Optimal. Leaf size=165 \[ -\frac {a^6 \tan (c+d x) \sec (c+d x)}{5 d (a-a \cos (c+d x))^3}-\frac {11 a^5 \tan (c+d x) \sec (c+d x)}{15 d (a-a \cos (c+d x))^2}+\frac {152 a^3 \tan (c+d x)}{15 d}+\frac {13 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {13 a^3 \tan (c+d x) \sec (c+d x)}{2 d}-\frac {76 a^6 \tan (c+d x) \sec (c+d x)}{15 d \left (a^3-a^3 \cos (c+d x)\right )} \]
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Rubi [A] time = 0.44, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3872, 2869, 2766, 2978, 2748, 3768, 3770, 3767, 8} \[ \frac {152 a^3 \tan (c+d x)}{15 d}+\frac {13 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {13 a^3 \tan (c+d x) \sec (c+d x)}{2 d}-\frac {76 a^6 \tan (c+d x) \sec (c+d x)}{15 d \left (a^3-a^3 \cos (c+d x)\right )}-\frac {a^6 \tan (c+d x) \sec (c+d x)}{5 d (a-a \cos (c+d x))^3}-\frac {11 a^5 \tan (c+d x) \sec (c+d x)}{15 d (a-a \cos (c+d x))^2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2748
Rule 2766
Rule 2869
Rule 2978
Rule 3767
Rule 3768
Rule 3770
Rule 3872
Rubi steps
\begin {align*} \int \csc ^6(c+d x) (a+a \sec (c+d x))^3 \, dx &=-\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 ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{15 d}\\ &=\frac {13 a^3 \tanh ^{-1}(\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*}
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Mathematica [B] time = 1.22, size = 353, normalized size = 2.14 \[ -\frac {a^3 (\cos (c+d x)+1)^3 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \sec ^2(c+d x) \left (24960 \cos ^2(c+d x) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )+\csc \left (\frac {c}{2}\right ) \sec (c) \left (4329 \sin \left (c-\frac {d x}{2}\right )-1989 \sin \left (c+\frac {d x}{2}\right )-3575 \sin \left (2 c+\frac {d x}{2}\right )+475 \sin \left (c+\frac {3 d x}{2}\right )+2005 \sin \left (2 c+\frac {3 d x}{2}\right )+2275 \sin \left (3 c+\frac {3 d x}{2}\right )-2673 \sin \left (c+\frac {5 d x}{2}\right )+105 \sin \left (2 c+\frac {5 d x}{2}\right )-1593 \sin \left (3 c+\frac {5 d x}{2}\right )-975 \sin \left (4 c+\frac {5 d x}{2}\right )+1325 \sin \left (2 c+\frac {7 d x}{2}\right )-255 \sin \left (3 c+\frac {7 d x}{2}\right )+875 \sin \left (4 c+\frac {7 d x}{2}\right )+195 \sin \left (5 c+\frac {7 d x}{2}\right )-304 \sin \left (3 c+\frac {9 d x}{2}\right )+90 \sin \left (4 c+\frac {9 d x}{2}\right )-214 \sin \left (5 c+\frac {9 d x}{2}\right )-1235 \sin \left (\frac {d x}{2}\right )+3805 \sin \left (\frac {3 d x}{2}\right )\right ) \csc ^5\left (\frac {1}{2} (c+d x)\right )\right )}{30720 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 225, normalized size = 1.36 \[ -\frac {608 \, a^{3} \cos \left (d x + c\right )^{5} - 826 \, a^{3} \cos \left (d x + c\right )^{4} - 476 \, a^{3} \cos \left (d x + c\right )^{3} + 868 \, a^{3} \cos \left (d x + c\right )^{2} - 120 \, a^{3} \cos \left (d x + c\right ) - 30 \, a^{3} - 195 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{3} + a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 195 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{3} + a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right )}{60 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 141, normalized size = 0.85 \[ \frac {390 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 390 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {60 \, {\left (5 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 7 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}} - \frac {465 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 40 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.05, size = 274, normalized size = 1.66 \[ -\frac {152 a^{3} \cot \left (d x +c \right )}{15 d}-\frac {a^{3} \cot \left (d x +c \right ) \left (\csc ^{4}\left (d x +c \right )\right )}{5 d}-\frac {4 a^{3} \cot \left (d x +c \right ) \left (\csc ^{2}\left (d x +c \right )\right )}{15 d}-\frac {3 a^{3}}{5 d \sin \left (d x +c \right )^{5}}-\frac {a^{3}}{d \sin \left (d x +c \right )^{3}}-\frac {13 a^{3}}{2 d \sin \left (d x +c \right )}+\frac {13 a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}-\frac {3 a^{3}}{5 d \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )}-\frac {6 a^{3}}{5 d \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {24 a^{3}}{5 d \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {a^{3}}{5 d \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )^{2}}-\frac {7 a^{3}}{15 d \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{2}}+\frac {7 a^{3}}{6 d \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 228, normalized size = 1.38 \[ -\frac {a^{3} {\left (\frac {2 \, {\left (105 \, \sin \left (d x + c\right )^{6} - 70 \, \sin \left (d x + c\right )^{4} - 14 \, \sin \left (d x + c\right )^{2} - 6\right )}}{\sin \left (d x + c\right )^{7} - \sin \left (d x + c\right )^{5}} - 105 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, a^{3} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{4} + 5 \, \sin \left (d x + c\right )^{2} + 3\right )}}{\sin \left (d x + c\right )^{5}} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 36 \, a^{3} {\left (\frac {15 \, \tan \left (d x + c\right )^{4} + 5 \, \tan \left (d x + c\right )^{2} + 1}{\tan \left (d x + c\right )^{5}} - 5 \, \tan \left (d x + c\right )\right )} + \frac {4 \, {\left (15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} + 3\right )} a^{3}}{\tan \left (d x + c\right )^{5}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.91, size = 136, normalized size = 0.82 \[ \frac {13\,a^3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {51\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-\frac {262\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {388\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{15}+\frac {34\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{15}+\frac {a^3}{5}}{d\,\left (4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\right )} \]
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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