Integrand size = 27, antiderivative size = 80 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {9 x}{2 a^3}+\frac {9 \cos (c+d x)}{2 a^3 d}+\frac {\cos ^5(c+d x)}{d (a+a \sin (c+d x))^3}+\frac {3 \cos ^3(c+d x)}{2 d \left (a^3+a^3 \sin (c+d x)\right )} \]
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Time = 0.11 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2938, 2758, 2761, 8} \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {9 \cos (c+d x)}{2 a^3 d}+\frac {3 \cos ^3(c+d x)}{2 d \left (a^3 \sin (c+d x)+a^3\right )}+\frac {9 x}{2 a^3}+\frac {\cos ^5(c+d x)}{d (a \sin (c+d x)+a)^3} \]
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Rule 8
Rule 2758
Rule 2761
Rule 2938
Rubi steps \begin{align*} \text {integral}& = \frac {\cos ^5(c+d x)}{d (a+a \sin (c+d x))^3}+\frac {3 \int \frac {\cos ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx}{a} \\ & = \frac {\cos ^5(c+d x)}{d (a+a \sin (c+d x))^3}+\frac {3 \cos ^3(c+d x)}{2 d \left (a^3+a^3 \sin (c+d x)\right )}+\frac {9 \int \frac {\cos ^2(c+d x)}{a+a \sin (c+d x)} \, dx}{2 a^2} \\ & = \frac {9 \cos (c+d x)}{2 a^3 d}+\frac {\cos ^5(c+d x)}{d (a+a \sin (c+d x))^3}+\frac {3 \cos ^3(c+d x)}{2 d \left (a^3+a^3 \sin (c+d x)\right )}+\frac {9 \int 1 \, dx}{2 a^3} \\ & = \frac {9 x}{2 a^3}+\frac {9 \cos (c+d x)}{2 a^3 d}+\frac {\cos ^5(c+d x)}{d (a+a \sin (c+d x))^3}+\frac {3 \cos ^3(c+d x)}{2 d \left (a^3+a^3 \sin (c+d x)\right )} \\ \end{align*}
Time = 0.54 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.79 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {180 d x \cos \left (\frac {d x}{2}\right )+59 \cos \left (c+\frac {d x}{2}\right )+55 \cos \left (c+\frac {3 d x}{2}\right )+5 \cos \left (3 c+\frac {5 d x}{2}\right )-381 \sin \left (\frac {d x}{2}\right )+180 d x \sin \left (c+\frac {d x}{2}\right )+55 \sin \left (2 c+\frac {3 d x}{2}\right )-5 \sin \left (2 c+\frac {5 d x}{2}\right )}{40 a^3 d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )} \]
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Time = 0.37 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.84
method | result | size |
parallelrisch | \(\frac {36 d x \cos \left (d x +c \right )+56 \cos \left (d x +c \right )-33 \sin \left (d x +c \right )+12 \cos \left (2 d x +2 c \right )-\sin \left (3 d x +3 c \right )+44}{8 d \,a^{3} \cos \left (d x +c \right )}\) | \(67\) |
risch | \(\frac {9 x}{2 a^{3}}+\frac {3 \,{\mathrm e}^{i \left (d x +c \right )}}{2 d \,a^{3}}+\frac {3 \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d \,a^{3}}+\frac {8}{d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}-\frac {\sin \left (2 d x +2 c \right )}{4 d \,a^{3}}\) | \(81\) |
derivativedivides | \(\frac {\frac {4 \left (\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}+\frac {3}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+9 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {8}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{d \,a^{3}}\) | \(92\) |
default | \(\frac {\frac {4 \left (\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}+\frac {3}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+9 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {8}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{d \,a^{3}}\) | \(92\) |
norman | \(\frac {\frac {909 x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {315 x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {1395 x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {135 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {1215 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {585 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {1395 x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {1215 x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {909 x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {585 x \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {14}{a d}+\frac {315 x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {135 x \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {9 x}{2 a}+\frac {296 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {167 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {45 x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {9 x \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {43 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {61 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}+\frac {1136 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {45 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}+\frac {807 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {360 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {1034 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {881 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {1083 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {614 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {9 \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {533 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {130 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d 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 = 0.27 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.25 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\cos \left (d x + c\right )^{3} + 9 \, d x + {\left (9 \, d x + 13\right )} \cos \left (d x + c\right ) + 6 \, \cos \left (d x + c\right )^{2} + {\left (9 \, d x - \cos \left (d x + c\right )^{2} + 5 \, \cos \left (d x + c\right ) - 8\right )} \sin \left (d x + c\right ) + 8}{2 \, {\left (a^{3} d \cos \left (d x + c\right ) + a^{3} d \sin \left (d x + c\right ) + a^{3} d\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1244 vs. \(2 (71) = 142\).
Time = 23.08 (sec) , antiderivative size = 1244, normalized size of antiderivative = 15.55 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 225 vs. \(2 (74) = 148\).
Time = 0.30 (sec) , antiderivative size = 225, normalized size of antiderivative = 2.81 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {\frac {5 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {21 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {7 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {9 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 14}{a^{3} + \frac {a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {2 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {2 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}} + \frac {9 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{d} \]
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Time = 0.32 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.14 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {9 \, {\left (d x + c\right )}}{a^{3}} + \frac {2 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{3}} + \frac {16}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}}}{2 \, d} \]
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Time = 11.78 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.18 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {9\,x}{2\,a^3}+\frac {9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+14}{a^3\,d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^2} \]
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