Integrand size = 27, antiderivative size = 49 \[ \int \frac {\cos (c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\log (1+\sin (c+d x))}{a d}-\frac {\sin (c+d x)}{a d}+\frac {\sin ^2(c+d x)}{2 a d} \]
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Time = 0.05 (sec) , antiderivative size = 49, 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} \[ \int \frac {\cos (c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\sin ^2(c+d x)}{2 a d}-\frac {\sin (c+d x)}{a d}+\frac {\log (\sin (c+d x)+1)}{a d} \]
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Rule 12
Rule 45
Rule 2912
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^2}{a^2 (a+x)} \, dx,x,a \sin (c+d x)\right )}{a d} \\ & = \frac {\text {Subst}\left (\int \frac {x^2}{a+x} \, dx,x,a \sin (c+d x)\right )}{a^3 d} \\ & = \frac {\text {Subst}\left (\int \left (-a+x+\frac {a^2}{a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 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} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.78 \[ \int \frac {\cos (c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {2 \log (1+\sin (c+d x))-2 \sin (c+d x)+\sin ^2(c+d x)}{2 a d} \]
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Time = 0.10 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.73
method | result | size |
derivativedivides | \(\frac {\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\sin \left (d x +c \right )+\ln \left (1+\sin \left (d x +c \right )\right )}{d a}\) | \(36\) |
default | \(\frac {\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\sin \left (d x +c \right )+\ln \left (1+\sin \left (d x +c \right )\right )}{d a}\) | \(36\) |
parallelrisch | \(\frac {8 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-4 \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1-\cos \left (2 d x +2 c \right )-4 \sin \left (d x +c \right )}{4 d a}\) | \(58\) |
risch | \(-\frac {i x}{a}+\frac {i {\mathrm e}^{i \left (d x +c \right )}}{2 d a}-\frac {i {\mathrm e}^{-i \left (d x +c \right )}}{2 d a}-\frac {2 i c}{a d}+\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{a d}-\frac {\cos \left (2 d x +2 c \right )}{4 a d}\) | \(93\) |
norman | \(\frac {\frac {2}{a d}+\frac {2 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {4 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {4 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {6 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {6 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {2 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a d}-\frac {\ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}\) | \(177\) |
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Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.73 \[ \int \frac {\cos (c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\cos \left (d x + c\right )^{2} - 2 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 2 \, \sin \left (d x + c\right )}{2 \, a d} \]
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Time = 0.32 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.08 \[ \int \frac {\cos (c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\begin {cases} \frac {\log {\left (\sin {\left (c + d x \right )} + 1 \right )}}{a d} + \frac {\sin ^{2}{\left (c + d x \right )}}{2 a d} - \frac {\sin {\left (c + d x \right )}}{a d} & \text {for}\: d \neq 0 \\\frac {x \sin ^{2}{\left (c \right )} \cos {\left (c \right )}}{a \sin {\left (c \right )} + a} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.84 \[ \int \frac {\cos (c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right )}{a} + \frac {2 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a}}{2 \, d} \]
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Time = 0.30 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.92 \[ \int \frac {\cos (c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {2 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} + \frac {a \sin \left (d x + c\right )^{2} - 2 \, a \sin \left (d x + c\right )}{a^{2}}}{2 \, d} \]
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Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.71 \[ \int \frac {\cos (c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )-\sin \left (c+d\,x\right )+\frac {{\sin \left (c+d\,x\right )}^2}{2}}{a\,d} \]
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