Integrand size = 29, antiderivative size = 62 \[ \int \frac {\cos ^3(c+d x) \cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\csc (c+d x)}{a d}-\frac {\log (\sin (c+d x))}{a d}-\frac {\sin (c+d x)}{a d}+\frac {\sin ^2(c+d x)}{2 a d} \]
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Time = 0.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2915, 12, 76} \[ \int \frac {\cos ^3(c+d x) \cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\sin ^2(c+d x)}{2 a d}-\frac {\sin (c+d x)}{a d}-\frac {\csc (c+d x)}{a d}-\frac {\log (\sin (c+d x))}{a d} \]
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Rule 12
Rule 76
Rule 2915
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a^2 (a-x)^2 (a+x)}{x^2} \, dx,x,a \sin (c+d x)\right )}{a^5 d} \\ & = \frac {\text {Subst}\left (\int \frac {(a-x)^2 (a+x)}{x^2} \, dx,x,a \sin (c+d x)\right )}{a^3 d} \\ & = \frac {\text {Subst}\left (\int \left (-a+\frac {a^3}{x^2}-\frac {a^2}{x}+x\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d} \\ & = -\frac {\csc (c+d x)}{a d}-\frac {\log (\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 = 0.04 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.74 \[ \int \frac {\cos ^3(c+d x) \cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {2 \csc (c+d x)+2 \log (\sin (c+d x))+2 \sin (c+d x)-\sin ^2(c+d x)}{2 a d} \]
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Time = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.74
method | result | size |
derivativedivides | \(\frac {\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\sin \left (d x +c \right )-\ln \left (\sin \left (d x +c \right )\right )-\frac {1}{\sin \left (d x +c \right )}}{d a}\) | \(46\) |
default | \(\frac {\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\sin \left (d x +c \right )-\ln \left (\sin \left (d x +c \right )\right )-\frac {1}{\sin \left (d x +c \right )}}{d a}\) | \(46\) |
parallelrisch | \(\frac {4 \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (4 \cos \left (d x +c \right )-4\right ) \cot \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )-\cos \left (2 d x +2 c \right )+1}{4 d a}\) | \(88\) |
risch | \(\frac {i x}{a}-\frac {{\mathrm e}^{2 i \left (d x +c \right )}}{8 a d}+\frac {i {\mathrm e}^{i \left (d x +c \right )}}{2 d a}-\frac {i {\mathrm e}^{-i \left (d x +c \right )}}{2 d a}-\frac {{\mathrm e}^{-2 i \left (d x +c \right )}}{8 a d}+\frac {2 i c}{a d}-\frac {2 i {\mathrm e}^{i \left (d x +c \right )}}{d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d a}\) | \(140\) |
norman | \(\frac {-\frac {1}{2 a d}-\frac {\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d a}-\frac {7 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}-\frac {7 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}-\frac {7 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}-\frac {7 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}-\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d a}-\frac {\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}\) | \(223\) |
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Time = 0.26 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.05 \[ \int \frac {\cos ^3(c+d x) \cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {4 \, \cos \left (d x + c\right )^{2} - {\left (2 \, \cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - 4 \, \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) - 8}{4 \, a d \sin \left (d x + c\right )} \]
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Timed out. \[ \int \frac {\cos ^3(c+d x) \cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.84 \[ \int \frac {\cos ^3(c+d x) \cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right )}{a} - \frac {2 \, \log \left (\sin \left (d x + c\right )\right )}{a} - \frac {2}{a \sin \left (d x + c\right )}}{2 \, d} \]
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Time = 0.40 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.05 \[ \int \frac {\cos ^3(c+d x) \cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {2 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} - \frac {a \sin \left (d x + c\right )^{2} - 2 \, a \sin \left (d x + c\right )}{a^{2}} - \frac {2 \, {\left (\sin \left (d x + c\right ) - 1\right )}}{a \sin \left (d x + c\right )}}{2 \, d} \]
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Time = 9.74 (sec) , antiderivative size = 146, normalized size of antiderivative = 2.35 \[ \int \frac {\cos ^3(c+d x) \cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a\,d}-\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1}{d\,\left (2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+4\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+2\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a\,d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a\,d} \]
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