Integrand size = 29, antiderivative size = 97 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {11 x}{2 a^3}-\frac {3 \cos (c+d x)}{a^3 d}+\frac {\cos (c+d x) \sin (c+d x)}{2 a^3 d}+\frac {2 \cos (c+d x)}{3 a^3 d (1+\sin (c+d x))^2}-\frac {19 \cos (c+d x)}{3 a^3 d (1+\sin (c+d x))} \]
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Time = 0.21 (sec) , antiderivative size = 97, 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 = {2953, 3045, 2718, 2715, 8, 2729, 2727} \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {3 \cos (c+d x)}{a^3 d}+\frac {\sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac {19 \cos (c+d x)}{3 a^3 d (\sin (c+d x)+1)}+\frac {2 \cos (c+d x)}{3 a^3 d (\sin (c+d x)+1)^2}-\frac {11 x}{2 a^3} \]
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Rule 8
Rule 2715
Rule 2718
Rule 2727
Rule 2729
Rule 2953
Rule 3045
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\sin ^3(c+d x) (a-a \sin (c+d x))}{(a+a \sin (c+d x))^2} \, dx}{a^2} \\ & = \frac {\int \left (-\frac {5}{a}+\frac {3 \sin (c+d x)}{a}-\frac {\sin ^2(c+d x)}{a}-\frac {2}{a (1+\sin (c+d x))^2}+\frac {7}{a (1+\sin (c+d x))}\right ) \, dx}{a^2} \\ & = -\frac {5 x}{a^3}-\frac {\int \sin ^2(c+d x) \, dx}{a^3}-\frac {2 \int \frac {1}{(1+\sin (c+d x))^2} \, dx}{a^3}+\frac {3 \int \sin (c+d x) \, dx}{a^3}+\frac {7 \int \frac {1}{1+\sin (c+d x)} \, dx}{a^3} \\ & = -\frac {5 x}{a^3}-\frac {3 \cos (c+d x)}{a^3 d}+\frac {\cos (c+d x) \sin (c+d x)}{2 a^3 d}+\frac {2 \cos (c+d x)}{3 a^3 d (1+\sin (c+d x))^2}-\frac {7 \cos (c+d x)}{a^3 d (1+\sin (c+d x))}-\frac {\int 1 \, dx}{2 a^3}-\frac {2 \int \frac {1}{1+\sin (c+d x)} \, dx}{3 a^3} \\ & = -\frac {11 x}{2 a^3}-\frac {3 \cos (c+d x)}{a^3 d}+\frac {\cos (c+d x) \sin (c+d x)}{2 a^3 d}+\frac {2 \cos (c+d x)}{3 a^3 d (1+\sin (c+d x))^2}-\frac {19 \cos (c+d x)}{3 a^3 d (1+\sin (c+d x))} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(197\) vs. \(2(97)=194\).
Time = 1.12 (sec) , antiderivative size = 197, normalized size of antiderivative = 2.03 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {1980 d x \cos \left (\frac {d x}{2}\right )-1326 \cos \left (c+\frac {d x}{2}\right )+2012 \cos \left (c+\frac {3 d x}{2}\right )-660 d x \cos \left (2 c+\frac {3 d x}{2}\right )-135 \cos \left (3 c+\frac {5 d x}{2}\right )+15 \cos \left (3 c+\frac {7 d x}{2}\right )-3216 \sin \left (\frac {d x}{2}\right )+1980 d x \sin \left (c+\frac {d x}{2}\right )+660 d x \sin \left (c+\frac {3 d x}{2}\right )+498 \sin \left (2 c+\frac {3 d x}{2}\right )+135 \sin \left (2 c+\frac {5 d x}{2}\right )+15 \sin \left (4 c+\frac {7 d x}{2}\right )}{240 a^3 d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3} \]
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Time = 0.55 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.27
method | result | size |
derivativedivides | \(\frac {-\frac {2 \left (\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+3\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-11 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {8}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {10}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{d \,a^{3}}\) | \(123\) |
default | \(\frac {-\frac {2 \left (\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+3\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-11 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {8}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {10}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{d \,a^{3}}\) | \(123\) |
risch | \(-\frac {11 x}{2 a^{3}}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 d \,a^{3}}-\frac {3 \,{\mathrm e}^{i \left (d x +c \right )}}{2 d \,a^{3}}-\frac {3 \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d \,a^{3}}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d \,a^{3}}-\frac {2 \left (36 i {\mathrm e}^{i \left (d x +c \right )}+21 \,{\mathrm e}^{2 i \left (d x +c \right )}-19\right )}{3 d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{3}}\) | \(125\) |
parallelrisch | \(\frac {-132 d x \cos \left (3 d x +3 c \right )-396 d x \cos \left (d x +c \right )+161 \sin \left (3 d x +3 c \right )-208 \cos \left (3 d x +3 c \right )-624 \cos \left (d x +c \right )+30 \sin \left (d x +c \right )-480 \cos \left (2 d x +2 c \right )-36 \cos \left (4 d x +4 c \right )+3 \sin \left (5 d x +5 c \right )-316}{24 d \,a^{3} \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right )}\) | \(125\) |
norman | \(\frac {-\frac {1111 x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {385 x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {1705 x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {165 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {1485 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {715 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {1705 x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {1485 x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {1111 x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {715 x \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {52}{3 a d}-\frac {385 x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {165 x \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {11 x}{2 a}-\frac {1112 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {205 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {55 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {227 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 d a}-\frac {4208 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {55 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}-\frac {3001 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {1336 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {1284 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {3263 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {4019 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {756 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {11 \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {663 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {484 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {55 x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {11 x \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5} a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}\) | \(565\) |
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Time = 0.28 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.68 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {3 \, \cos \left (d x + c\right )^{4} - {\left (33 \, d x - 53\right )} \cos \left (d x + c\right )^{2} - 12 \, \cos \left (d x + c\right )^{3} + 66 \, d x + {\left (33 \, d x + 64\right )} \cos \left (d x + c\right ) + {\left (3 \, \cos \left (d x + c\right )^{3} + 66 \, d x + {\left (33 \, d x + 68\right )} \cos \left (d x + c\right ) + 15 \, \cos \left (d x + c\right )^{2} + 4\right )} \sin \left (d x + c\right ) - 4}{6 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d \cos \left (d x + c\right ) - 2 \, a^{3} d - {\left (a^{3} d \cos \left (d x + c\right ) + 2 \, a^{3} d\right )} \sin \left (d x + c\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 2264 vs. \(2 (90) = 180\).
Time = 23.09 (sec) , antiderivative size = 2264, normalized size of antiderivative = 23.34 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 314 vs. \(2 (89) = 178\).
Time = 0.28 (sec) , antiderivative size = 314, normalized size of antiderivative = 3.24 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {\frac {123 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {161 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {210 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {154 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {99 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {33 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 52}{a^{3} + \frac {3 \, a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {5 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {7 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {7 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {5 \, a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a^{3} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}} + \frac {33 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{3 \, d} \]
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Time = 0.46 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.21 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {33 \, {\left (d x + c\right )}}{a^{3}} + \frac {6 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{3}} + \frac {4 \, {\left (15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 36 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 17\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{3}}}{6 \, d} \]
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Time = 13.45 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.25 \[ \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {11\,x}{2\,a^3}-\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+33\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {154\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}+70\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {161\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+41\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {52}{3}}{a^3\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^3\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^2} \]
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