Optimal. Leaf size=160 \[ -\frac {a \cot ^9(c+d x)}{9 d}-\frac {a \cot ^7(c+d x)}{7 d}+\frac {3 a \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac {a \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac {a \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {3 a \cot (c+d x) \csc (c+d x)}{256 d} \]
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Rubi [A] time = 0.21, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 10, 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, 14} \[ -\frac {a \cot ^9(c+d x)}{9 d}-\frac {a \cot ^7(c+d x)}{7 d}+\frac {3 a \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac {a \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac {a \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {3 a \cot (c+d x) \csc (c+d x)}{256 d} \]
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 ^5(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cot ^6(c+d x) \csc ^4(c+d x) \, dx+a \int \cot ^6(c+d x) \csc ^5(c+d x) \, dx\\ &=-\frac {a \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {1}{2} a \int \cot ^4(c+d x) \csc ^5(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 {a \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac {1}{16} (3 a) \int \cot ^2(c+d x) \csc ^5(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 {a \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {a \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {1}{32} a \int \csc ^5(c+d x) \, dx\\ &=-\frac {a \cot ^7(c+d x)}{7 d}-\frac {a \cot ^9(c+d x)}{9 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{128 d}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {a \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {1}{128} (3 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 {3 a \cot (c+d x) \csc (c+d x)}{256 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{128 d}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {a \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac {1}{256} (3 a) \int \csc (c+d x) \, dx\\ &=\frac {3 a \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac {a \cot ^7(c+d x)}{7 d}-\frac {a \cot ^9(c+d x)}{9 d}+\frac {3 a \cot (c+d x) \csc (c+d x)}{256 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{128 d}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac {a \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac {a \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}\\ \end {align*}
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Mathematica [B] time = 0.09, size = 341, normalized size = 2.13 \[ \frac {2 a \cot (c+d x)}{63 d}-\frac {a \csc ^{10}\left (\frac {1}{2} (c+d x)\right )}{10240 d}+\frac {3 a \csc ^8\left (\frac {1}{2} (c+d x)\right )}{4096 d}-\frac {3 a \csc ^6\left (\frac {1}{2} (c+d x)\right )}{2048 d}-\frac {a \csc ^4\left (\frac {1}{2} (c+d x)\right )}{1024 d}+\frac {3 a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{1024 d}+\frac {a \sec ^{10}\left (\frac {1}{2} (c+d x)\right )}{10240 d}-\frac {3 a \sec ^8\left (\frac {1}{2} (c+d x)\right )}{4096 d}+\frac {3 a \sec ^6\left (\frac {1}{2} (c+d x)\right )}{2048 d}+\frac {a \sec ^4\left (\frac {1}{2} (c+d x)\right )}{1024 d}-\frac {3 a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{1024 d}-\frac {3 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{256 d}+\frac {3 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{256 d}-\frac {a \cot (c+d x) \csc ^8(c+d x)}{9 d}+\frac {19 a \cot (c+d x) \csc ^6(c+d x)}{63 d}-\frac {5 a \cot (c+d x) \csc ^4(c+d x)}{21 d}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{63 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.66, size = 289, normalized size = 1.81 \[ -\frac {1890 \, a \cos \left (d x + c\right )^{9} - 8820 \, a \cos \left (d x + c\right )^{7} - 16128 \, a \cos \left (d x + c\right )^{5} + 8820 \, a \cos \left (d x + c\right )^{3} - 1890 \, a \cos \left (d x + c\right ) - 945 \, {\left (a \cos \left (d x + c\right )^{10} - 5 \, a \cos \left (d x + c\right )^{8} + 10 \, a \cos \left (d x + c\right )^{6} - 10 \, a \cos \left (d x + c\right )^{4} + 5 \, a \cos \left (d x + c\right )^{2} - a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 945 \, {\left (a \cos \left (d x + c\right )^{10} - 5 \, a \cos \left (d x + c\right )^{8} + 10 \, a \cos \left (d x + c\right )^{6} - 10 \, a \cos \left (d x + c\right )^{4} + 5 \, a \cos \left (d x + c\right )^{2} - a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2560 \, {\left (2 \, a \cos \left (d x + c\right )^{9} - 9 \, a \cos \left (d x + c\right )^{7}\right )} \sin \left (d x + c\right )}{161280 \, {\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 284, normalized size = 1.78 \[ \frac {126 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 280 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 315 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 1080 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 630 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 2520 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 6720 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1260 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15120 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 15120 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {44286 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 15120 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 1260 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 6720 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 2520 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 630 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1080 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 315 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 280 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 126 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10}}}{1290240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 218, normalized size = 1.36 \[ -\frac {a \left (\cos ^{7}\left (d x +c \right )\right )}{9 d \sin \left (d x +c \right )^{9}}-\frac {2 a \left (\cos ^{7}\left (d x +c \right )\right )}{63 d \sin \left (d x +c \right )^{7}}-\frac {a \left (\cos ^{7}\left (d x +c \right )\right )}{10 d \sin \left (d x +c \right )^{10}}-\frac {3 a \left (\cos ^{7}\left (d x +c \right )\right )}{80 d \sin \left (d x +c \right )^{8}}-\frac {a \left (\cos ^{7}\left (d x +c \right )\right )}{160 d \sin \left (d x +c \right )^{6}}+\frac {a \left (\cos ^{7}\left (d x +c \right )\right )}{640 d \sin \left (d x +c \right )^{4}}-\frac {3 a \left (\cos ^{7}\left (d x +c \right )\right )}{1280 d \sin \left (d x +c \right )^{2}}-\frac {3 a \left (\cos ^{5}\left (d x +c \right )\right )}{1280 d}-\frac {a \left (\cos ^{3}\left (d x +c \right )\right )}{256 d}-\frac {3 a \cos \left (d x +c \right )}{256 d}-\frac {3 a \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 158, normalized size = 0.99 \[ -\frac {63 \, a {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} - 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {2560 \, {\left (9 \, \tan \left (d x + c\right )^{2} + 7\right )} a}{\tan \left (d x + c\right )^{9}}}{161280 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.40, size = 319, normalized size = 1.99 \[ \frac {3\,a\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{256\,d}-\frac {3\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{256\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{192\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{512\,d}+\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2048\,d}+\frac {3\,a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3584\,d}+\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4096\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4608\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{192\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{512\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2048\,d}-\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3584\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4096\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4608\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}-\frac {3\,a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{256\,d} \]
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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