Integrand size = 19, antiderivative size = 165 \[ \int \csc ^{10}(c+d x) (a+a \sec (c+d x)) \, dx=\frac {a \text {arctanh}(\sin (c+d x))}{d}-\frac {a \cot (c+d x)}{d}-\frac {4 a \cot ^3(c+d x)}{3 d}-\frac {6 a \cot ^5(c+d x)}{5 d}-\frac {4 a \cot ^7(c+d x)}{7 d}-\frac {a \cot ^9(c+d x)}{9 d}-\frac {a \csc (c+d x)}{d}-\frac {a \csc ^3(c+d x)}{3 d}-\frac {a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^9(c+d x)}{9 d} \]
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Time = 0.16 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3957, 2917, 2701, 308, 213, 3852} \[ \int \csc ^{10}(c+d x) (a+a \sec (c+d x)) \, dx=\frac {a \text {arctanh}(\sin (c+d x))}{d}-\frac {a \cot ^9(c+d x)}{9 d}-\frac {4 a \cot ^7(c+d x)}{7 d}-\frac {6 a \cot ^5(c+d x)}{5 d}-\frac {4 a \cot ^3(c+d x)}{3 d}-\frac {a \cot (c+d x)}{d}-\frac {a \csc ^9(c+d x)}{9 d}-\frac {a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^3(c+d x)}{3 d}-\frac {a \csc (c+d x)}{d} \]
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Rule 213
Rule 308
Rule 2701
Rule 2917
Rule 3852
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int (-a-a \cos (c+d x)) \csc ^{10}(c+d x) \sec (c+d x) \, dx \\ & = a \int \csc ^{10}(c+d x) \, dx+a \int \csc ^{10}(c+d x) \sec (c+d x) \, dx \\ & = -\frac {a \text {Subst}\left (\int \frac {x^{10}}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}-\frac {a \text {Subst}\left (\int \left (1+4 x^2+6 x^4+4 x^6+x^8\right ) \, dx,x,\cot (c+d x)\right )}{d} \\ & = -\frac {a \cot (c+d x)}{d}-\frac {4 a \cot ^3(c+d x)}{3 d}-\frac {6 a \cot ^5(c+d x)}{5 d}-\frac {4 a \cot ^7(c+d x)}{7 d}-\frac {a \cot ^9(c+d x)}{9 d}-\frac {a \text {Subst}\left (\int \left (1+x^2+x^4+x^6+x^8+\frac {1}{-1+x^2}\right ) \, dx,x,\csc (c+d x)\right )}{d} \\ & = -\frac {a \cot (c+d x)}{d}-\frac {4 a \cot ^3(c+d x)}{3 d}-\frac {6 a \cot ^5(c+d x)}{5 d}-\frac {4 a \cot ^7(c+d x)}{7 d}-\frac {a \cot ^9(c+d x)}{9 d}-\frac {a \csc (c+d x)}{d}-\frac {a \csc ^3(c+d x)}{3 d}-\frac {a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^9(c+d x)}{9 d}-\frac {a \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d} \\ & = \frac {a \text {arctanh}(\sin (c+d x))}{d}-\frac {a \cot (c+d x)}{d}-\frac {4 a \cot ^3(c+d x)}{3 d}-\frac {6 a \cot ^5(c+d x)}{5 d}-\frac {4 a \cot ^7(c+d x)}{7 d}-\frac {a \cot ^9(c+d x)}{9 d}-\frac {a \csc (c+d x)}{d}-\frac {a \csc ^3(c+d x)}{3 d}-\frac {a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^9(c+d x)}{9 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.30 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.82 \[ \int \csc ^{10}(c+d x) (a+a \sec (c+d x)) \, dx=-\frac {128 a \cot (c+d x)}{315 d}-\frac {64 a \cot (c+d x) \csc ^2(c+d x)}{315 d}-\frac {16 a \cot (c+d x) \csc ^4(c+d x)}{105 d}-\frac {8 a \cot (c+d x) \csc ^6(c+d x)}{63 d}-\frac {a \cot (c+d x) \csc ^8(c+d x)}{9 d}-\frac {a \csc ^9(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {9}{2},1,-\frac {7}{2},\sin ^2(c+d x)\right )}{9 d} \]
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Time = 1.03 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.75
method | result | size |
derivativedivides | \(\frac {a \left (-\frac {1}{9 \sin \left (d x +c \right )^{9}}-\frac {1}{7 \sin \left (d x +c \right )^{7}}-\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a \left (-\frac {128}{315}-\frac {\csc \left (d x +c \right )^{8}}{9}-\frac {8 \csc \left (d x +c \right )^{6}}{63}-\frac {16 \csc \left (d x +c \right )^{4}}{105}-\frac {64 \csc \left (d x +c \right )^{2}}{315}\right ) \cot \left (d x +c \right )}{d}\) | \(123\) |
default | \(\frac {a \left (-\frac {1}{9 \sin \left (d x +c \right )^{9}}-\frac {1}{7 \sin \left (d x +c \right )^{7}}-\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a \left (-\frac {128}{315}-\frac {\csc \left (d x +c \right )^{8}}{9}-\frac {8 \csc \left (d x +c \right )^{6}}{63}-\frac {16 \csc \left (d x +c \right )^{4}}{105}-\frac {64 \csc \left (d x +c \right )^{2}}{315}\right ) \cot \left (d x +c \right )}{d}\) | \(123\) |
parallelrisch | \(-\frac {a \left (\cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+\frac {90 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}+\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}+\frac {414 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}+18 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+390 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+138 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+2304 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+1170 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2304 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-2304 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )\right )}{2304 d}\) | \(147\) |
risch | \(-\frac {2 i a \left (315 \,{\mathrm e}^{15 i \left (d x +c \right )}-630 \,{\mathrm e}^{14 i \left (d x +c \right )}-1995 \,{\mathrm e}^{13 i \left (d x +c \right )}+4620 \,{\mathrm e}^{12 i \left (d x +c \right )}+5103 \,{\mathrm e}^{11 i \left (d x +c \right )}-14826 \,{\mathrm e}^{10 i \left (d x +c \right )}-6303 \,{\mathrm e}^{9 i \left (d x +c \right )}+27432 \,{\mathrm e}^{8 i \left (d x +c \right )}+2657 \,{\mathrm e}^{7 i \left (d x +c \right )}-16618 \,{\mathrm e}^{6 i \left (d x +c \right )}-273 \,{\mathrm e}^{5 i \left (d x +c \right )}+6412 \,{\mathrm e}^{4 i \left (d x +c \right )}-203 \,{\mathrm e}^{3 i \left (d x +c \right )}-1398 \,{\mathrm e}^{2 i \left (d x +c \right )}+59 \,{\mathrm e}^{i \left (d x +c \right )}+128\right )}{315 d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{9} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}\) | \(239\) |
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Leaf count of result is larger than twice the leaf count of optimal. 366 vs. \(2 (149) = 298\).
Time = 0.27 (sec) , antiderivative size = 366, normalized size of antiderivative = 2.22 \[ \int \csc ^{10}(c+d x) (a+a \sec (c+d x)) \, dx=-\frac {256 \, a \cos \left (d x + c\right )^{8} + 374 \, a \cos \left (d x + c\right )^{7} - 1526 \, a \cos \left (d x + c\right )^{6} - 1204 \, a \cos \left (d x + c\right )^{5} + 3220 \, a \cos \left (d x + c\right )^{4} + 1316 \, a \cos \left (d x + c\right )^{3} - 2996 \, a \cos \left (d x + c\right )^{2} - 315 \, {\left (a \cos \left (d x + c\right )^{7} - a \cos \left (d x + c\right )^{6} - 3 \, a \cos \left (d x + c\right )^{5} + 3 \, a \cos \left (d x + c\right )^{4} + 3 \, a \cos \left (d x + c\right )^{3} - 3 \, a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) + a\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 315 \, {\left (a \cos \left (d x + c\right )^{7} - a \cos \left (d x + c\right )^{6} - 3 \, a \cos \left (d x + c\right )^{5} + 3 \, a \cos \left (d x + c\right )^{4} + 3 \, a \cos \left (d x + c\right )^{3} - 3 \, a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) + a\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) - 496 \, a \cos \left (d x + c\right ) + 1126 \, a}{630 \, {\left (d \cos \left (d x + c\right )^{7} - d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{5} + 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{3} - 3 \, d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + d\right )} \sin \left (d x + c\right )} \]
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Timed out. \[ \int \csc ^{10}(c+d x) (a+a \sec (c+d x)) \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.82 \[ \int \csc ^{10}(c+d x) (a+a \sec (c+d x)) \, dx=-\frac {a {\left (\frac {2 \, {\left (315 \, \sin \left (d x + c\right )^{8} + 105 \, \sin \left (d x + c\right )^{6} + 63 \, \sin \left (d x + c\right )^{4} + 45 \, \sin \left (d x + c\right )^{2} + 35\right )}}{\sin \left (d x + c\right )^{9}} - 315 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 315 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + \frac {2 \, {\left (315 \, \tan \left (d x + c\right )^{8} + 420 \, \tan \left (d x + c\right )^{6} + 378 \, \tan \left (d x + c\right )^{4} + 180 \, \tan \left (d x + c\right )^{2} + 35\right )} a}{\tan \left (d x + c\right )^{9}}}{630 \, d} \]
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Time = 0.34 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.99 \[ \int \csc ^{10}(c+d x) (a+a \sec (c+d x)) \, dx=-\frac {45 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 630 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4830 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 80640 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 80640 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 40950 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {80640 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 13650 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 2898 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 450 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 35 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9}}}{80640 \, d} \]
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Time = 14.79 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.96 \[ \int \csc ^{10}(c+d x) (a+a \sec (c+d x)) \, dx=\frac {2\,a\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {65\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d}-\frac {23\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{384\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{128\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{1792\,d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (256\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {130\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {46\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}+\frac {10\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{7}+\frac {a}{9}\right )}{256\,d} \]
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