Integrand size = 27, antiderivative size = 83 \[ \int \frac {\cos (c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {\log (1+\sin (c+d x))}{a^4 d}+\frac {1}{3 a d (a+a \sin (c+d x))^3}-\frac {3}{2 d \left (a^2+a^2 \sin (c+d x)\right )^2}+\frac {3}{d \left (a^4+a^4 \sin (c+d x)\right )} \]
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Time = 0.06 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2912, 12, 45} \[ \int \frac {\cos (c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {3}{d \left (a^4 \sin (c+d x)+a^4\right )}+\frac {\log (\sin (c+d x)+1)}{a^4 d}-\frac {3}{2 d \left (a^2 \sin (c+d x)+a^2\right )^2}+\frac {1}{3 a d (a \sin (c+d x)+a)^3} \]
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Rule 12
Rule 45
Rule 2912
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^3}{a^3 (a+x)^4} \, dx,x,a \sin (c+d x)\right )}{a d} \\ & = \frac {\text {Subst}\left (\int \frac {x^3}{(a+x)^4} \, dx,x,a \sin (c+d x)\right )}{a^4 d} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {a^3}{(a+x)^4}+\frac {3 a^2}{(a+x)^3}-\frac {3 a}{(a+x)^2}+\frac {1}{a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^4 d} \\ & = \frac {\log (1+\sin (c+d x))}{a^4 d}+\frac {1}{3 a d (a+a \sin (c+d x))^3}-\frac {3}{2 d \left (a^2+a^2 \sin (c+d x)\right )^2}+\frac {3}{d \left (a^4+a^4 \sin (c+d x)\right )} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.73 \[ \int \frac {\cos (c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {11+27 \sin (c+d x)+18 \sin ^2(c+d x)+6 \log (1+\sin (c+d x)) (1+\sin (c+d x))^3}{6 a^4 d (1+\sin (c+d x))^3} \]
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Time = 0.42 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.65
method | result | size |
derivativedivides | \(\frac {\frac {3}{1+\sin \left (d x +c \right )}+\ln \left (1+\sin \left (d x +c \right )\right )+\frac {1}{3 \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {3}{2 \left (1+\sin \left (d x +c \right )\right )^{2}}}{d \,a^{4}}\) | \(54\) |
default | \(\frac {\frac {3}{1+\sin \left (d x +c \right )}+\ln \left (1+\sin \left (d x +c \right )\right )+\frac {1}{3 \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {3}{2 \left (1+\sin \left (d x +c \right )\right )^{2}}}{d \,a^{4}}\) | \(54\) |
risch | \(-\frac {i x}{a^{4}}-\frac {2 i c}{d \,a^{4}}+\frac {2 i \left (-40 \,{\mathrm e}^{3 i \left (d x +c \right )}-27 i {\mathrm e}^{2 i \left (d x +c \right )}+27 i {\mathrm e}^{4 i \left (d x +c \right )}+9 \,{\mathrm e}^{5 i \left (d x +c \right )}+9 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{6}}+\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d \,a^{4}}\) | \(121\) |
parallelrisch | \(\frac {\left (-36 \cos \left (2 d x +2 c \right )+90 \sin \left (d x +c \right )-6 \sin \left (3 d x +3 c \right )+60\right ) \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (72 \cos \left (2 d x +2 c \right )-180 \sin \left (d x +c \right )+12 \sin \left (3 d x +3 c \right )-120\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-30 \cos \left (2 d x +2 c \right )+57 \sin \left (d x +c \right )-11 \sin \left (3 d x +3 c \right )+30}{6 d \,a^{4} \left (-10+6 \cos \left (2 d x +2 c \right )+\sin \left (3 d x +3 c \right )-15 \sin \left (d x +c \right )\right )}\) | \(163\) |
norman | \(\frac {-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}-\frac {2 \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {12 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {12 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {228 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {228 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {230 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {230 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {110 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {110 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {416 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {416 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {566 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {566 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {2 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d \,a^{4}}-\frac {\ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{4}}\) | \(341\) |
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Time = 0.28 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.35 \[ \int \frac {\cos (c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {18 \, \cos \left (d x + c\right )^{2} + 6 \, {\left (3 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 4\right )} \sin \left (d x + c\right ) - 4\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 27 \, \sin \left (d x + c\right ) - 29}{6 \, {\left (3 \, a^{4} d \cos \left (d x + c\right )^{2} - 4 \, a^{4} d + {\left (a^{4} d \cos \left (d x + c\right )^{2} - 4 \, a^{4} d\right )} \sin \left (d x + c\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 466 vs. \(2 (70) = 140\).
Time = 0.85 (sec) , antiderivative size = 466, normalized size of antiderivative = 5.61 \[ \int \frac {\cos (c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\begin {cases} \frac {6 \log {\left (\sin {\left (c + d x \right )} + 1 \right )} \sin ^{3}{\left (c + d x \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin {\left (c + d x \right )} + 6 a^{4} d} + \frac {18 \log {\left (\sin {\left (c + d x \right )} + 1 \right )} \sin ^{2}{\left (c + d x \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin {\left (c + d x \right )} + 6 a^{4} d} + \frac {18 \log {\left (\sin {\left (c + d x \right )} + 1 \right )} \sin {\left (c + d x \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin {\left (c + d x \right )} + 6 a^{4} d} + \frac {6 \log {\left (\sin {\left (c + d x \right )} + 1 \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin {\left (c + d x \right )} + 6 a^{4} d} + \frac {18 \sin ^{2}{\left (c + d x \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin {\left (c + d x \right )} + 6 a^{4} d} + \frac {27 \sin {\left (c + d x \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin {\left (c + d x \right )} + 6 a^{4} d} + \frac {11}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin {\left (c + d x \right )} + 6 a^{4} d} & \text {for}\: d \neq 0 \\\frac {x \sin ^{3}{\left (c \right )} \cos {\left (c \right )}}{\left (a \sin {\left (c \right )} + a\right )^{4}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {\frac {18 \, \sin \left (d x + c\right )^{2} + 27 \, \sin \left (d x + c\right ) + 11}{a^{4} \sin \left (d x + c\right )^{3} + 3 \, a^{4} \sin \left (d x + c\right )^{2} + 3 \, a^{4} \sin \left (d x + c\right ) + a^{4}} + \frac {6 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{4}}}{6 \, d} \]
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Time = 0.35 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.66 \[ \int \frac {\cos (c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {\frac {6 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{4}} + \frac {18 \, \sin \left (d x + c\right )^{2} + 27 \, \sin \left (d x + c\right ) + 11}{a^{4} {\left (\sin \left (d x + c\right ) + 1\right )}^{3}}}{6 \, d} \]
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Time = 0.07 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.65 \[ \int \frac {\cos (c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )}{a^4\,d}+\frac {3\,{\sin \left (c+d\,x\right )}^2+\frac {9\,\sin \left (c+d\,x\right )}{2}+\frac {11}{6}}{a^4\,d\,{\left (\sin \left (c+d\,x\right )+1\right )}^3} \]
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