Integrand size = 27, antiderivative size = 138 \[ \int \cot ^2(c+d x) \csc (c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {1}{2} b \left (6 a^2-b^2\right ) x+\frac {a \left (a^2-6 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 d}+\frac {15 a b^2 \cos (c+d x)}{2 d}+\frac {5 b^3 \cos (c+d x) \sin (c+d x)}{2 d}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^2}{2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 d} \]
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Time = 0.32 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2968, 3127, 3126, 3112, 3102, 2814, 3855} \[ \int \cot ^2(c+d x) \csc (c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {a \left (a^2-6 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 d}-\frac {1}{2} b x \left (6 a^2-b^2\right )+\frac {15 a b^2 \cos (c+d x)}{2 d}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^2}{2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 d}+\frac {5 b^3 \sin (c+d x) \cos (c+d x)}{2 d} \]
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Rule 2814
Rule 2968
Rule 3102
Rule 3112
Rule 3126
Rule 3127
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \int \csc ^3(c+d x) (a+b \sin (c+d x))^3 \left (1-\sin ^2(c+d x)\right ) \, dx \\ & = -\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 d}+\frac {1}{2} \int \csc ^2(c+d x) (a+b \sin (c+d x))^2 \left (3 b-a \sin (c+d x)-4 b \sin ^2(c+d x)\right ) \, dx \\ & = -\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^2}{2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 d}+\frac {1}{2} \int \csc (c+d x) (a+b \sin (c+d x)) \left (-a^2+6 b^2-5 a b \sin (c+d x)-10 b^2 \sin ^2(c+d x)\right ) \, dx \\ & = \frac {5 b^3 \cos (c+d x) \sin (c+d x)}{2 d}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^2}{2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 d}+\frac {1}{4} \int \csc (c+d x) \left (-2 a \left (a^2-6 b^2\right )-2 b \left (6 a^2-b^2\right ) \sin (c+d x)-30 a b^2 \sin ^2(c+d x)\right ) \, dx \\ & = \frac {15 a b^2 \cos (c+d x)}{2 d}+\frac {5 b^3 \cos (c+d x) \sin (c+d x)}{2 d}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^2}{2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 d}+\frac {1}{4} \int \csc (c+d x) \left (-2 a \left (a^2-6 b^2\right )-2 b \left (6 a^2-b^2\right ) \sin (c+d x)\right ) \, dx \\ & = -\frac {1}{2} b \left (6 a^2-b^2\right ) x+\frac {15 a b^2 \cos (c+d x)}{2 d}+\frac {5 b^3 \cos (c+d x) \sin (c+d x)}{2 d}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^2}{2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 d}-\frac {1}{2} \left (a \left (a^2-6 b^2\right )\right ) \int \csc (c+d x) \, dx \\ & = -\frac {1}{2} b \left (6 a^2-b^2\right ) x+\frac {a \left (a^2-6 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 d}+\frac {15 a b^2 \cos (c+d x)}{2 d}+\frac {5 b^3 \cos (c+d x) \sin (c+d x)}{2 d}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^2}{2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 d} \\ \end{align*}
Time = 1.91 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.39 \[ \int \cot ^2(c+d x) \csc (c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {-24 a^2 b c+4 b^3 c-24 a^2 b d x+4 b^3 d x+24 a b^2 \cos (c+d x)-12 a^2 b \cot \left (\frac {1}{2} (c+d x)\right )-a^3 \csc ^2\left (\frac {1}{2} (c+d x)\right )+4 a^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-24 a b^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-4 a^3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+24 a b^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+a^3 \sec ^2\left (\frac {1}{2} (c+d x)\right )+2 b^3 \sin (2 (c+d x))+12 a^2 b \tan \left (\frac {1}{2} (c+d x)\right )}{8 d} \]
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Time = 0.44 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.96
method | result | size |
derivativedivides | \(\frac {a^{3} \left (-\frac {\cos ^{3}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{2}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+3 a^{2} b \left (-\cot \left (d x +c \right )-d x -c \right )+3 a \,b^{2} \left (\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+b^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(132\) |
default | \(\frac {a^{3} \left (-\frac {\cos ^{3}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{2}-\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+3 a^{2} b \left (-\cot \left (d x +c \right )-d x -c \right )+3 a \,b^{2} \left (\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+b^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(132\) |
parallelrisch | \(\frac {4 \left (-a^{3}+6 a \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a^{3} \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-4\right ) \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+12 \csc \left (\frac {d x}{2}+\frac {c}{2}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b +2 a^{3} \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-24 a^{2} b d x +4 b^{3} d x -24 a^{2} b \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+24 \cos \left (d x +c \right ) a \,b^{2}+2 b^{3} \sin \left (2 d x +2 c \right )+24 a \,b^{2}}{8 d}\) | \(161\) |
risch | \(-3 a^{2} b x +\frac {b^{3} x}{2}-\frac {i b^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}+\frac {3 \,{\mathrm e}^{i \left (d x +c \right )} a \,b^{2}}{2 d}+\frac {3 \,{\mathrm e}^{-i \left (d x +c \right )} a \,b^{2}}{2 d}+\frac {i b^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {i a^{2} \left (i a \,{\mathrm e}^{3 i \left (d x +c \right )}+i a \,{\mathrm e}^{i \left (d x +c \right )}+6 b \,{\mathrm e}^{2 i \left (d x +c \right )}-6 b \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}+\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{d}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}-\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{d}\) | \(233\) |
norman | \(\frac {\left (-9 a^{2} b +\frac {3}{2} b^{3}\right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-9 a^{2} b +\frac {3}{2} b^{3}\right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-3 a^{2} b +\frac {1}{2} b^{3}\right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-3 a^{2} b +\frac {1}{2} b^{3}\right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {b \left (3 a^{2}-b^{2}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {a^{3}}{8 d}+\frac {a^{3} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {\left (3 a^{3}-24 a \,b^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {\left (7 a^{3}-48 a \,b^{2}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {\left (11 a^{3}-96 a \,b^{2}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {3 a^{2} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d}+\frac {3 a^{2} b \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {b \left (3 a^{2}-b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {a \left (a^{2}-6 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}\) | \(359\) |
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Time = 0.30 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.57 \[ \int \cot ^2(c+d x) \csc (c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {12 \, a b^{2} \cos \left (d x + c\right )^{3} - 2 \, {\left (6 \, a^{2} b - b^{3}\right )} d x \cos \left (d x + c\right )^{2} + 2 \, {\left (6 \, a^{2} b - b^{3}\right )} d x + 2 \, {\left (a^{3} - 6 \, a b^{2}\right )} \cos \left (d x + c\right ) - {\left (a^{3} - 6 \, a b^{2} - {\left (a^{3} - 6 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left (a^{3} - 6 \, a b^{2} - {\left (a^{3} - 6 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2 \, {\left (b^{3} \cos \left (d x + c\right )^{3} + {\left (6 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \]
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Timed out. \[ \int \cot ^2(c+d x) \csc (c+d x) (a+b \sin (c+d x))^3 \, dx=\text {Timed out} \]
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Time = 0.27 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.93 \[ \int \cot ^2(c+d x) \csc (c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {12 \, {\left (d x + c + \frac {1}{\tan \left (d x + c\right )}\right )} a^{2} b - {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} b^{3} - a^{3} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + \log \left (\cos \left (d x + c\right ) + 1\right ) - \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 6 \, a b^{2} {\left (2 \, \cos \left (d x + c\right ) - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{4 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (126) = 252\).
Time = 0.37 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.97 \[ \int \cot ^2(c+d x) \csc (c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, {\left (6 \, a^{2} b - b^{3}\right )} {\left (d x + c\right )} - 4 \, {\left (a^{3} - 6 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {2 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 12 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 12 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 8 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 24 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 24 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{3}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}^{2}}}{8 \, d} \]
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Time = 9.95 (sec) , antiderivative size = 585, normalized size of antiderivative = 4.24 \[ \int \cot ^2(c+d x) \csc (c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (3\,a\,b^2-\frac {a^3}{2}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (6\,a^2\,b+4\,b^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (24\,a\,b^2-\frac {a^3}{2}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (24\,a\,b^2-a^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (12\,a^2\,b-4\,b^3\right )+\frac {a^3}{2}+6\,a^2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}+\frac {3\,a^2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d}+\frac {b\,\mathrm {atan}\left (\frac {\frac {b\,\left (6\,a^2-b^2\right )\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (6\,a\,b^2-a^3\right )-6\,a^2\,b+b^3-b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (6\,a^2-b^2\right )\,3{}\mathrm {i}\right )}{2}+\frac {b\,\left (6\,a^2-b^2\right )\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (6\,a\,b^2-a^3\right )-6\,a^2\,b+b^3+b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (6\,a^2-b^2\right )\,3{}\mathrm {i}\right )}{2}}{2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (36\,a^4\,b^2-12\,a^2\,b^4+b^6\right )+6\,a\,b^5+6\,a^5\,b-37\,a^3\,b^3-\frac {b\,\left (6\,a^2-b^2\right )\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (6\,a\,b^2-a^3\right )-6\,a^2\,b+b^3-b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (6\,a^2-b^2\right )\,3{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}+\frac {b\,\left (6\,a^2-b^2\right )\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (6\,a\,b^2-a^3\right )-6\,a^2\,b+b^3+b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (6\,a^2-b^2\right )\,3{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}}\right )\,\left (6\,a^2-b^2\right )}{d} \]
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