Integrand size = 27, antiderivative size = 70 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {x}{a^2}-\frac {2 \cos ^3(c+d x)}{3 a^2 d}-\frac {\cos (c+d x) \sin (c+d x)}{a^2 d}-\frac {\cos ^5(c+d x)}{d (a+a \sin (c+d x))^2} \]
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Time = 0.08 (sec) , antiderivative size = 70, 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, 2761, 2715, 8} \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {2 \cos ^3(c+d x)}{3 a^2 d}-\frac {\sin (c+d x) \cos (c+d x)}{a^2 d}-\frac {x}{a^2}-\frac {\cos ^5(c+d x)}{d (a \sin (c+d x)+a)^2} \]
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Rule 8
Rule 2715
Rule 2761
Rule 2938
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos ^5(c+d x)}{d (a+a \sin (c+d x))^2}-\frac {2 \int \frac {\cos ^4(c+d x)}{a+a \sin (c+d x)} \, dx}{a} \\ & = -\frac {2 \cos ^3(c+d x)}{3 a^2 d}-\frac {\cos ^5(c+d x)}{d (a+a \sin (c+d x))^2}-\frac {2 \int \cos ^2(c+d x) \, dx}{a^2} \\ & = -\frac {2 \cos ^3(c+d x)}{3 a^2 d}-\frac {\cos (c+d x) \sin (c+d x)}{a^2 d}-\frac {\cos ^5(c+d x)}{d (a+a \sin (c+d x))^2}-\frac {\int 1 \, dx}{a^2} \\ & = -\frac {x}{a^2}-\frac {2 \cos ^3(c+d x)}{3 a^2 d}-\frac {\cos (c+d x) \sin (c+d x)}{a^2 d}-\frac {\cos ^5(c+d x)}{d (a+a \sin (c+d x))^2} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(204\) vs. \(2(70)=140\).
Time = 0.49 (sec) , antiderivative size = 204, normalized size of antiderivative = 2.91 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {-2 (1+12 d x) \cos \left (\frac {c}{2}\right )-21 \cos \left (\frac {c}{2}+d x\right )-21 \cos \left (\frac {3 c}{2}+d x\right )+6 \cos \left (\frac {3 c}{2}+2 d x\right )-6 \cos \left (\frac {5 c}{2}+2 d x\right )+\cos \left (\frac {5 c}{2}+3 d x\right )+\cos \left (\frac {7 c}{2}+3 d x\right )+2 \sin \left (\frac {c}{2}\right )-24 d x \sin \left (\frac {c}{2}\right )+21 \sin \left (\frac {c}{2}+d x\right )-21 \sin \left (\frac {3 c}{2}+d x\right )+6 \sin \left (\frac {3 c}{2}+2 d x\right )+6 \sin \left (\frac {5 c}{2}+2 d x\right )-\sin \left (\frac {5 c}{2}+3 d x\right )+\sin \left (\frac {7 c}{2}+3 d x\right )}{24 a^2 d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right )} \]
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Time = 0.28 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.61
method | result | size |
parallelrisch | \(\frac {-12 d x +\cos \left (3 d x +3 c \right )-21 \cos \left (d x +c \right )+6 \sin \left (2 d x +2 c \right )-20}{12 d \,a^{2}}\) | \(43\) |
risch | \(-\frac {x}{a^{2}}-\frac {7 \cos \left (d x +c \right )}{4 a^{2} d}+\frac {\cos \left (3 d x +3 c \right )}{12 d \,a^{2}}+\frac {\sin \left (2 d x +2 c \right )}{2 d \,a^{2}}\) | \(56\) |
derivativedivides | \(\frac {\frac {4 \left (-\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {5}{6}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2}}\) | \(90\) |
default | \(\frac {\frac {4 \left (-\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {5}{6}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2}}\) | \(90\) |
norman | \(\frac {-\frac {35 x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {16 x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {40 x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {8 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {40 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {25 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {35 x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {25 x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {16 x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {8 x \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {10}{3 a d}-\frac {3 x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {x \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {x}{a}-\frac {8 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {56 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}-\frac {184 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {3 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a}-\frac {32 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {112 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {50 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {200 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {72 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {154 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {16 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {2 \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 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}\) | \(493\) |
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Time = 0.25 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.61 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\cos \left (d x + c\right )^{3} - 3 \, d x + 3 \, \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 6 \, \cos \left (d x + c\right )}{3 \, a^{2} d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 694 vs. \(2 (63) = 126\).
Time = 12.02 (sec) , antiderivative size = 694, normalized size of antiderivative = 9.91 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx=\begin {cases} - \frac {3 d x \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a^{2} d} - \frac {9 d x \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a^{2} d} - \frac {9 d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a^{2} d} - \frac {3 d x}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a^{2} d} - \frac {6 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a^{2} d} - \frac {6 \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a^{2} d} - \frac {24 \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a^{2} d} + \frac {6 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a^{2} d} - \frac {10}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a^{2} d} & \text {for}\: d \neq 0 \\\frac {x \sin {\left (c \right )} \cos ^{4}{\left (c \right )}}{\left (a \sin {\left (c \right )} + a\right )^{2}} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (68) = 136\).
Time = 0.31 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.63 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {2 \, {\left (\frac {\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {12 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {3 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 5}{a^{2} + \frac {3 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} - \frac {3 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}\right )}}{3 \, d} \]
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Time = 0.31 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.26 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {3 \, {\left (d x + c\right )}}{a^{2}} + \frac {2 \, {\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 12 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} a^{2}}}{3 \, d} \]
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Time = 9.76 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.79 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\cos \left (3\,c+3\,d\,x\right )}{12\,a^2\,d}-\frac {7\,\cos \left (c+d\,x\right )}{4\,a^2\,d}-\frac {x}{a^2}+\frac {\sin \left (2\,c+2\,d\,x\right )}{2\,a^2\,d} \]
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