^{2}}\, a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,b^{7}}+\frac {6 i \sqrt {a^{2}-b^{2}}\, a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,b^{7}}-\frac {3 i \sqrt {a^{2}-b^{2}}\, a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,b^{5}}-\frac {\cos \left (5 d x +5 c \right )}{80 d \,b^{2}}-\frac {a \sin \left (4 d x +4 c \right )}{16 b^{3} d}+\frac {\cos \left (3 d x +3 c \right ) a^{2}}{4 d \,b^{4}}-\frac {\cos \left (3 d x +3 c \right )}{16 d \,b^{2}}\) | \(575\) |
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Time = 0.33 (sec) , antiderivative size = 649, normalized size of antiderivative = 2.11 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\left [\frac {6 \, a b^{5} \cos \left (d x + c\right )^{5} - 5 \, {\left (4 \, a^{3} b^{3} - a b^{5}\right )} \cos \left (d x + c\right )^{3} - 15 \, {\left (8 \, a^{6} - 8 \, a^{4} b^{2} + a^{2} b^{4}\right )} d x - 30 \, {\left (2 \, a^{5} - a^{3} b^{2} + {\left (2 \, a^{4} b - 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 ) - 15 \, {\left (8 \, a^{5} b - 8 \, a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right ) - {\left (4 \, b^{6} \cos \left (d x + c\right )^{5} - 10 \, a^{2} b^{4} \cos \left (d x + c\right )^{3} + 15 \, {\left (8 \, a^{5} b - 8 \, a^{3} b^{3} + a b^{5}\right )} d x + 15 \, {\left (4 \, a^{4} b^{2} - 3 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{20 \, {\left (b^{8} d \sin \left (d x + c\right ) + a b^{7} d\right )}}, \frac {6 \, a b^{5} \cos \left (d x + c\right )^{5} - 5 \, {\left (4 \, a^{3} b^{3} - a b^{5}\right )} \cos \left (d x + c\right )^{3} - 15 \, {\left (8 \, a^{6} - 8 \, a^{4} b^{2} + a^{2} b^{4}\right )} d x - 60 \, {\left (2 \, a^{5} - a^{3} b^{2} + {\left (2 \, a^{4} b - 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 ) - 15 \, {\left (8 \, a^{5} b - 8 \, a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right ) - {\left (4 \, b^{6} \cos \left (d x + c\right )^{5} - 10 \, a^{2} b^{4} \cos \left (d x + c\right )^{3} + 15 \, {\left (8 \, a^{5} b - 8 \, a^{3} b^{3} + a b^{5}\right )} d x + 15 \, {\left (4 \, a^{4} b^{2} - 3 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{20 \, {\left (b^{8} d \sin \left (d x + c\right ) + a b^{7} d\right )}}\right ] \]
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Timed out. \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.35 (sec) , antiderivative size = 536, normalized size of antiderivative = 1.75 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {\frac {15 \, {\left (8 \, a^{5} - 8 \, a^{3} b^{2} + a b^{4}\right )} {\left (d x + c\right )}}{b^{7}} - \frac {120 \, {\left (2 \, a^{6} - 3 \, a^{4} b^{2} + a^{2} b^{4}\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^{7}} + \frac {40 \, {\left (a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{5} - a^{3} b^{2}\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^{6}} + \frac {2 \, {\left (40 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 25 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 100 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 120 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 20 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 80 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 10 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 400 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 360 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 600 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 440 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 40 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 80 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 10 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 400 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 280 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 40 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 25 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 100 \, a^{4} - 80 \, a^{2} b^{2} + 4 \, b^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5} b^{6}}}{20 \, d} \]
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Time = 14.77 (sec) , antiderivative size = 2390, normalized size of antiderivative = 7.79 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \]
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