Integrand size = 21, antiderivative size = 165 \[ \int \csc ^6(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {13 a^3 \text {arctanh}(\sin (c+d x))}{2 d}+\frac {152 a^3 \tan (c+d x)}{15 d}+\frac {13 a^3 \sec (c+d x) \tan (c+d x)}{2 d}-\frac {a^6 \sec (c+d x) \tan (c+d x)}{5 d (a-a \cos (c+d x))^3}-\frac {11 a^5 \sec (c+d x) \tan (c+d x)}{15 d (a-a \cos (c+d x))^2}-\frac {76 a^6 \sec (c+d x) \tan (c+d x)}{15 d \left (a^3-a^3 \cos (c+d x)\right )} \]
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Time = 0.52 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3957, 2948, 2845, 3057, 2827, 3853, 3855, 3852, 8} \[ \int \csc ^6(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {a^6 \tan (c+d x) \sec (c+d x)}{5 d (a-a \cos (c+d x))^3}-\frac {11 a^5 \tan (c+d x) \sec (c+d x)}{15 d (a-a \cos (c+d x))^2}+\frac {13 a^3 \text {arctanh}(\sin (c+d x))}{2 d}+\frac {152 a^3 \tan (c+d x)}{15 d}+\frac {13 a^3 \tan (c+d x) \sec (c+d x)}{2 d}-\frac {76 a^6 \tan (c+d x) \sec (c+d x)}{15 d \left (a^3-a^3 \cos (c+d x)\right )} \]
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Rule 8
Rule 2827
Rule 2845
Rule 2948
Rule 3057
Rule 3852
Rule 3853
Rule 3855
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int (-a-a \cos (c+d x))^3 \csc ^6(c+d x) \sec ^3(c+d x) \, dx \\ & = -\left (a^6 \int \frac {\sec ^3(c+d x)}{(-a+a \cos (c+d x))^3} \, dx\right ) \\ & = -\frac {a^6 \sec (c+d x) \tan (c+d x)}{5 d (a-a \cos (c+d x))^3}-\frac {1}{5} a^4 \int \frac {(-7 a-4 a \cos (c+d x)) \sec ^3(c+d x)}{(-a+a \cos (c+d x))^2} \, dx \\ & = -\frac {a^6 \sec (c+d x) \tan (c+d x)}{5 d (a-a \cos (c+d x))^3}-\frac {11 a^5 \sec (c+d x) \tan (c+d x)}{15 d (a-a \cos (c+d x))^2}-\frac {1}{15} a^2 \int \frac {\left (43 a^2+33 a^2 \cos (c+d x)\right ) \sec ^3(c+d x)}{-a+a \cos (c+d x)} \, dx \\ & = -\frac {a^6 \sec (c+d x) \tan (c+d x)}{5 d (a-a \cos (c+d x))^3}-\frac {11 a^5 \sec (c+d x) \tan (c+d x)}{15 d (a-a \cos (c+d x))^2}-\frac {76 a^4 \sec (c+d x) \tan (c+d x)}{15 d (a-a \cos (c+d x))}-\frac {1}{15} \int \left (-195 a^3-152 a^3 \cos (c+d x)\right ) \sec ^3(c+d x) \, dx \\ & = -\frac {a^6 \sec (c+d x) \tan (c+d x)}{5 d (a-a \cos (c+d x))^3}-\frac {11 a^5 \sec (c+d x) \tan (c+d x)}{15 d (a-a \cos (c+d x))^2}-\frac {76 a^4 \sec (c+d x) \tan (c+d x)}{15 d (a-a \cos (c+d x))}+\frac {1}{15} \left (152 a^3\right ) \int \sec ^2(c+d x) \, dx+\left (13 a^3\right ) \int \sec ^3(c+d x) \, dx \\ & = \frac {13 a^3 \sec (c+d x) \tan (c+d x)}{2 d}-\frac {a^6 \sec (c+d x) \tan (c+d x)}{5 d (a-a \cos (c+d x))^3}-\frac {11 a^5 \sec (c+d x) \tan (c+d x)}{15 d (a-a \cos (c+d x))^2}-\frac {76 a^4 \sec (c+d x) \tan (c+d x)}{15 d (a-a \cos (c+d x))}+\frac {1}{2} \left (13 a^3\right ) \int \sec (c+d x) \, dx-\frac {\left (152 a^3\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{15 d} \\ & = \frac {13 a^3 \text {arctanh}(\sin (c+d x))}{2 d}+\frac {152 a^3 \tan (c+d x)}{15 d}+\frac {13 a^3 \sec (c+d x) \tan (c+d x)}{2 d}-\frac {a^6 \sec (c+d x) \tan (c+d x)}{5 d (a-a \cos (c+d x))^3}-\frac {11 a^5 \sec (c+d x) \tan (c+d x)}{15 d (a-a \cos (c+d x))^2}-\frac {76 a^4 \sec (c+d x) \tan (c+d x)}{15 d (a-a \cos (c+d x))} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(353\) vs. \(2(165)=330\).
Time = 2.01 (sec) , antiderivative size = 353, normalized size of antiderivative = 2.14 \[ \int \csc ^6(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {a^3 (1+\cos (c+d x))^3 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \sec ^2(c+d x) \left (24960 \cos ^2(c+d x) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+\csc \left (\frac {c}{2}\right ) \csc ^5\left (\frac {1}{2} (c+d x)\right ) \sec (c) \left (-1235 \sin \left (\frac {d x}{2}\right )+3805 \sin \left (\frac {3 d x}{2}\right )+4329 \sin \left (c-\frac {d x}{2}\right )-1989 \sin \left (c+\frac {d x}{2}\right )-3575 \sin \left (2 c+\frac {d x}{2}\right )+475 \sin \left (c+\frac {3 d x}{2}\right )+2005 \sin \left (2 c+\frac {3 d x}{2}\right )+2275 \sin \left (3 c+\frac {3 d x}{2}\right )-2673 \sin \left (c+\frac {5 d x}{2}\right )+105 \sin \left (2 c+\frac {5 d x}{2}\right )-1593 \sin \left (3 c+\frac {5 d x}{2}\right )-975 \sin \left (4 c+\frac {5 d x}{2}\right )+1325 \sin \left (2 c+\frac {7 d x}{2}\right )-255 \sin \left (3 c+\frac {7 d x}{2}\right )+875 \sin \left (4 c+\frac {7 d x}{2}\right )+195 \sin \left (5 c+\frac {7 d x}{2}\right )-304 \sin \left (3 c+\frac {9 d x}{2}\right )+90 \sin \left (4 c+\frac {9 d x}{2}\right )-214 \sin \left (5 c+\frac {9 d x}{2}\right )\right )\right )}{30720 d} \]
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Time = 1.40 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.83
method | result | size |
parallelrisch | \(\frac {777 a^{3} \left (\frac {520 \left (-\cos \left (2 d x +2 c \right )-1\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{777}+\frac {520 \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{777}+\cot \left (\frac {d x}{2}+\frac {c}{2}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (\cos \left (d x +c \right )-\frac {174 \cos \left (2 d x +2 c \right )}{259}+\frac {239 \cos \left (3 d x +3 c \right )}{777}-\frac {152 \cos \left (4 d x +4 c \right )}{2331}-\frac {1354}{2331}\right )\right )}{80 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(137\) |
norman | \(\frac {-\frac {a^{3}}{20 d}-\frac {17 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{30 d}-\frac {97 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{15 d}+\frac {131 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{6 d}-\frac {51 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{4 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}-\frac {13 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {13 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(154\) |
risch | \(-\frac {i a^{3} \left (195 \,{\mathrm e}^{8 i \left (d x +c \right )}-975 \,{\mathrm e}^{7 i \left (d x +c \right )}+2275 \,{\mathrm e}^{6 i \left (d x +c \right )}-3575 \,{\mathrm e}^{5 i \left (d x +c \right )}+4329 \,{\mathrm e}^{4 i \left (d x +c \right )}-3805 \,{\mathrm e}^{3 i \left (d x +c \right )}+2673 \,{\mathrm e}^{2 i \left (d x +c \right )}-1325 \,{\mathrm e}^{i \left (d x +c \right )}+304\right )}{15 d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{5} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {13 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}-\frac {13 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}\) | \(169\) |
derivativedivides | \(\frac {a^{3} \left (-\frac {1}{5 \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )^{2}}-\frac {7}{15 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{2}}+\frac {7}{6 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {7}{2 \sin \left (d x +c \right )}+\frac {7 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{3} \left (-\frac {1}{5 \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )}-\frac {2}{5 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {8}{5 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {16 \cot \left (d x +c \right )}{5}\right )+3 a^{3} \left (-\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{3} \left (-\frac {8}{15}-\frac {\csc \left (d x +c \right )^{4}}{5}-\frac {4 \csc \left (d x +c \right )^{2}}{15}\right ) \cot \left (d x +c \right )}{d}\) | \(241\) |
default | \(\frac {a^{3} \left (-\frac {1}{5 \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )^{2}}-\frac {7}{15 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{2}}+\frac {7}{6 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {7}{2 \sin \left (d x +c \right )}+\frac {7 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{3} \left (-\frac {1}{5 \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )}-\frac {2}{5 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {8}{5 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {16 \cot \left (d x +c \right )}{5}\right )+3 a^{3} \left (-\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{3} \left (-\frac {8}{15}-\frac {\csc \left (d x +c \right )^{4}}{5}-\frac {4 \csc \left (d x +c \right )^{2}}{15}\right ) \cot \left (d x +c \right )}{d}\) | \(241\) |
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Time = 0.29 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.36 \[ \int \csc ^6(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {608 \, a^{3} \cos \left (d x + c\right )^{5} - 826 \, a^{3} \cos \left (d x + c\right )^{4} - 476 \, a^{3} \cos \left (d x + c\right )^{3} + 868 \, a^{3} \cos \left (d x + c\right )^{2} - 120 \, a^{3} \cos \left (d x + c\right ) - 30 \, a^{3} - 195 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{3} + a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 195 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{3} + a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right )}{60 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )} \]
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Timed out. \[ \int \csc ^6(c+d x) (a+a \sec (c+d x))^3 \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.38 \[ \int \csc ^6(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {a^{3} {\left (\frac {2 \, {\left (105 \, \sin \left (d x + c\right )^{6} - 70 \, \sin \left (d x + c\right )^{4} - 14 \, \sin \left (d x + c\right )^{2} - 6\right )}}{\sin \left (d x + c\right )^{7} - \sin \left (d x + c\right )^{5}} - 105 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, a^{3} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{4} + 5 \, \sin \left (d x + c\right )^{2} + 3\right )}}{\sin \left (d x + c\right )^{5}} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 36 \, a^{3} {\left (\frac {15 \, \tan \left (d x + c\right )^{4} + 5 \, \tan \left (d x + c\right )^{2} + 1}{\tan \left (d x + c\right )^{5}} - 5 \, \tan \left (d x + c\right )\right )} + \frac {4 \, {\left (15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} + 3\right )} a^{3}}{\tan \left (d x + c\right )^{5}}}{60 \, d} \]
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Time = 0.40 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.85 \[ \int \csc ^6(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {390 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 390 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {60 \, {\left (5 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 7 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}} - \frac {465 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 40 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{60 \, d} \]
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Time = 17.71 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.82 \[ \int \csc ^6(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {13\,a^3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {51\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-\frac {262\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {388\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{15}+\frac {34\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{15}+\frac {a^3}{5}}{d\,\left (4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\right )} \]
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