Integrand size = 21, antiderivative size = 279 \[ \int \csc ^6(c+d x) (a+b \sec (c+d x))^3 \, dx=\frac {3 a^2 b \text {arctanh}(\sin (c+d x))}{d}+\frac {7 b^3 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {a^3 \cot (c+d x)}{d}-\frac {9 a b^2 \cot (c+d x)}{d}-\frac {2 a^3 \cot ^3(c+d x)}{3 d}-\frac {3 a b^2 \cot ^3(c+d x)}{d}-\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {3 a b^2 \cot ^5(c+d x)}{5 d}-\frac {3 a^2 b \csc (c+d x)}{d}-\frac {7 b^3 \csc (c+d x)}{2 d}-\frac {a^2 b \csc ^3(c+d x)}{d}-\frac {7 b^3 \csc ^3(c+d x)}{6 d}-\frac {3 a^2 b \csc ^5(c+d x)}{5 d}-\frac {7 b^3 \csc ^5(c+d x)}{10 d}+\frac {b^3 \csc ^5(c+d x) \sec ^2(c+d x)}{2 d}+\frac {3 a b^2 \tan (c+d x)}{d} \]
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Time = 0.39 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3957, 2991, 3852, 2701, 308, 213, 2700, 276, 294} \[ \int \csc ^6(c+d x) (a+b \sec (c+d x))^3 \, dx=-\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {2 a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot (c+d x)}{d}+\frac {3 a^2 b \text {arctanh}(\sin (c+d x))}{d}-\frac {3 a^2 b \csc ^5(c+d x)}{5 d}-\frac {a^2 b \csc ^3(c+d x)}{d}-\frac {3 a^2 b \csc (c+d x)}{d}+\frac {3 a b^2 \tan (c+d x)}{d}-\frac {3 a b^2 \cot ^5(c+d x)}{5 d}-\frac {3 a b^2 \cot ^3(c+d x)}{d}-\frac {9 a b^2 \cot (c+d x)}{d}+\frac {7 b^3 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {7 b^3 \csc ^5(c+d x)}{10 d}-\frac {7 b^3 \csc ^3(c+d x)}{6 d}-\frac {7 b^3 \csc (c+d x)}{2 d}+\frac {b^3 \csc ^5(c+d x) \sec ^2(c+d x)}{2 d} \]
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Rule 213
Rule 276
Rule 294
Rule 308
Rule 2700
Rule 2701
Rule 2991
Rule 3852
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int (-b-a \cos (c+d x))^3 \csc ^6(c+d x) \sec ^3(c+d x) \, dx \\ & = \int \left (a^3 \csc ^6(c+d x)+3 a^2 b \csc ^6(c+d x) \sec (c+d x)+3 a b^2 \csc ^6(c+d x) \sec ^2(c+d x)+b^3 \csc ^6(c+d x) \sec ^3(c+d x)\right ) \, dx \\ & = a^3 \int \csc ^6(c+d x) \, dx+\left (3 a^2 b\right ) \int \csc ^6(c+d x) \sec (c+d x) \, dx+\left (3 a b^2\right ) \int \csc ^6(c+d x) \sec ^2(c+d x) \, dx+b^3 \int \csc ^6(c+d x) \sec ^3(c+d x) \, dx \\ & = -\frac {a^3 \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{d}-\frac {\left (3 a^2 b\right ) \text {Subst}\left (\int \frac {x^6}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}+\frac {\left (3 a b^2\right ) \text {Subst}\left (\int \frac {\left (1+x^2\right )^3}{x^6} \, dx,x,\tan (c+d x)\right )}{d}-\frac {b^3 \text {Subst}\left (\int \frac {x^8}{\left (-1+x^2\right )^2} \, dx,x,\csc (c+d x)\right )}{d} \\ & = -\frac {a^3 \cot (c+d x)}{d}-\frac {2 a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot ^5(c+d x)}{5 d}+\frac {b^3 \csc ^5(c+d x) \sec ^2(c+d x)}{2 d}-\frac {\left (3 a^2 b\right ) \text {Subst}\left (\int \left (1+x^2+x^4+\frac {1}{-1+x^2}\right ) \, dx,x,\csc (c+d x)\right )}{d}+\frac {\left (3 a b^2\right ) \text {Subst}\left (\int \left (1+\frac {1}{x^6}+\frac {3}{x^4}+\frac {3}{x^2}\right ) \, dx,x,\tan (c+d x)\right )}{d}-\frac {\left (7 b^3\right ) \text {Subst}\left (\int \frac {x^6}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{2 d} \\ & = -\frac {a^3 \cot (c+d x)}{d}-\frac {9 a b^2 \cot (c+d x)}{d}-\frac {2 a^3 \cot ^3(c+d x)}{3 d}-\frac {3 a b^2 \cot ^3(c+d x)}{d}-\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {3 a b^2 \cot ^5(c+d x)}{5 d}-\frac {3 a^2 b \csc (c+d x)}{d}-\frac {a^2 b \csc ^3(c+d x)}{d}-\frac {3 a^2 b \csc ^5(c+d x)}{5 d}+\frac {b^3 \csc ^5(c+d x) \sec ^2(c+d x)}{2 d}+\frac {3 a b^2 \tan (c+d x)}{d}-\frac {\left (3 a^2 b\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}-\frac {\left (7 b^3\right ) \text {Subst}\left (\int \left (1+x^2+x^4+\frac {1}{-1+x^2}\right ) \, dx,x,\csc (c+d x)\right )}{2 d} \\ & = \frac {3 a^2 b \text {arctanh}(\sin (c+d x))}{d}-\frac {a^3 \cot (c+d x)}{d}-\frac {9 a b^2 \cot (c+d x)}{d}-\frac {2 a^3 \cot ^3(c+d x)}{3 d}-\frac {3 a b^2 \cot ^3(c+d x)}{d}-\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {3 a b^2 \cot ^5(c+d x)}{5 d}-\frac {3 a^2 b \csc (c+d x)}{d}-\frac {7 b^3 \csc (c+d x)}{2 d}-\frac {a^2 b \csc ^3(c+d x)}{d}-\frac {7 b^3 \csc ^3(c+d x)}{6 d}-\frac {3 a^2 b \csc ^5(c+d x)}{5 d}-\frac {7 b^3 \csc ^5(c+d x)}{10 d}+\frac {b^3 \csc ^5(c+d x) \sec ^2(c+d x)}{2 d}+\frac {3 a b^2 \tan (c+d x)}{d}-\frac {\left (7 b^3\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{2 d} \\ & = \frac {3 a^2 b \text {arctanh}(\sin (c+d x))}{d}+\frac {7 b^3 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {a^3 \cot (c+d x)}{d}-\frac {9 a b^2 \cot (c+d x)}{d}-\frac {2 a^3 \cot ^3(c+d x)}{3 d}-\frac {3 a b^2 \cot ^3(c+d x)}{d}-\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {3 a b^2 \cot ^5(c+d x)}{5 d}-\frac {3 a^2 b \csc (c+d x)}{d}-\frac {7 b^3 \csc (c+d x)}{2 d}-\frac {a^2 b \csc ^3(c+d x)}{d}-\frac {7 b^3 \csc ^3(c+d x)}{6 d}-\frac {3 a^2 b \csc ^5(c+d x)}{5 d}-\frac {7 b^3 \csc ^5(c+d x)}{10 d}+\frac {b^3 \csc ^5(c+d x) \sec ^2(c+d x)}{2 d}+\frac {3 a b^2 \tan (c+d x)}{d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(812\) vs. \(2(279)=558\).
Time = 2.16 (sec) , antiderivative size = 812, normalized size of antiderivative = 2.91 \[ \int \csc ^6(c+d x) (a+b \sec (c+d x))^3 \, dx=-\frac {\csc ^9\left (\frac {1}{2} (c+d x)\right ) \sec ^5\left (\frac {1}{2} (c+d x)\right ) \left (1176 a^2 b+412 b^3+80 a \left (5 a^2+18 b^2\right ) \cos (c+d x)+66 \left (6 a^2 b+7 b^3\right ) \cos (2 (c+d x))+16 a^3 \cos (3 (c+d x))+288 a b^2 \cos (3 (c+d x))-600 a^2 b \cos (4 (c+d x))-700 b^3 \cos (4 (c+d x))-48 a^3 \cos (5 (c+d x))-864 a b^2 \cos (5 (c+d x))+180 a^2 b \cos (6 (c+d x))+210 b^3 \cos (6 (c+d x))+16 a^3 \cos (7 (c+d x))+288 a b^2 \cos (7 (c+d x))+450 a^2 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+525 b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-450 a^2 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-525 b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+90 a^2 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+105 b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-90 a^2 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-105 b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-270 a^2 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))-315 b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))+270 a^2 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))+315 b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))+90 a^2 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))+105 b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))-90 a^2 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))-105 b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))\right )}{61440 d \left (-1+\cot ^2\left (\frac {1}{2} (c+d x)\right )\right )^2} \]
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Time = 1.87 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.87
method | result | size |
derivativedivides | \(\frac {a^{3} \left (-\frac {8}{15}-\frac {\csc \left (d x +c \right )^{4}}{5}-\frac {4 \csc \left (d x +c \right )^{2}}{15}\right ) \cot \left (d x +c \right )+3 a^{2} b \left (-\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+3 a \,b^{2} \left (-\frac {1}{5 \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )}-\frac {2}{5 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {8}{5 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {16 \cot \left (d x +c \right )}{5}\right )+b^{3} \left (-\frac {1}{5 \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )^{2}}-\frac {7}{15 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{2}}+\frac {7}{6 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {7}{2 \sin \left (d x +c \right )}+\frac {7 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(243\) |
default | \(\frac {a^{3} \left (-\frac {8}{15}-\frac {\csc \left (d x +c \right )^{4}}{5}-\frac {4 \csc \left (d x +c \right )^{2}}{15}\right ) \cot \left (d x +c \right )+3 a^{2} b \left (-\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+3 a \,b^{2} \left (-\frac {1}{5 \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )}-\frac {2}{5 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {8}{5 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {16 \cot \left (d x +c \right )}{5}\right )+b^{3} \left (-\frac {1}{5 \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )^{2}}-\frac {7}{15 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{2}}+\frac {7}{6 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {7}{2 \sin \left (d x +c \right )}+\frac {7 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(243\) |
parallelrisch | \(\frac {-46080 \left (1+\cos \left (2 d x +2 c \right )\right ) b \left (a^{2}+\frac {7 b^{2}}{6}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+46080 \left (1+\cos \left (2 d x +2 c \right )\right ) b \left (a^{2}+\frac {7 b^{2}}{6}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+24 \csc \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \left (\left (-\frac {33}{4} a^{2} b -\frac {77}{8} b^{3}\right ) \cos \left (2 d x +2 c \right )+\left (-\frac {1}{3} a^{3}-6 a \,b^{2}\right ) \cos \left (3 d x +3 c \right )+\left (\frac {25}{2} a^{2} b +\frac {175}{12} b^{3}\right ) \cos \left (4 d x +4 c \right )+\left (a^{3}+18 a \,b^{2}\right ) \cos \left (5 d x +5 c \right )+\left (-\frac {15}{4} a^{2} b -\frac {35}{8} b^{3}\right ) \cos \left (6 d x +6 c \right )+\left (-\frac {1}{3} a^{3}-6 a \,b^{2}\right ) \cos \left (7 d x +7 c \right )+\left (-\frac {25}{3} a^{3}-30 a \,b^{2}\right ) \cos \left (d x +c \right )-\frac {49 a^{2} b}{2}-\frac {103 b^{3}}{12}\right )}{15360 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(275\) |
norman | \(\frac {-\frac {a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}}{160 d}+\frac {\left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{160 d}+\frac {\left (19 a^{3}-87 a^{2} b +117 a \,b^{2}-49 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{480 d}-\frac {\left (19 a^{3}+87 a^{2} b +117 a \,b^{2}+49 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{480 d}-\frac {\left (85 a^{3}-177 a^{2} b +1575 a \,b^{2}-259 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{96 d}+\frac {\left (85 a^{3}+177 a^{2} b +1575 a \,b^{2}+259 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{96 d}+\frac {\left (103 a^{3}-789 a^{2} b +1449 a \,b^{2}-763 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{480 d}-\frac {\left (103 a^{3}+789 a^{2} b +1449 a \,b^{2}+763 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{480 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}-\frac {b \left (6 a^{2}+7 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {b \left (6 a^{2}+7 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(383\) |
risch | \(-\frac {i \left (90 a^{2} b \,{\mathrm e}^{13 i \left (d x +c \right )}+105 b^{3} {\mathrm e}^{13 i \left (d x +c \right )}-300 a^{2} b \,{\mathrm e}^{11 i \left (d x +c \right )}-350 b^{3} {\mathrm e}^{11 i \left (d x +c \right )}+198 a^{2} b \,{\mathrm e}^{9 i \left (d x +c \right )}+231 b^{3} {\mathrm e}^{9 i \left (d x +c \right )}+160 a^{3} {\mathrm e}^{8 i \left (d x +c \right )}+1176 a^{2} b \,{\mathrm e}^{7 i \left (d x +c \right )}+412 b^{3} {\mathrm e}^{7 i \left (d x +c \right )}+240 a^{3} {\mathrm e}^{6 i \left (d x +c \right )}+1440 a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+198 a^{2} b \,{\mathrm e}^{5 i \left (d x +c \right )}+231 b^{3} {\mathrm e}^{5 i \left (d x +c \right )}+16 a^{3} {\mathrm e}^{4 i \left (d x +c \right )}+288 a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-300 a^{2} b \,{\mathrm e}^{3 i \left (d x +c \right )}-350 b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-48 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-864 a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+90 a^{2} b \,{\mathrm e}^{i \left (d x +c \right )}+105 b^{3} {\mathrm e}^{i \left (d x +c \right )}+16 a^{3}+288 a \,b^{2}\right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{2}}{d}+\frac {7 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}-\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{2}}{d}-\frac {7 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}\) | \(436\) |
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Time = 0.27 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.27 \[ \int \csc ^6(c+d x) (a+b \sec (c+d x))^3 \, dx=-\frac {32 \, {\left (a^{3} + 18 \, a b^{2}\right )} \cos \left (d x + c\right )^{7} + 30 \, {\left (6 \, a^{2} b + 7 \, b^{3}\right )} \cos \left (d x + c\right )^{6} - 80 \, {\left (a^{3} + 18 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} - 70 \, {\left (6 \, a^{2} b + 7 \, b^{3}\right )} \cos \left (d x + c\right )^{4} - 180 \, a b^{2} \cos \left (d x + c\right ) + 60 \, {\left (a^{3} + 18 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} - 30 \, b^{3} + 46 \, {\left (6 \, a^{2} b + 7 \, b^{3}\right )} \cos \left (d x + c\right )^{2} - 15 \, {\left ({\left (6 \, a^{2} b + 7 \, b^{3}\right )} \cos \left (d x + c\right )^{6} - 2 \, {\left (6 \, a^{2} b + 7 \, b^{3}\right )} \cos \left (d x + c\right )^{4} + {\left (6 \, a^{2} b + 7 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 15 \, {\left ({\left (6 \, a^{2} b + 7 \, b^{3}\right )} \cos \left (d x + c\right )^{6} - 2 \, {\left (6 \, a^{2} b + 7 \, b^{3}\right )} \cos \left (d x + c\right )^{4} + {\left (6 \, a^{2} b + 7 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right )}{60 \, {\left (d \cos \left (d x + c\right )^{6} - 2 \, d \cos \left (d x + c\right )^{4} + d \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )} \]
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Timed out. \[ \int \csc ^6(c+d x) (a+b \sec (c+d x))^3 \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.82 \[ \int \csc ^6(c+d x) (a+b \sec (c+d x))^3 \, dx=-\frac {b^{3} {\left (\frac {2 \, {\left (105 \, \sin \left (d x + c\right )^{6} - 70 \, \sin \left (d x + c\right )^{4} - 14 \, \sin \left (d x + c\right )^{2} - 6\right )}}{\sin \left (d x + c\right )^{7} - \sin \left (d x + c\right )^{5}} - 105 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, a^{2} b {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{4} + 5 \, \sin \left (d x + c\right )^{2} + 3\right )}}{\sin \left (d x + c\right )^{5}} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 36 \, a b^{2} {\left (\frac {15 \, \tan \left (d x + c\right )^{4} + 5 \, \tan \left (d x + c\right )^{2} + 1}{\tan \left (d x + c\right )^{5}} - 5 \, \tan \left (d x + c\right )\right )} + \frac {4 \, {\left (15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} + 3\right )} a^{3}}{\tan \left (d x + c\right )^{5}}}{60 \, d} \]
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Time = 0.42 (sec) , antiderivative size = 498, normalized size of antiderivative = 1.78 \[ \int \csc ^6(c+d x) (a+b \sec (c+d x))^3 \, dx=\frac {3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 25 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 105 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 135 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 55 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 150 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 990 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1710 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 870 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 240 \, {\left (6 \, a^{2} b + 7 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 240 \, {\left (6 \, a^{2} b + 7 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {480 \, {\left (6 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b^{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 {150 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 990 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1710 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 870 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 25 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 105 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 135 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 55 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3 \, a^{3} + 9 \, a^{2} b + 9 \, a b^{2} + 3 \, b^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{480 \, d} \]
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Time = 13.68 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.30 \[ \int \csc ^6(c+d x) (a+b \sec (c+d x))^3 \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\left (a-b\right )}^3}{160\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {19\,a^3}{15}+\frac {29\,a^2\,b}{5}+\frac {39\,a\,b^2}{5}+\frac {49\,b^3}{15}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (10\,a^3+66\,a^2\,b+306\,a\,b^2+26\,b^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {55\,a^3}{3}+125\,a^2\,b+411\,a\,b^2+\frac {433\,b^3}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {103\,a^3}{15}+\frac {263\,a^2\,b}{5}+\frac {483\,a\,b^2}{5}+\frac {763\,b^3}{15}\right )+\frac {3\,a\,b^2}{5}+\frac {3\,a^2\,b}{5}+\frac {a^3}{5}+\frac {b^3}{5}}{d\,\left (32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\right )}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {21\,a\,b^2}{16}-\frac {3\,a^2\,b}{8}-\frac {a^3}{16}-\frac {7\,b^3}{8}+\frac {3\,{\left (a-b\right )}^2\,\left (a-4\,b\right )}{16}+\frac {3\,{\left (a-b\right )}^3}{16}\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {{\left (a-b\right )}^2\,\left (a-4\,b\right )}{48}+\frac {{\left (a-b\right )}^3}{32}\right )}{d}-\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (a^2\,b\,6{}\mathrm {i}+b^3\,7{}\mathrm {i}\right )\,1{}\mathrm {i}}{d} \]
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