Optimal. Leaf size=168 \[ -\frac {a^2 \cot ^9(c+d x)}{9 d}-\frac {3 a^2 \cot ^7(c+d x)}{7 d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}-\frac {3 a^2 \tanh ^{-1}(\cos (c+d x))}{64 d}-\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{4 d}+\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{8 d}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{32 d}-\frac {3 a^2 \cot (c+d x) \csc (c+d x)}{64 d} \]
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Rubi [A] time = 0.27, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2873, 2607, 14, 2611, 3768, 3770, 270} \[ -\frac {a^2 \cot ^9(c+d x)}{9 d}-\frac {3 a^2 \cot ^7(c+d x)}{7 d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}-\frac {3 a^2 \tanh ^{-1}(\cos (c+d x))}{64 d}-\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{4 d}+\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{8 d}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{32 d}-\frac {3 a^2 \cot (c+d x) \csc (c+d x)}{64 d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 270
Rule 2607
Rule 2611
Rule 2873
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \cot ^4(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^2 \, dx &=\int \left (a^2 \cot ^4(c+d x) \csc ^4(c+d x)+2 a^2 \cot ^4(c+d x) \csc ^5(c+d x)+a^2 \cot ^4(c+d x) \csc ^6(c+d x)\right ) \, dx\\ &=a^2 \int \cot ^4(c+d x) \csc ^4(c+d x) \, dx+a^2 \int \cot ^4(c+d x) \csc ^6(c+d x) \, dx+\left (2 a^2\right ) \int \cot ^4(c+d x) \csc ^5(c+d x) \, dx\\ &=-\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{4 d}-\frac {1}{4} \left (3 a^2\right ) \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx+\frac {a^2 \operatorname {Subst}\left (\int x^4 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d}+\frac {a^2 \operatorname {Subst}\left (\int x^4 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{d}\\ &=\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{8 d}-\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{4 d}+\frac {1}{8} a^2 \int \csc ^5(c+d x) \, dx+\frac {a^2 \operatorname {Subst}\left (\int \left (x^4+x^6\right ) \, dx,x,-\cot (c+d x)\right )}{d}+\frac {a^2 \operatorname {Subst}\left (\int \left (x^4+2 x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac {2 a^2 \cot ^5(c+d x)}{5 d}-\frac {3 a^2 \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot ^9(c+d x)}{9 d}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{32 d}+\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{8 d}-\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{4 d}+\frac {1}{32} \left (3 a^2\right ) \int \csc ^3(c+d x) \, dx\\ &=-\frac {2 a^2 \cot ^5(c+d x)}{5 d}-\frac {3 a^2 \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot ^9(c+d x)}{9 d}-\frac {3 a^2 \cot (c+d x) \csc (c+d x)}{64 d}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{32 d}+\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{8 d}-\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{4 d}+\frac {1}{64} \left (3 a^2\right ) \int \csc (c+d x) \, dx\\ &=-\frac {3 a^2 \tanh ^{-1}(\cos (c+d x))}{64 d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}-\frac {3 a^2 \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot ^9(c+d x)}{9 d}-\frac {3 a^2 \cot (c+d x) \csc (c+d x)}{64 d}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{32 d}+\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{8 d}-\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 1.37, size = 313, normalized size = 1.86 \[ -\frac {a^2 \csc ^9(c+d x) \left (212940 \sin (2 (c+d x))+195300 \sin (4 (c+d x))+16380 \sin (6 (c+d x))-1890 \sin (8 (c+d x))+451584 \cos (c+d x)+155904 \cos (3 (c+d x))-20736 \cos (5 (c+d x))-14976 \cos (7 (c+d x))+1664 \cos (9 (c+d x))-119070 \sin (c+d x) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+79380 \sin (3 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-34020 \sin (5 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+8505 \sin (7 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-945 \sin (9 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+119070 \sin (c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-79380 \sin (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+34020 \sin (5 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-8505 \sin (7 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+945 \sin (9 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )}{5160960 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 304, normalized size = 1.81 \[ -\frac {3328 \, a^{2} \cos \left (d x + c\right )^{9} - 14976 \, a^{2} \cos \left (d x + c\right )^{7} + 16128 \, a^{2} \cos \left (d x + c\right )^{5} + 945 \, {\left (a^{2} \cos \left (d x + c\right )^{8} - 4 \, a^{2} \cos \left (d x + c\right )^{6} + 6 \, a^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 945 \, {\left (a^{2} \cos \left (d x + c\right )^{8} - 4 \, a^{2} \cos \left (d x + c\right )^{6} + 6 \, a^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 630 \, {\left (3 \, a^{2} \cos \left (d x + c\right )^{7} - 11 \, a^{2} \cos \left (d x + c\right )^{5} - 11 \, a^{2} \cos \left (d x + c\right )^{3} + 3 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{40320 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 261, normalized size = 1.55 \[ \frac {70 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 315 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 450 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1008 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2520 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 3360 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15120 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 11340 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {42774 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 11340 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 3360 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 2520 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1008 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 450 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 315 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 70 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9}}}{322560 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.41, size = 224, normalized size = 1.33 \[ -\frac {13 a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{63 d \sin \left (d x +c \right )^{7}}-\frac {26 a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{315 d \sin \left (d x +c \right )^{5}}-\frac {a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{4 d \sin \left (d x +c \right )^{8}}-\frac {a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{8 d \sin \left (d x +c \right )^{6}}-\frac {a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{32 d \sin \left (d x +c \right )^{4}}+\frac {a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{64 d \sin \left (d x +c \right )^{2}}+\frac {a^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{64 d}+\frac {3 a^{2} \cos \left (d x +c \right )}{64 d}+\frac {3 a^{2} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{64 d}-\frac {a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{9 d \sin \left (d x +c \right )^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 177, normalized size = 1.05 \[ \frac {315 \, a^{2} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{7} - 11 \, \cos \left (d x + c\right )^{5} - 11 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {1152 \, {\left (7 \, \tan \left (d x + c\right )^{2} + 5\right )} a^{2}}{\tan \left (d x + c\right )^{7}} - \frac {128 \, {\left (63 \, \tan \left (d x + c\right )^{4} + 90 \, \tan \left (d x + c\right )^{2} + 35\right )} a^{2}}{\tan \left (d x + c\right )^{9}}}{40320 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.07, size = 387, normalized size = 2.30 \[ \frac {a^2\,\left (70\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}-70\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}+315\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}-315\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+450\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-1008\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-2520\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-3360\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+11340\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-11340\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+3360\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+2520\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+1008\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-450\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+15120\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\right )}{322560\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9} \]
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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