Integrand size = 27, antiderivative size = 67 \[ \int \frac {\cos (c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\log (1+\sin (c+d x))}{a d}+\frac {\sin (c+d x)}{a d}-\frac {\sin ^2(c+d x)}{2 a d}+\frac {\sin ^3(c+d x)}{3 a d} \]
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Time = 0.05 (sec) , antiderivative size = 67, 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} \[ \int \frac {\cos (c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\sin ^3(c+d x)}{3 a d}-\frac {\sin ^2(c+d x)}{2 a d}+\frac {\sin (c+d x)}{a d}-\frac {\log (\sin (c+d x)+1)}{a d} \]
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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)} \, dx,x,a \sin (c+d x)\right )}{a d} \\ & = \frac {\text {Subst}\left (\int \frac {x^3}{a+x} \, dx,x,a \sin (c+d x)\right )}{a^4 d} \\ & = \frac {\text {Subst}\left (\int \left (a^2-a x+x^2-\frac {a^3}{a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^4 d} \\ & = -\frac {\log (1+\sin (c+d x))}{a d}+\frac {\sin (c+d x)}{a d}-\frac {\sin ^2(c+d x)}{2 a d}+\frac {\sin ^3(c+d x)}{3 a d} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.75 \[ \int \frac {\cos (c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {-6 \log (1+\sin (c+d x))+6 \sin (c+d x)-3 \sin ^2(c+d x)+2 \sin ^3(c+d x)}{6 a d} \]
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Time = 0.15 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.69
method | result | size |
derivativedivides | \(\frac {\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}+\sin \left (d x +c \right )-\ln \left (1+\sin \left (d x +c \right )\right )}{d a}\) | \(46\) |
default | \(\frac {\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}+\sin \left (d x +c \right )-\ln \left (1+\sin \left (d x +c \right )\right )}{d a}\) | \(46\) |
parallelrisch | \(\frac {12 \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-24 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-3-\sin \left (3 d x +3 c \right )+15 \sin \left (d x +c \right )+3 \cos \left (2 d x +2 c \right )}{12 d a}\) | \(69\) |
risch | \(\frac {i x}{a}-\frac {5 i {\mathrm e}^{i \left (d x +c \right )}}{8 d a}+\frac {5 i {\mathrm e}^{-i \left (d x +c \right )}}{8 d a}+\frac {2 i c}{a d}-\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{a d}-\frac {\sin \left (3 d x +3 c \right )}{12 d a}+\frac {\cos \left (2 d x +2 c \right )}{4 a d}\) | \(110\) |
norman | \(\frac {-\frac {5}{3 a d}-\frac {20 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {20 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {5 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 d a}+\frac {\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )}{3 d a}-\frac {16 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {16 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {2 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a d}\) | \(212\) |
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Time = 0.29 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.72 \[ \int \frac {\cos (c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {3 \, \cos \left (d x + c\right )^{2} - 2 \, {\left (\cos \left (d x + c\right )^{2} - 4\right )} \sin \left (d x + c\right ) - 6 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{6 \, a d} \]
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Time = 0.39 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.99 \[ \int \frac {\cos (c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\begin {cases} - \frac {\log {\left (\sin {\left (c + d x \right )} + 1 \right )}}{a d} + \frac {\sin ^{3}{\left (c + d x \right )}}{3 a d} - \frac {\sin ^{2}{\left (c + d x \right )}}{2 a d} + \frac {\sin {\left (c + d x \right )}}{a d} & \text {for}\: d \neq 0 \\\frac {x \sin ^{3}{\left (c \right )} \cos {\left (c \right )}}{a \sin {\left (c \right )} + a} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.79 \[ \int \frac {\cos (c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {2 \, \sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )^{2} + 6 \, \sin \left (d x + c\right )}{a} - \frac {6 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a}}{6 \, d} \]
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Time = 0.30 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.96 \[ \int \frac {\cos (c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {6 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {2 \, a^{2} \sin \left (d x + c\right )^{3} - 3 \, a^{2} \sin \left (d x + c\right )^{2} + 6 \, a^{2} \sin \left (d x + c\right )}{a^{3}}}{6 \, d} \]
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Time = 0.06 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.84 \[ \int \frac {\cos (c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )}{a}-\frac {\sin \left (c+d\,x\right )}{a}+\frac {{\sin \left (c+d\,x\right )}^2}{2\,a}-\frac {{\sin \left (c+d\,x\right )}^3}{3\,a}}{d} \]
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