Optimal. Leaf size=165 \[ -\frac {a \cot ^9(c+d x)}{9 d}-\frac {4 a \cot ^7(c+d x)}{7 d}-\frac {6 a \cot ^5(c+d x)}{5 d}-\frac {4 a \cot ^3(c+d x)}{3 d}-\frac {a \cot (c+d x)}{d}-\frac {a \csc ^9(c+d x)}{9 d}-\frac {a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^3(c+d x)}{3 d}-\frac {a \csc (c+d x)}{d}+\frac {a \tanh ^{-1}(\sin (c+d x))}{d} \]
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Rubi [A] time = 0.13, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3872, 2838, 2621, 302, 207, 3767} \[ -\frac {a \cot ^9(c+d x)}{9 d}-\frac {4 a \cot ^7(c+d x)}{7 d}-\frac {6 a \cot ^5(c+d x)}{5 d}-\frac {4 a \cot ^3(c+d x)}{3 d}-\frac {a \cot (c+d x)}{d}-\frac {a \csc ^9(c+d x)}{9 d}-\frac {a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^3(c+d x)}{3 d}-\frac {a \csc (c+d x)}{d}+\frac {a \tanh ^{-1}(\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 207
Rule 302
Rule 2621
Rule 2838
Rule 3767
Rule 3872
Rubi steps
\begin {align*} \int \csc ^{10}(c+d x) (a+a \sec (c+d x)) \, dx &=-\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 \operatorname {Subst}\left (\int \frac {x^{10}}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}-\frac {a \operatorname {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 \operatorname {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 \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}\\ &=\frac {a \tanh ^{-1}(\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*}
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Mathematica [C] time = 0.06, size = 135, normalized size = 0.82 \[ -\frac {a \csc ^9(c+d x) \, _2F_1\left (-\frac {9}{2},1;-\frac {7}{2};\sin ^2(c+d x)\right )}{9 d}-\frac {128 a \cot (c+d x)}{315 d}-\frac {a \cot (c+d x) \csc ^8(c+d x)}{9 d}-\frac {8 a \cot (c+d x) \csc ^6(c+d x)}{63 d}-\frac {16 a \cot (c+d x) \csc ^4(c+d x)}{105 d}-\frac {64 a \cot (c+d x) \csc ^2(c+d x)}{315 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.69, size = 366, normalized size = 2.22 \[ -\frac {256 \, a \cos \left (d x + c\right )^{8} + 374 \, a \cos \left (d x + c\right )^{7} - 1526 \, a \cos \left (d x + c\right )^{6} - 1204 \, a \cos \left (d x + c\right )^{5} + 3220 \, a \cos \left (d x + c\right )^{4} + 1316 \, a \cos \left (d x + c\right )^{3} - 2996 \, a \cos \left (d x + c\right )^{2} - 315 \, {\left (a \cos \left (d x + c\right )^{7} - a \cos \left (d x + c\right )^{6} - 3 \, a \cos \left (d x + c\right )^{5} + 3 \, a \cos \left (d x + c\right )^{4} + 3 \, a \cos \left (d x + c\right )^{3} - 3 \, a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) + a\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 315 \, {\left (a \cos \left (d x + c\right )^{7} - a \cos \left (d x + c\right )^{6} - 3 \, a \cos \left (d x + c\right )^{5} + 3 \, a \cos \left (d x + c\right )^{4} + 3 \, a \cos \left (d x + c\right )^{3} - 3 \, a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) + a\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) - 496 \, a \cos \left (d x + c\right ) + 1126 \, a}{630 \, {\left (d \cos \left (d x + c\right )^{7} - d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{5} + 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{3} - 3 \, d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 164, normalized size = 0.99 \[ -\frac {45 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 630 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4830 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 80640 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 80640 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 40950 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {80640 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 13650 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 2898 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 450 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 35 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9}}}{80640 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.81, size = 183, normalized size = 1.11 \[ -\frac {128 a \cot \left (d x +c \right )}{315 d}-\frac {a \cot \left (d x +c \right ) \left (\csc ^{8}\left (d x +c \right )\right )}{9 d}-\frac {8 a \cot \left (d x +c \right ) \left (\csc ^{6}\left (d x +c \right )\right )}{63 d}-\frac {16 a \cot \left (d x +c \right ) \left (\csc ^{4}\left (d x +c \right )\right )}{105 d}-\frac {64 a \cot \left (d x +c \right ) \left (\csc ^{2}\left (d x +c \right )\right )}{315 d}-\frac {a}{9 d \sin \left (d x +c \right )^{9}}-\frac {a}{7 d \sin \left (d x +c \right )^{7}}-\frac {a}{5 d \sin \left (d x +c \right )^{5}}-\frac {a}{3 d \sin \left (d x +c \right )^{3}}-\frac {a}{d \sin \left (d x +c \right )}+\frac {a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 136, normalized size = 0.82 \[ -\frac {a {\left (\frac {2 \, {\left (315 \, \sin \left (d x + c\right )^{8} + 105 \, \sin \left (d x + c\right )^{6} + 63 \, \sin \left (d x + c\right )^{4} + 45 \, \sin \left (d x + c\right )^{2} + 35\right )}}{\sin \left (d x + c\right )^{9}} - 315 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 315 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + \frac {2 \, {\left (315 \, \tan \left (d x + c\right )^{8} + 420 \, \tan \left (d x + c\right )^{6} + 378 \, \tan \left (d x + c\right )^{4} + 180 \, \tan \left (d x + c\right )^{2} + 35\right )} a}{\tan \left (d x + c\right )^{9}}}{630 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.64, size = 159, normalized size = 0.96 \[ \frac {2\,a\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {65\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d}-\frac {23\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{384\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{128\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{1792\,d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (256\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {130\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {46\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}+\frac {10\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{7}+\frac {a}{9}\right )}{256\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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