Optimal. Leaf size=122 \[ -\frac {a \cot ^7(c+d x)}{7 d}+\frac {5 a \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac {a \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac {5 a \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {5 a \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a \cot (c+d x) \csc (c+d x)}{128 d} \]
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Rubi [A] time = 0.18, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2838, 2611, 3768, 3770, 2607, 30} \[ -\frac {a \cot ^7(c+d x)}{7 d}+\frac {5 a \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac {a \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac {5 a \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {5 a \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a \cot (c+d x) \csc (c+d x)}{128 d} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2607
Rule 2611
Rule 2838
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)) \, dx &=a \int \cot ^6(c+d x) \csc ^2(c+d x) \, dx+a \int \cot ^6(c+d x) \csc ^3(c+d x) \, dx\\ &=-\frac {a \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {1}{8} (5 a) \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx+\frac {a \operatorname {Subst}\left (\int x^6 \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac {a \cot ^7(c+d x)}{7 d}+\frac {5 a \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac {1}{16} (5 a) \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx\\ &=-\frac {a \cot ^7(c+d x)}{7 d}-\frac {5 a \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {1}{64} (5 a) \int \csc ^3(c+d x) \, dx\\ &=-\frac {a \cot ^7(c+d x)}{7 d}+\frac {5 a \cot (c+d x) \csc (c+d x)}{128 d}-\frac {5 a \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {1}{128} (5 a) \int \csc (c+d x) \, dx\\ &=\frac {5 a \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac {a \cot ^7(c+d x)}{7 d}+\frac {5 a \cot (c+d x) \csc (c+d x)}{128 d}-\frac {5 a \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 215, normalized size = 1.76 \[ -\frac {a \cot ^7(c+d x)}{7 d}-\frac {a \csc ^8\left (\frac {1}{2} (c+d x)\right )}{2048 d}+\frac {7 a \csc ^6\left (\frac {1}{2} (c+d x)\right )}{1536 d}-\frac {15 a \csc ^4\left (\frac {1}{2} (c+d x)\right )}{1024 d}+\frac {5 a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{512 d}+\frac {a \sec ^8\left (\frac {1}{2} (c+d x)\right )}{2048 d}-\frac {7 a \sec ^6\left (\frac {1}{2} (c+d x)\right )}{1536 d}+\frac {15 a \sec ^4\left (\frac {1}{2} (c+d x)\right )}{1024 d}-\frac {5 a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{512 d}-\frac {5 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{128 d}+\frac {5 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{128 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.78, size = 225, normalized size = 1.84 \[ -\frac {768 \, a \cos \left (d x + c\right )^{7} \sin \left (d x + c\right ) + 210 \, a \cos \left (d x + c\right )^{7} + 1022 \, a \cos \left (d x + c\right )^{5} - 770 \, a \cos \left (d x + c\right )^{3} + 210 \, a \cos \left (d x + c\right ) - 105 \, {\left (a \cos \left (d x + c\right )^{8} - 4 \, a \cos \left (d x + c\right )^{6} + 6 \, a \cos \left (d x + c\right )^{4} - 4 \, a \cos \left (d x + c\right )^{2} + a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 105 \, {\left (a \cos \left (d x + c\right )^{8} - 4 \, a \cos \left (d x + c\right )^{6} + 6 \, a \cos \left (d x + c\right )^{4} - 4 \, a \cos \left (d x + c\right )^{2} + a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{5376 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.28, size = 256, normalized size = 2.10 \[ \frac {21 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 48 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 112 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 336 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 168 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1008 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 336 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1680 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 1680 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {4566 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 1680 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 336 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1008 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 168 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 336 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 112 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 48 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 21 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8}}}{43008 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 174, normalized size = 1.43 \[ -\frac {a \left (\cos ^{7}\left (d x +c \right )\right )}{7 d \sin \left (d x +c \right )^{7}}-\frac {a \left (\cos ^{7}\left (d x +c \right )\right )}{8 d \sin \left (d x +c \right )^{8}}-\frac {a \left (\cos ^{7}\left (d x +c \right )\right )}{48 d \sin \left (d x +c \right )^{6}}+\frac {a \left (\cos ^{7}\left (d x +c \right )\right )}{192 d \sin \left (d x +c \right )^{4}}-\frac {a \left (\cos ^{7}\left (d x +c \right )\right )}{128 d \sin \left (d x +c \right )^{2}}-\frac {a \left (\cos ^{5}\left (d x +c \right )\right )}{128 d}-\frac {5 a \left (\cos ^{3}\left (d x +c \right )\right )}{384 d}-\frac {5 a \cos \left (d x +c \right )}{128 d}-\frac {5 a \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 126, normalized size = 1.03 \[ -\frac {7 \, a {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{7} + 73 \, \cos \left (d x + c\right )^{5} - 55 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {768 \, a}{\tan \left (d x + c\right )^{7}}}{5376 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.15, size = 285, normalized size = 2.34 \[ \frac {5\,a\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d}-\frac {5\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}-\frac {3\,a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{128\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{256\,d}+\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{128\,d}+\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{896\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2048\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}+\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{128\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{256\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{128\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{896\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2048\,d}-\frac {5\,a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{128\,d} \]
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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