Integrand size = 25, antiderivative size = 52 \[ \int \cot ^2(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx=-a x+\frac {a \text {arctanh}(\cos (c+d x))}{2 d}-\frac {a \cot (c+d x)}{d}-\frac {a \cot (c+d x) \csc (c+d x)}{2 d} \]
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Time = 0.06 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2917, 2691, 3855, 3554, 8} \[ \int \cot ^2(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \text {arctanh}(\cos (c+d x))}{2 d}-\frac {a \cot (c+d x)}{d}-\frac {a \cot (c+d x) \csc (c+d x)}{2 d}-a x \]
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Rule 8
Rule 2691
Rule 2917
Rule 3554
Rule 3855
Rubi steps \begin{align*} \text {integral}& = a \int \cot ^2(c+d x) \, dx+a \int \cot ^2(c+d x) \csc (c+d x) \, dx \\ & = -\frac {a \cot (c+d x)}{d}-\frac {a \cot (c+d x) \csc (c+d x)}{2 d}-\frac {1}{2} a \int \csc (c+d x) \, dx-a \int 1 \, dx \\ & = -a x+\frac {a \text {arctanh}(\cos (c+d x))}{2 d}-\frac {a \cot (c+d x)}{d}-\frac {a \cot (c+d x) \csc (c+d x)}{2 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.13 (sec) , antiderivative size = 109, normalized size of antiderivative = 2.10 \[ \int \cot ^2(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {a \cot (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-\tan ^2(c+d x)\right )}{d}+\frac {a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}-\frac {a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 d} \]
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Time = 0.18 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.35
method | result | size |
parallelrisch | \(-\frac {a \left (-\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+8 d x +4 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+4 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}\) | \(70\) |
derivativedivides | \(\frac {a \left (-\cot \left (d x +c \right )-d x -c \right )+a \left (-\frac {\cos ^{3}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{2}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )}{d}\) | \(71\) |
default | \(\frac {a \left (-\cot \left (d x +c \right )-d x -c \right )+a \left (-\frac {\cos ^{3}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{2}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )}{d}\) | \(71\) |
risch | \(-a x +\frac {a \left ({\mathrm e}^{3 i \left (d x +c \right )}+{\mathrm e}^{i \left (d x +c \right )}-2 i {\mathrm e}^{2 i \left (d x +c \right )}+2 i\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}+\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}\) | \(93\) |
norman | \(\frac {-\frac {a}{8 d}-\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d}+\frac {a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-a x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}\) | \(148\) |
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Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (48) = 96\).
Time = 0.28 (sec) , antiderivative size = 114, normalized size of antiderivative = 2.19 \[ \int \cot ^2(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx=-\frac {4 \, a d x \cos \left (d x + c\right )^{2} - 4 \, a d x - 4 \, a \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 \, a \cos \left (d x + c\right ) - {\left (a \cos \left (d x + c\right )^{2} - a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left (a \cos \left (d x + c\right )^{2} - a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{4 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \]
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\[ \int \cot ^2(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx=a \left (\int \cos ^{2}{\left (c + d x \right )} \csc ^{3}{\left (c + d x \right )}\, dx + \int \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )} \csc ^{3}{\left (c + d x \right )}\, dx\right ) \]
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Time = 0.29 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.27 \[ \int \cot ^2(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx=-\frac {4 \, {\left (d x + c + \frac {1}{\tan \left (d x + c\right )}\right )} a - a {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + \log \left (\cos \left (d x + c\right ) + 1\right ) - \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{4 \, d} \]
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Time = 0.38 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.83 \[ \int \cot ^2(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, {\left (d x + c\right )} a - 4 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 4 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {6 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}}{8 \, d} \]
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Time = 9.30 (sec) , antiderivative size = 145, normalized size of antiderivative = 2.79 \[ \int \cot ^2(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx=\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d}-\frac {a\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d}-\frac {2\,a\,\mathrm {atan}\left (\frac {2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {a\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{2\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d} \]
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