Integrand size = 32, antiderivative size = 86 \[ \int \csc ^5(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=\frac {5 a^3 A \text {arctanh}(\cos (c+d x))}{8 d}-\frac {2 a^3 A \cot ^3(c+d x)}{3 d}-\frac {3 a^3 A \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a^3 A \cot (c+d x) \csc ^3(c+d x)}{4 d} \]
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Time = 0.12 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {3045, 3855, 3852, 8, 3853} \[ \int \csc ^5(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=\frac {5 a^3 A \text {arctanh}(\cos (c+d x))}{8 d}-\frac {2 a^3 A \cot ^3(c+d x)}{3 d}-\frac {a^3 A \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {3 a^3 A \cot (c+d x) \csc (c+d x)}{8 d} \]
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Rule 8
Rule 3045
Rule 3852
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \int \left (-a^3 A \csc (c+d x)-2 a^3 A \csc ^2(c+d x)+2 a^3 A \csc ^4(c+d x)+a^3 A \csc ^5(c+d x)\right ) \, dx \\ & = -\left (\left (a^3 A\right ) \int \csc (c+d x) \, dx\right )+\left (a^3 A\right ) \int \csc ^5(c+d x) \, dx-\left (2 a^3 A\right ) \int \csc ^2(c+d x) \, dx+\left (2 a^3 A\right ) \int \csc ^4(c+d x) \, dx \\ & = \frac {a^3 A \text {arctanh}(\cos (c+d x))}{d}-\frac {a^3 A \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {1}{4} \left (3 a^3 A\right ) \int \csc ^3(c+d x) \, dx+\frac {\left (2 a^3 A\right ) \text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}-\frac {\left (2 a^3 A\right ) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{d} \\ & = \frac {a^3 A \text {arctanh}(\cos (c+d x))}{d}-\frac {2 a^3 A \cot ^3(c+d x)}{3 d}-\frac {3 a^3 A \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a^3 A \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {1}{8} \left (3 a^3 A\right ) \int \csc (c+d x) \, dx \\ & = \frac {5 a^3 A \text {arctanh}(\cos (c+d x))}{8 d}-\frac {2 a^3 A \cot ^3(c+d x)}{3 d}-\frac {3 a^3 A \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a^3 A \cot (c+d x) \csc ^3(c+d x)}{4 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(210\) vs. \(2(86)=172\).
Time = 0.23 (sec) , antiderivative size = 210, normalized size of antiderivative = 2.44 \[ \int \csc ^5(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=a^3 A \left (\frac {\cot \left (\frac {1}{2} (c+d x)\right )}{3 d}-\frac {3 \csc ^2\left (\frac {1}{2} (c+d x)\right )}{32 d}-\frac {\cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{12 d}-\frac {\csc ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {5 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}-\frac {5 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}+\frac {3 \sec ^2\left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {\sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{3 d}+\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{12 d}\right ) \]
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Time = 1.28 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.33
method | result | size |
derivativedivides | \(\frac {-A \,a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )+2 A \,a^{3} \cot \left (d x +c \right )+2 A \,a^{3} \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{3}\right ) \cot \left (d x +c \right )+A \,a^{3} \left (\left (-\frac {\left (\csc ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \csc \left (d x +c \right )}{8}\right ) \cot \left (d x +c \right )+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )}{d}\) | \(114\) |
default | \(\frac {-A \,a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )+2 A \,a^{3} \cot \left (d x +c \right )+2 A \,a^{3} \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{3}\right ) \cot \left (d x +c \right )+A \,a^{3} \left (\left (-\frac {\left (\csc ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \csc \left (d x +c \right )}{8}\right ) \cot \left (d x +c \right )+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )}{d}\) | \(114\) |
parallelrisch | \(-\frac {a^{3} \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )-\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {16 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {16 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+8 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-16 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+40 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right ) A}{64 d}\) | \(121\) |
risch | \(\frac {A \,a^{3} \left (9 \,{\mathrm e}^{7 i \left (d x +c \right )}-33 \,{\mathrm e}^{5 i \left (d x +c \right )}+48 i {\mathrm e}^{6 i \left (d x +c \right )}-33 \,{\mathrm e}^{3 i \left (d x +c \right )}-48 i {\mathrm e}^{4 i \left (d x +c \right )}+9 \,{\mathrm e}^{i \left (d x +c \right )}+16 i {\mathrm e}^{2 i \left (d x +c \right )}-16 i\right )}{12 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}-\frac {5 A \,a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d}+\frac {5 A \,a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d}\) | \(149\) |
norman | \(\frac {-\frac {A \,a^{3}}{64 d}-\frac {57 A \,a^{3} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}-\frac {19 A \,a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}-\frac {27 A \,a^{3} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}-\frac {49 A \,a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}-\frac {A \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{12 d}-\frac {3 A \,a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}-\frac {A \,a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}+\frac {A \,a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {A \,a^{3} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {A \,a^{3} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {A \,a^{3} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {A \,a^{3} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}+\frac {3 A \,a^{3} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}+\frac {A \,a^{3} \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}+\frac {A \,a^{3} \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {5 A \,a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}\) | \(356\) |
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Leaf count of result is larger than twice the leaf count of optimal. 166 vs. \(2 (78) = 156\).
Time = 0.29 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.93 \[ \int \csc ^5(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=-\frac {32 \, A a^{3} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - 18 \, A a^{3} \cos \left (d x + c\right )^{3} + 30 \, A a^{3} \cos \left (d x + c\right ) - 15 \, {\left (A a^{3} \cos \left (d x + c\right )^{4} - 2 \, A a^{3} \cos \left (d x + c\right )^{2} + A a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 15 \, {\left (A a^{3} \cos \left (d x + c\right )^{4} - 2 \, A a^{3} \cos \left (d x + c\right )^{2} + A a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{48 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
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Timed out. \[ \int \csc ^5(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.69 \[ \int \csc ^5(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=\frac {3 \, A a^{3} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{3} - 5 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 24 \, A a^{3} {\left (\log \left (\cos \left (d x + c\right ) + 1\right ) - \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {96 \, A a^{3}}{\tan \left (d x + c\right )} - \frac {32 \, {\left (3 \, \tan \left (d x + c\right )^{2} + 1\right )} A a^{3}}{\tan \left (d x + c\right )^{3}}}{48 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (78) = 156\).
Time = 0.32 (sec) , antiderivative size = 174, normalized size of antiderivative = 2.02 \[ \int \csc ^5(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=\frac {3 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 16 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 24 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 120 \, A a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 48 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {250 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 48 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 16 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, A a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \]
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Time = 13.20 (sec) , antiderivative size = 244, normalized size of antiderivative = 2.84 \[ \int \csc ^5(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=-\frac {A\,a^3\,\left (3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-16\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+16\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-24\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+48\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-48\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+24\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+120\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}{192\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4} \]
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