Optimal. Leaf size=238 \[ -\frac {3 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^5(c+d x)}{5 d}+\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot (c+d x)}{d}+\frac {125 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac {a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^3 \cot ^5(c+d x) \csc (c+d x)}{2 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}+\frac {5 a^3 \cot ^3(c+d x) \csc (c+d x)}{8 d}-\frac {5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {115 a^3 \cot (c+d x) \csc (c+d x)}{128 d}-a^3 x \]
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Rubi [A] time = 0.36, antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2873, 3473, 8, 2611, 3770, 2607, 30, 3768} \[ -\frac {3 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^5(c+d x)}{5 d}+\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot (c+d x)}{d}+\frac {125 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac {a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^3 \cot ^5(c+d x) \csc (c+d x)}{2 d}+\frac {5 a^3 \cot ^3(c+d x) \csc (c+d x)}{8 d}-\frac {5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {115 a^3 \cot (c+d x) \csc (c+d x)}{128 d}-a^3 x \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 2607
Rule 2611
Rule 2873
Rule 3473
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx &=\int \left (a^3 \cot ^6(c+d x)+3 a^3 \cot ^6(c+d x) \csc (c+d x)+3 a^3 \cot ^6(c+d x) \csc ^2(c+d x)+a^3 \cot ^6(c+d x) \csc ^3(c+d x)\right ) \, dx\\ &=a^3 \int \cot ^6(c+d x) \, dx+a^3 \int \cot ^6(c+d x) \csc ^3(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^6(c+d x) \csc (c+d x) \, dx+\left (3 a^3\right ) \int \cot ^6(c+d x) \csc ^2(c+d x) \, dx\\ &=-\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^5(c+d x) \csc (c+d x)}{2 d}-\frac {a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {1}{8} \left (5 a^3\right ) \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx-a^3 \int \cot ^4(c+d x) \, dx-\frac {1}{2} \left (5 a^3\right ) \int \cot ^4(c+d x) \csc (c+d x) \, dx+\frac {\left (3 a^3\right ) \operatorname {Subst}\left (\int x^6 \, dx,x,-\cot (c+d x)\right )}{d}\\ &=\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {3 a^3 \cot ^7(c+d x)}{7 d}+\frac {5 a^3 \cot ^3(c+d x) \csc (c+d x)}{8 d}-\frac {a^3 \cot ^5(c+d x) \csc (c+d x)}{2 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac {1}{16} \left (5 a^3\right ) \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx+a^3 \int \cot ^2(c+d x) \, dx+\frac {1}{8} \left (15 a^3\right ) \int \cot ^2(c+d x) \csc (c+d x) \, dx\\ &=-\frac {a^3 \cot (c+d x)}{d}+\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {3 a^3 \cot ^7(c+d x)}{7 d}-\frac {15 a^3 \cot (c+d x) \csc (c+d x)}{16 d}+\frac {5 a^3 \cot ^3(c+d x) \csc (c+d x)}{8 d}-\frac {a^3 \cot ^5(c+d x) \csc (c+d x)}{2 d}-\frac {5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {1}{64} \left (5 a^3\right ) \int \csc ^3(c+d x) \, dx-\frac {1}{16} \left (15 a^3\right ) \int \csc (c+d x) \, dx-a^3 \int 1 \, dx\\ &=-a^3 x+\frac {15 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac {a^3 \cot (c+d x)}{d}+\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {3 a^3 \cot ^7(c+d x)}{7 d}-\frac {115 a^3 \cot (c+d x) \csc (c+d x)}{128 d}+\frac {5 a^3 \cot ^3(c+d x) \csc (c+d x)}{8 d}-\frac {a^3 \cot ^5(c+d x) \csc (c+d x)}{2 d}-\frac {5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {1}{128} \left (5 a^3\right ) \int \csc (c+d x) \, dx\\ &=-a^3 x+\frac {125 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac {a^3 \cot (c+d x)}{d}+\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {3 a^3 \cot ^7(c+d x)}{7 d}-\frac {115 a^3 \cot (c+d x) \csc (c+d x)}{128 d}+\frac {5 a^3 \cot ^3(c+d x) \csc (c+d x)}{8 d}-\frac {a^3 \cot ^5(c+d x) \csc (c+d x)}{2 d}-\frac {5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}\\ \end {align*}
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Mathematica [A] time = 1.21, size = 279, normalized size = 1.17 \[ \frac {a^3 \left (118784 \tan \left (\frac {1}{2} (c+d x)\right )-118784 \cot \left (\frac {1}{2} (c+d x)\right )-108780 \csc ^2\left (\frac {1}{2} (c+d x)\right )+105 \sec ^8\left (\frac {1}{2} (c+d x)\right )+700 \sec ^6\left (\frac {1}{2} (c+d x)\right )-17010 \sec ^4\left (\frac {1}{2} (c+d x)\right )+108780 \sec ^2\left (\frac {1}{2} (c+d x)\right )-210000 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+210000 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-15 (24 \sin (c+d x)+7) \csc ^8\left (\frac {1}{2} (c+d x)\right )+4 (732 \sin (c+d x)-175) \csc ^6\left (\frac {1}{2} (c+d x)\right )+(17010-4496 \sin (c+d x)) \csc ^4\left (\frac {1}{2} (c+d x)\right )+71936 \sin ^4\left (\frac {1}{2} (c+d x)\right ) \csc ^3(c+d x)+720 \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^6\left (\frac {1}{2} (c+d x)\right )-5856 \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^4\left (\frac {1}{2} (c+d x)\right )-215040 c-215040 d x\right )}{215040 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 362, normalized size = 1.52 \[ -\frac {26880 \, a^{3} d x \cos \left (d x + c\right )^{8} - 107520 \, a^{3} d x \cos \left (d x + c\right )^{6} - 54390 \, a^{3} \cos \left (d x + c\right )^{7} + 161280 \, a^{3} d x \cos \left (d x + c\right )^{4} + 127750 \, a^{3} \cos \left (d x + c\right )^{5} - 107520 \, a^{3} d x \cos \left (d x + c\right )^{2} - 96250 \, a^{3} \cos \left (d x + c\right )^{3} + 26880 \, a^{3} d x + 26250 \, a^{3} \cos \left (d x + c\right ) - 13125 \, {\left (a^{3} \cos \left (d x + c\right )^{8} - 4 \, a^{3} \cos \left (d x + c\right )^{6} + 6 \, a^{3} \cos \left (d x + c\right )^{4} - 4 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 13125 \, {\left (a^{3} \cos \left (d x + c\right )^{8} - 4 \, a^{3} \cos \left (d x + c\right )^{6} + 6 \, a^{3} \cos \left (d x + c\right )^{4} - 4 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 256 \, {\left (116 \, a^{3} \cos \left (d x + c\right )^{7} - 406 \, a^{3} \cos \left (d x + c\right )^{5} + 350 \, a^{3} \cos \left (d x + c\right )^{3} - 105 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{26880 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.47, size = 302, normalized size = 1.27 \[ \frac {105 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 720 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1120 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 3696 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 14280 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 560 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 77280 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 215040 \, {\left (d x + c\right )} a^{3} - 210000 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 122640 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {570750 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 122640 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 77280 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 560 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 14280 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 3696 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1120 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 720 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 105 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8}}}{215040 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.45, size = 253, normalized size = 1.06 \[ -\frac {a^{3} \left (\cot ^{5}\left (d x +c \right )\right )}{5 d}+\frac {a^{3} \left (\cot ^{3}\left (d x +c \right )\right )}{3 d}-\frac {a^{3} \cot \left (d x +c \right )}{d}-a^{3} x -\frac {a^{3} c}{d}-\frac {25 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{48 d \sin \left (d x +c \right )^{6}}+\frac {25 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{192 d \sin \left (d x +c \right )^{4}}-\frac {25 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{128 d \sin \left (d x +c \right )^{2}}-\frac {25 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{128 d}-\frac {125 a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{384 d}-\frac {125 a^{3} \cos \left (d x +c \right )}{128 d}-\frac {125 a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128 d}-\frac {3 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{7 d \sin \left (d x +c \right )^{7}}-\frac {a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{8 d \sin \left (d x +c \right )^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 265, normalized size = 1.11 \[ -\frac {1792 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a^{3} + 35 \, a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{7} + 73 \, \cos \left (d x + c\right )^{5} - 55 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 840 \, a^{3} {\left (\frac {2 \, {\left (33 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {11520 \, a^{3}}{\tan \left (d x + c\right )^{7}}}{26880 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.30, size = 389, normalized size = 1.63 \[ \frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{384\,d}-\frac {23\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64\,d}+\frac {17\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{256\,d}+\frac {11\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{640\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{192\,d}-\frac {3\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{896\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2048\,d}+\frac {23\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{384\,d}-\frac {17\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{256\,d}-\frac {11\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{640\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{192\,d}+\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{896\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2048\,d}-\frac {2\,a^3\,\mathrm {atan}\left (\frac {128\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+125\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{125\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-128\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {125\,a^3\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{128\,d}-\frac {73\,a^3\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d}+\frac {73\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d} \]
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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