Optimal. Leaf size=201 \[ \frac {a^2 \tan (c+d x)}{d}-\frac {2 a^2 \cot ^9(c+d x)}{9 d}-\frac {9 a^2 \cot ^7(c+d x)}{7 d}-\frac {16 a^2 \cot ^5(c+d x)}{5 d}-\frac {14 a^2 \cot ^3(c+d x)}{3 d}-\frac {6 a^2 \cot (c+d x)}{d}-\frac {2 a^2 \csc ^9(c+d x)}{9 d}-\frac {2 a^2 \csc ^7(c+d x)}{7 d}-\frac {2 a^2 \csc ^5(c+d x)}{5 d}-\frac {2 a^2 \csc ^3(c+d x)}{3 d}-\frac {2 a^2 \csc (c+d x)}{d}+\frac {2 a^2 \tanh ^{-1}(\sin (c+d x))}{d} \]
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Rubi [A] time = 0.26, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3872, 2873, 3767, 2621, 302, 207, 2620, 270} \[ \frac {a^2 \tan (c+d x)}{d}-\frac {2 a^2 \cot ^9(c+d x)}{9 d}-\frac {9 a^2 \cot ^7(c+d x)}{7 d}-\frac {16 a^2 \cot ^5(c+d x)}{5 d}-\frac {14 a^2 \cot ^3(c+d x)}{3 d}-\frac {6 a^2 \cot (c+d x)}{d}-\frac {2 a^2 \csc ^9(c+d x)}{9 d}-\frac {2 a^2 \csc ^7(c+d x)}{7 d}-\frac {2 a^2 \csc ^5(c+d x)}{5 d}-\frac {2 a^2 \csc ^3(c+d x)}{3 d}-\frac {2 a^2 \csc (c+d x)}{d}+\frac {2 a^2 \tanh ^{-1}(\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 207
Rule 270
Rule 302
Rule 2620
Rule 2621
Rule 2873
Rule 3767
Rule 3872
Rubi steps
\begin {align*} \int \csc ^{10}(c+d x) (a+a \sec (c+d x))^2 \, dx &=\int (-a-a \cos (c+d x))^2 \csc ^{10}(c+d x) \sec ^2(c+d x) \, dx\\ &=\int \left (a^2 \csc ^{10}(c+d x)+2 a^2 \csc ^{10}(c+d x) \sec (c+d x)+a^2 \csc ^{10}(c+d x) \sec ^2(c+d x)\right ) \, dx\\ &=a^2 \int \csc ^{10}(c+d x) \, dx+a^2 \int \csc ^{10}(c+d x) \sec ^2(c+d x) \, dx+\left (2 a^2\right ) \int \csc ^{10}(c+d x) \sec (c+d x) \, dx\\ &=\frac {a^2 \operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^5}{x^{10}} \, dx,x,\tan (c+d x)\right )}{d}-\frac {a^2 \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 {\left (2 a^2\right ) \operatorname {Subst}\left (\int \frac {x^{10}}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}\\ &=-\frac {a^2 \cot (c+d x)}{d}-\frac {4 a^2 \cot ^3(c+d x)}{3 d}-\frac {6 a^2 \cot ^5(c+d x)}{5 d}-\frac {4 a^2 \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot ^9(c+d x)}{9 d}+\frac {a^2 \operatorname {Subst}\left (\int \left (1+\frac {1}{x^{10}}+\frac {5}{x^8}+\frac {10}{x^6}+\frac {10}{x^4}+\frac {5}{x^2}\right ) \, dx,x,\tan (c+d x)\right )}{d}-\frac {\left (2 a^2\right ) \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 {6 a^2 \cot (c+d x)}{d}-\frac {14 a^2 \cot ^3(c+d x)}{3 d}-\frac {16 a^2 \cot ^5(c+d x)}{5 d}-\frac {9 a^2 \cot ^7(c+d x)}{7 d}-\frac {2 a^2 \cot ^9(c+d x)}{9 d}-\frac {2 a^2 \csc (c+d x)}{d}-\frac {2 a^2 \csc ^3(c+d x)}{3 d}-\frac {2 a^2 \csc ^5(c+d x)}{5 d}-\frac {2 a^2 \csc ^7(c+d x)}{7 d}-\frac {2 a^2 \csc ^9(c+d x)}{9 d}+\frac {a^2 \tan (c+d x)}{d}-\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}\\ &=\frac {2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac {6 a^2 \cot (c+d x)}{d}-\frac {14 a^2 \cot ^3(c+d x)}{3 d}-\frac {16 a^2 \cot ^5(c+d x)}{5 d}-\frac {9 a^2 \cot ^7(c+d x)}{7 d}-\frac {2 a^2 \cot ^9(c+d x)}{9 d}-\frac {2 a^2 \csc (c+d x)}{d}-\frac {2 a^2 \csc ^3(c+d x)}{3 d}-\frac {2 a^2 \csc ^5(c+d x)}{5 d}-\frac {2 a^2 \csc ^7(c+d x)}{7 d}-\frac {2 a^2 \csc ^9(c+d x)}{9 d}+\frac {a^2 \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [B] time = 6.95, size = 1050, normalized size = 5.22 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 3.26, size = 406, normalized size = 2.02 \[ -\frac {1408 \, a^{2} \cos \left (d x + c\right )^{8} - 2186 \, a^{2} \cos \left (d x + c\right )^{7} - 3372 \, a^{2} \cos \left (d x + c\right )^{6} + 6200 \, a^{2} \cos \left (d x + c\right )^{5} + 2060 \, a^{2} \cos \left (d x + c\right )^{4} - 5784 \, a^{2} \cos \left (d x + c\right )^{3} + 268 \, a^{2} \cos \left (d x + c\right )^{2} + 1756 \, a^{2} \cos \left (d x + c\right ) - 315 \, {\left (a^{2} \cos \left (d x + c\right )^{7} - 2 \, a^{2} \cos \left (d x + c\right )^{6} - a^{2} \cos \left (d x + c\right )^{5} + 4 \, a^{2} \cos \left (d x + c\right )^{4} - a^{2} \cos \left (d x + c\right )^{3} - 2 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2} \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 315 \, {\left (a^{2} \cos \left (d x + c\right )^{7} - 2 \, a^{2} \cos \left (d x + c\right )^{6} - a^{2} \cos \left (d x + c\right )^{5} + 4 \, a^{2} \cos \left (d x + c\right )^{4} - a^{2} \cos \left (d x + c\right )^{3} - 2 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2} \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) - 315 \, a^{2}}{315 \, {\left (d \cos \left (d x + c\right )^{7} - 2 \, d \cos \left (d x + c\right )^{6} - d \cos \left (d x + c\right )^{5} + 4 \, d \cos \left (d x + c\right )^{4} - d \cos \left (d x + c\right )^{3} - 2 \, d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.61, size = 200, normalized size = 1.00 \[ \frac {63 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1155 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 80640 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 80640 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 17955 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {80640 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} - \frac {139545 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 19635 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 3591 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 495 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 35 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9}}}{40320 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.03, size = 326, normalized size = 1.62 \[ -\frac {1408 a^{2} \cot \left (d x +c \right )}{315 d}-\frac {a^{2} \cot \left (d x +c \right ) \left (\csc ^{8}\left (d x +c \right )\right )}{9 d}-\frac {8 a^{2} \cot \left (d x +c \right ) \left (\csc ^{6}\left (d x +c \right )\right )}{63 d}-\frac {16 a^{2} \cot \left (d x +c \right ) \left (\csc ^{4}\left (d x +c \right )\right )}{105 d}-\frac {64 a^{2} \cot \left (d x +c \right ) \left (\csc ^{2}\left (d x +c \right )\right )}{315 d}-\frac {2 a^{2}}{9 d \sin \left (d x +c \right )^{9}}-\frac {2 a^{2}}{7 d \sin \left (d x +c \right )^{7}}-\frac {2 a^{2}}{5 d \sin \left (d x +c \right )^{5}}-\frac {2 a^{2}}{3 d \sin \left (d x +c \right )^{3}}-\frac {2 a^{2}}{d \sin \left (d x +c \right )}+\frac {2 a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}-\frac {a^{2}}{9 d \sin \left (d x +c \right )^{9} \cos \left (d x +c \right )}-\frac {10 a^{2}}{63 d \sin \left (d x +c \right )^{7} \cos \left (d x +c \right )}-\frac {16 a^{2}}{63 d \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )}-\frac {32 a^{2}}{63 d \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {128 a^{2}}{63 d \sin \left (d x +c \right ) \cos \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 204, normalized size = 1.01 \[ -\frac {a^{2} {\left (\frac {2 \, {\left (315 \, \sin \left (d x + c\right )^{8} + 105 \, \sin \left (d x + c\right )^{6} + 63 \, \sin \left (d x + c\right )^{4} + 45 \, \sin \left (d x + c\right )^{2} + 35\right )}}{\sin \left (d x + c\right )^{9}} - 315 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 315 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 5 \, a^{2} {\left (\frac {315 \, \tan \left (d x + c\right )^{8} + 210 \, \tan \left (d x + c\right )^{6} + 126 \, \tan \left (d x + c\right )^{4} + 45 \, \tan \left (d x + c\right )^{2} + 7}{\tan \left (d x + c\right )^{9}} - 63 \, \tan \left (d x + c\right )\right )} + \frac {{\left (315 \, \tan \left (d x + c\right )^{8} + 420 \, \tan \left (d x + c\right )^{6} + 378 \, \tan \left (d x + c\right )^{4} + 180 \, \tan \left (d x + c\right )^{2} + 35\right )} a^{2}}{\tan \left (d x + c\right )^{9}}}{315 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.97, size = 194, normalized size = 0.97 \[ \frac {11\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{384\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{640\,d}+\frac {4\,a^2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {-699\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {1142\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}+\frac {764\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{15}+\frac {344\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{35}+\frac {92\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{63}+\frac {a^2}{9}}{d\,\left (128\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-128\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\right )}+\frac {57\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d} \]
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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