Integrand size = 25, antiderivative size = 53 \[ \int \cos (c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \csc (c+d x)}{d}+\frac {a \log (\sin (c+d x))}{d}-\frac {a \sin (c+d x)}{d}-\frac {a \sin ^2(c+d x)}{2 d} \]
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Time = 0.04 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2915, 12, 76} \[ \int \cos (c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \sin ^2(c+d x)}{2 d}-\frac {a \sin (c+d x)}{d}-\frac {a \csc (c+d x)}{d}+\frac {a \log (\sin (c+d x))}{d} \]
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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) (a+x)^2}{x^2} \, dx,x,a \sin (c+d x)\right )}{a^3 d} \\ & = \frac {\text {Subst}\left (\int \frac {(a-x) (a+x)^2}{x^2} \, dx,x,a \sin (c+d x)\right )}{a 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 d} \\ & = -\frac {a \csc (c+d x)}{d}+\frac {a \log (\sin (c+d x))}{d}-\frac {a \sin (c+d x)}{d}-\frac {a \sin ^2(c+d x)}{2 d} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00 \[ \int \cos (c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \csc (c+d x)}{d}+\frac {a \log (\sin (c+d x))}{d}-\frac {a \sin (c+d x)}{d}-\frac {a \sin ^2(c+d x)}{2 d} \]
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Time = 0.17 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.23
method | result | size |
derivativedivides | \(\frac {a \left (\frac {\left (\cos ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+a \left (-\frac {\cos ^{4}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )\right )}{d}\) | \(65\) |
default | \(\frac {a \left (\frac {\left (\cos ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+a \left (-\frac {\cos ^{4}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )\right )}{d}\) | \(65\) |
parallelrisch | \(\frac {a \left (4 \cot \left (\frac {d x}{2}+\frac {c}{2}\right ) \cos \left (d x +c \right )-2 \sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )+\cos \left (2 d x +2 c \right )+4 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{4 d}\) | \(92\) |
risch | \(-i a x +\frac {a \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 d}+\frac {i a \,{\mathrm e}^{i \left (d x +c \right )}}{2 d}-\frac {i a \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {2 i a c}{d}-\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}+\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(123\) |
norman | \(\frac {-\frac {a}{2 d}-\frac {7 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {7 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {2 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {a \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\) | \(139\) |
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Time = 0.27 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.28 \[ \int \cos (c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {4 \, a \cos \left (d x + c\right )^{2} + 4 \, a \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) + {\left (2 \, a \cos \left (d x + c\right )^{2} - a\right )} \sin \left (d x + c\right ) - 8 \, a}{4 \, d \sin \left (d x + c\right )} \]
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\[ \int \cos (c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx=a \left (\int \cos ^{3}{\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}\, dx + \int \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}\, dx\right ) \]
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Time = 0.21 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.87 \[ \int \cos (c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \sin \left (d x + c\right )^{2} - 2 \, a \log \left (\sin \left (d x + c\right )\right ) + 2 \, a \sin \left (d x + c\right ) + \frac {2 \, a}{\sin \left (d x + c\right )}}{2 \, d} \]
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Time = 0.32 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.89 \[ \int \cos (c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \sin \left (d x + c\right )^{2} - 2 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 2 \, a \sin \left (d x + c\right ) + \frac {2 \, a}{\sin \left (d x + c\right )}}{2 \, d} \]
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Time = 9.63 (sec) , antiderivative size = 140, normalized size of antiderivative = 2.64 \[ \int \cos (c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {a\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}-\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d}-\frac {5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+6\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a}{d\,\left (2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )} \]
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