Integrand size = 32, antiderivative size = 78 \[ \int \csc ^3(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=-2 a^3 A x-\frac {a^3 A \text {arctanh}(\cos (c+d x))}{2 d}+\frac {a^3 A \cos (c+d x)}{d}-\frac {2 a^3 A \cot (c+d x)}{d}-\frac {a^3 A \cot (c+d x) \csc (c+d x)}{2 d} \]
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Time = 0.09 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3045, 3852, 8, 3853, 3855, 2718} \[ \int \csc ^3(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=-\frac {a^3 A \text {arctanh}(\cos (c+d x))}{2 d}+\frac {a^3 A \cos (c+d x)}{d}-\frac {2 a^3 A \cot (c+d x)}{d}-\frac {a^3 A \cot (c+d x) \csc (c+d x)}{2 d}-2 a^3 A x \]
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Rule 8
Rule 2718
Rule 3045
Rule 3852
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \int \left (-2 a^3 A+2 a^3 A \csc ^2(c+d x)+a^3 A \csc ^3(c+d x)-a^3 A \sin (c+d x)\right ) \, dx \\ & = -2 a^3 A x+\left (a^3 A\right ) \int \csc ^3(c+d x) \, dx-\left (a^3 A\right ) \int \sin (c+d x) \, dx+\left (2 a^3 A\right ) \int \csc ^2(c+d x) \, dx \\ & = -2 a^3 A x+\frac {a^3 A \cos (c+d x)}{d}-\frac {a^3 A \cot (c+d x) \csc (c+d x)}{2 d}+\frac {1}{2} \left (a^3 A\right ) \int \csc (c+d x) \, dx-\frac {\left (2 a^3 A\right ) \text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{d} \\ & = -2 a^3 A x-\frac {a^3 A \text {arctanh}(\cos (c+d x))}{2 d}+\frac {a^3 A \cos (c+d x)}{d}-\frac {2 a^3 A \cot (c+d x)}{d}-\frac {a^3 A \cot (c+d x) \csc (c+d x)}{2 d} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.82 \[ \int \csc ^3(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=-2 a^3 A x+\frac {a^3 A \cos (c) \cos (d x)}{d}-\frac {2 a^3 A \cot (c+d x)}{d}-\frac {a^3 A \csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {a^3 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {a^3 A \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {a^3 A \sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {a^3 A \sin (c) \sin (d x)}{d} \]
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Time = 1.21 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(\frac {A \,a^{3} \cos \left (d x +c \right )-2 A \,a^{3} \left (d x +c \right )-2 A \,a^{3} \cot \left (d x +c \right )+A \,a^{3} \left (-\frac {\csc \left (d x +c \right ) \cot \left (d x +c \right )}{2}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )}{d}\) | \(78\) |
default | \(\frac {A \,a^{3} \cos \left (d x +c \right )-2 A \,a^{3} \left (d x +c \right )-2 A \,a^{3} \cot \left (d x +c \right )+A \,a^{3} \left (-\frac {\csc \left (d x +c \right ) \cot \left (d x +c \right )}{2}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )}{d}\) | \(78\) |
parallelrisch | \(\frac {A \,a^{3} \left (-16 d x +4 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (d x +c \right )+23 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 \sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )-17 \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-16 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}\) | \(126\) |
risch | \(-2 a^{3} A x +\frac {A \,a^{3} {\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {A \,a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {A \,a^{3} \left ({\mathrm e}^{3 i \left (d x +c \right )}+{\mathrm e}^{i \left (d x +c \right )}-4 i {\mathrm e}^{2 i \left (d x +c \right )}+4 i\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}-\frac {A \,a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}+\frac {A \,a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}\) | \(141\) |
norman | \(\frac {\frac {A \,a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {A \,a^{3} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {A \,a^{3}}{8 d}+\frac {3 A \,a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {5 A \,a^{3} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {27 A \,a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {A \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {3 A \,a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 A \,a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 A \,a^{3} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {3 A \,a^{3} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {A \,a^{3} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-2 a^{3} A x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 a^{3} A x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12 a^{3} A x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 a^{3} A x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a^{3} A x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {A \,a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}\) | \(364\) |
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Leaf count of result is larger than twice the leaf count of optimal. 152 vs. \(2 (74) = 148\).
Time = 0.30 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.95 \[ \int \csc ^3(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=-\frac {8 \, A a^{3} d x \cos \left (d x + c\right )^{2} - 4 \, A a^{3} \cos \left (d x + c\right )^{3} - 8 \, A a^{3} d x - 8 \, A a^{3} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 2 \, A a^{3} \cos \left (d x + c\right ) + {\left (A a^{3} \cos \left (d x + c\right )^{2} - A a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - {\left (A a^{3} \cos \left (d x + c\right )^{2} - A a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{4 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \]
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\[ \int \csc ^3(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=- A a^{3} \left (\int \left (- 2 \sin {\left (c + d x \right )} \csc ^{3}{\left (c + d x \right )}\right )\, dx + \int 2 \sin ^{3}{\left (c + d x \right )} \csc ^{3}{\left (c + d x \right )}\, dx + \int \sin ^{4}{\left (c + d x \right )} \csc ^{3}{\left (c + d x \right )}\, dx + \int \left (- \csc ^{3}{\left (c + d x \right )}\right )\, dx\right ) \]
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Time = 0.22 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.15 \[ \int \csc ^3(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=-\frac {8 \, {\left (d x + c\right )} A a^{3} - A a^{3} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 4 \, A a^{3} \cos \left (d x + c\right ) + \frac {8 \, A a^{3}}{\tan \left (d x + c\right )}}{4 \, d} \]
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Time = 0.31 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.76 \[ \int \csc ^3(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=\frac {A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 16 \, {\left (d x + c\right )} A a^{3} + 4 \, A a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 8 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {16 \, A a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} - \frac {6 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + A a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}}{8 \, d} \]
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Time = 13.78 (sec) , antiderivative size = 220, normalized size of antiderivative = 2.82 \[ \int \csc ^3(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=\frac {A\,a^3\,\left (\frac {\cos \left (c+d\,x\right )}{2}-4\,\mathrm {atan}\left (\frac {\sqrt {17}\,\left (4\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{17\,\cos \left (\frac {c}{2}-\mathrm {atan}\left (4\right )+\frac {d\,x}{2}\right )}\right )-\frac {\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{2}+\cos \left (2\,c+2\,d\,x\right )+\frac {\cos \left (3\,c+3\,d\,x\right )}{2}+2\,\sin \left (2\,c+2\,d\,x\right )+4\,\mathrm {atan}\left (\frac {\sqrt {17}\,\left (4\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{17\,\cos \left (\frac {c}{2}-\mathrm {atan}\left (4\right )+\frac {d\,x}{2}\right )}\right )\,\cos \left (2\,c+2\,d\,x\right )+\frac {\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\cos \left (2\,c+2\,d\,x\right )}{2}-1\right )}{2\,d\,\left ({\cos \left (c+d\,x\right )}^2-1\right )} \]
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