Integrand size = 27, antiderivative size = 44 \[ \int \csc (c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {a^2 \text {arctanh}(\cos (c+d x))}{d}+\frac {2 a^2 \sec (c+d x)}{d}+\frac {2 a^2 \tan (c+d x)}{d} \]
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Time = 0.10 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2952, 3852, 8, 2702, 327, 213, 2686} \[ \int \csc (c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {a^2 \text {arctanh}(\cos (c+d x))}{d}+\frac {2 a^2 \tan (c+d x)}{d}+\frac {2 a^2 \sec (c+d x)}{d} \]
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Rule 8
Rule 213
Rule 327
Rule 2686
Rule 2702
Rule 2952
Rule 3852
Rubi steps \begin{align*} \text {integral}& = \int \left (2 a^2 \sec ^2(c+d x)+a^2 \csc (c+d x) \sec ^2(c+d x)+a^2 \sec (c+d x) \tan (c+d x)\right ) \, dx \\ & = a^2 \int \csc (c+d x) \sec ^2(c+d x) \, dx+a^2 \int \sec (c+d x) \tan (c+d x) \, dx+\left (2 a^2\right ) \int \sec ^2(c+d x) \, dx \\ & = \frac {a^2 \text {Subst}(\int 1 \, dx,x,\sec (c+d x))}{d}+\frac {a^2 \text {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d}-\frac {\left (2 a^2\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d} \\ & = \frac {2 a^2 \sec (c+d x)}{d}+\frac {2 a^2 \tan (c+d x)}{d}+\frac {a^2 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d} \\ & = -\frac {a^2 \text {arctanh}(\cos (c+d x))}{d}+\frac {2 a^2 \sec (c+d x)}{d}+\frac {2 a^2 \tan (c+d x)}{d} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.57 \[ \int \csc (c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 \left (-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {4 \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}\right )}{d} \]
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Time = 0.19 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.02
method | result | size |
parallelrisch | \(\frac {\left (-4+\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )\right ) a^{2}}{d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}\) | \(45\) |
derivativedivides | \(\frac {\frac {a^{2}}{\cos \left (d x +c \right )}+2 a^{2} \tan \left (d x +c \right )+a^{2} \left (\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )}{d}\) | \(58\) |
default | \(\frac {\frac {a^{2}}{\cos \left (d x +c \right )}+2 a^{2} \tan \left (d x +c \right )+a^{2} \left (\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )}{d}\) | \(58\) |
risch | \(\frac {4 a^{2}}{d \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}+\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}-\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}\) | \(63\) |
norman | \(\frac {-\frac {4 a^{2}}{d}-\frac {4 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {8 a^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {4 a^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {4 a^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {8 a^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\) | \(152\) |
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Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (44) = 88\).
Time = 0.28 (sec) , antiderivative size = 126, normalized size of antiderivative = 2.86 \[ \int \csc (c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {4 \, a^{2} \cos \left (d x + c\right ) + 4 \, a^{2} \sin \left (d x + c\right ) + 4 \, a^{2} - {\left (a^{2} \cos \left (d x + c\right ) - a^{2} \sin \left (d x + c\right ) + a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left (a^{2} \cos \left (d x + c\right ) - a^{2} \sin \left (d x + c\right ) + a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{2 \, {\left (d \cos \left (d x + c\right ) - d \sin \left (d x + c\right ) + d\right )}} \]
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\[ \int \csc (c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=a^{2} \left (\int \csc {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 2 \sin {\left (c + d x \right )} \csc {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \sin ^{2}{\left (c + d x \right )} \csc {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx\right ) \]
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Time = 0.21 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.48 \[ \int \csc (c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^{2} {\left (\frac {2}{\cos \left (d x + c\right )} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 4 \, a^{2} \tan \left (d x + c\right ) + \frac {2 \, a^{2}}{\cos \left (d x + c\right )}}{2 \, d} \]
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Time = 0.29 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.86 \[ \int \csc (c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {4 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1}}{d} \]
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Time = 9.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.89 \[ \int \csc (c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {4\,a^2}{d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )} \]
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