Optimal. Leaf size=138 \[ -\frac {a \cot ^9(c+d x)}{9 d}-\frac {a \cot ^7(c+d x)}{7 d}+\frac {5 a \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac {a \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac {5 a \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {5 a \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a \cot (c+d x) \csc (c+d x)}{128 d} \]
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Rubi [A] time = 0.20, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2838, 2607, 14, 2611, 3768, 3770} \[ -\frac {a \cot ^9(c+d x)}{9 d}-\frac {a \cot ^7(c+d x)}{7 d}+\frac {5 a \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac {a \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac {5 a \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {5 a \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a \cot (c+d x) \csc (c+d x)}{128 d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2607
Rule 2611
Rule 2838
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cot ^6(c+d x) \csc ^3(c+d x) \, dx+a \int \cot ^6(c+d x) \csc ^4(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 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{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 \operatorname {Subst}\left (\int \left (x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac {a \cot ^7(c+d x)}{7 d}-\frac {a \cot ^9(c+d x)}{9 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 {a \cot ^9(c+d x)}{9 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 {a \cot ^9(c+d x)}{9 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 [B] time = 0.09, size = 301, normalized size = 2.18 \[ \frac {2 a \cot (c+d x)}{63 d}-\frac {a \csc ^8\left (\frac {1}{2} (c+d x)\right )}{2048 d}+\frac {7 a \csc ^6\left (\frac {1}{2} (c+d x)\right )}{1536 d}-\frac {15 a \csc ^4\left (\frac {1}{2} (c+d x)\right )}{1024 d}+\frac {5 a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{512 d}+\frac {a \sec ^8\left (\frac {1}{2} (c+d x)\right )}{2048 d}-\frac {7 a \sec ^6\left (\frac {1}{2} (c+d x)\right )}{1536 d}+\frac {15 a \sec ^4\left (\frac {1}{2} (c+d x)\right )}{1024 d}-\frac {5 a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{512 d}-\frac {5 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{128 d}+\frac {5 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{128 d}-\frac {a \cot (c+d x) \csc ^8(c+d x)}{9 d}+\frac {19 a \cot (c+d x) \csc ^6(c+d x)}{63 d}-\frac {5 a \cot (c+d x) \csc ^4(c+d x)}{21 d}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{63 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.60, size = 259, normalized size = 1.88 \[ \frac {512 \, a \cos \left (d x + c\right )^{9} - 2304 \, a \cos \left (d x + c\right )^{7} + 315 \, {\left (a \cos \left (d x + c\right )^{8} - 4 \, a \cos \left (d x + c\right )^{6} + 6 \, a \cos \left (d x + c\right )^{4} - 4 \, a \cos \left (d x + c\right )^{2} + a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 315 \, {\left (a \cos \left (d x + c\right )^{8} - 4 \, a \cos \left (d x + c\right )^{6} + 6 \, a \cos \left (d x + c\right )^{4} - 4 \, a \cos \left (d x + c\right )^{2} + a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 42 \, {\left (15 \, a \cos \left (d x + c\right )^{7} + 73 \, a \cos \left (d x + c\right )^{5} - 55 \, a \cos \left (d x + c\right )^{3} + 15 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{16128 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 256, normalized size = 1.86 \[ \frac {28 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 63 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 108 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 336 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 504 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 672 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1008 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 5040 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 1512 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {14258 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 1512 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 1008 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 672 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 504 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 336 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 108 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 63 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 28 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9}}}{129024 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 196, normalized size = 1.42 \[ -\frac {a \left (\cos ^{7}\left (d x +c \right )\right )}{8 d \sin \left (d x +c \right )^{8}}-\frac {a \left (\cos ^{7}\left (d x +c \right )\right )}{48 d \sin \left (d x +c \right )^{6}}+\frac {a \left (\cos ^{7}\left (d x +c \right )\right )}{192 d \sin \left (d x +c \right )^{4}}-\frac {a \left (\cos ^{7}\left (d x +c \right )\right )}{128 d \sin \left (d x +c \right )^{2}}-\frac {a \left (\cos ^{5}\left (d x +c \right )\right )}{128 d}-\frac {5 a \left (\cos ^{3}\left (d x +c \right )\right )}{384 d}-\frac {5 a \cos \left (d x +c \right )}{128 d}-\frac {5 a \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128 d}-\frac {a \left (\cos ^{7}\left (d x +c \right )\right )}{9 d \sin \left (d x +c \right )^{9}}-\frac {2 a \left (\cos ^{7}\left (d x +c \right )\right )}{63 d \sin \left (d x +c \right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 138, normalized size = 1.00 \[ -\frac {21 \, a {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{7} + 73 \, \cos \left (d x + c\right )^{5} - 55 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {256 \, {\left (9 \, \tan \left (d x + c\right )^{2} + 7\right )} a}{\tan \left (d x + c\right )^{9}}}{16128 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.19, size = 285, normalized size = 2.07 \[ \frac {3\,a\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{256\,d}-\frac {3\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{256\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{192\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{256\,d}+\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}+\frac {3\,a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3584\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2048\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4608\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{192\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{256\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}-\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3584\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2048\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4608\,d}-\frac {5\,a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{128\,d} \]
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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