Optimal. Leaf size=114 \[ -\frac {\cot ^5(c+d x)}{5 a^3 d}-\frac {5 \cot ^3(c+d x)}{3 a^3 d}-\frac {4 \cot (c+d x)}{a^3 d}+\frac {13 \tanh ^{-1}(\cos (c+d x))}{8 a^3 d}+\frac {3 \cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}+\frac {13 \cot (c+d x) \csc (c+d x)}{8 a^3 d} \]
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Rubi [A] time = 0.18, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2708, 2757, 3768, 3770, 3767} \[ -\frac {\cot ^5(c+d x)}{5 a^3 d}-\frac {5 \cot ^3(c+d x)}{3 a^3 d}-\frac {4 \cot (c+d x)}{a^3 d}+\frac {13 \tanh ^{-1}(\cos (c+d x))}{8 a^3 d}+\frac {3 \cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}+\frac {13 \cot (c+d x) \csc (c+d x)}{8 a^3 d} \]
Antiderivative was successfully verified.
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Rule 2708
Rule 2757
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\cot ^6(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\int \csc ^6(c+d x) (a-a \sin (c+d x))^3 \, dx}{a^6}\\ &=\frac {\int \left (-a^3 \csc ^3(c+d x)+3 a^3 \csc ^4(c+d x)-3 a^3 \csc ^5(c+d x)+a^3 \csc ^6(c+d x)\right ) \, dx}{a^6}\\ &=-\frac {\int \csc ^3(c+d x) \, dx}{a^3}+\frac {\int \csc ^6(c+d x) \, dx}{a^3}+\frac {3 \int \csc ^4(c+d x) \, dx}{a^3}-\frac {3 \int \csc ^5(c+d x) \, dx}{a^3}\\ &=\frac {\cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac {3 \cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}-\frac {\int \csc (c+d x) \, dx}{2 a^3}-\frac {9 \int \csc ^3(c+d x) \, dx}{4 a^3}-\frac {\operatorname {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{a^3 d}-\frac {3 \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{a^3 d}\\ &=\frac {\tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac {4 \cot (c+d x)}{a^3 d}-\frac {5 \cot ^3(c+d x)}{3 a^3 d}-\frac {\cot ^5(c+d x)}{5 a^3 d}+\frac {13 \cot (c+d x) \csc (c+d x)}{8 a^3 d}+\frac {3 \cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}-\frac {9 \int \csc (c+d x) \, dx}{8 a^3}\\ &=\frac {13 \tanh ^{-1}(\cos (c+d x))}{8 a^3 d}-\frac {4 \cot (c+d x)}{a^3 d}-\frac {5 \cot ^3(c+d x)}{3 a^3 d}-\frac {\cot ^5(c+d x)}{5 a^3 d}+\frac {13 \cot (c+d x) \csc (c+d x)}{8 a^3 d}+\frac {3 \cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}\\ \end {align*}
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Mathematica [A] time = 1.80, size = 189, normalized size = 1.66 \[ \frac {\csc ^5(c+d x) \left (1500 \sin (2 (c+d x))-390 \sin (4 (c+d x))-1600 \cos (c+d x)+1520 \cos (3 (c+d x))-304 \cos (5 (c+d x))-1950 \sin (c+d x) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+975 \sin (3 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-195 \sin (5 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+1950 \sin (c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-975 \sin (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+195 \sin (5 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )}{1920 a^3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 179, normalized size = 1.57 \[ -\frac {608 \, \cos \left (d x + c\right )^{5} - 1520 \, \cos \left (d x + c\right )^{3} - 195 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 195 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 30 \, {\left (13 \, \cos \left (d x + c\right )^{3} - 19 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 960 \, \cos \left (d x + c\right )}{240 \, {\left (a^{3} d \cos \left (d x + c\right )^{4} - 2 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 187, normalized size = 1.64 \[ -\frac {\frac {1560 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac {3562 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1380 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 480 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 170 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 45 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}} - \frac {6 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 45 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 170 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 480 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1380 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{15}}}{960 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.68, size = 208, normalized size = 1.82 \[ \frac {\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )}{160 d \,a^{3}}-\frac {3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a^{3} d}+\frac {17 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d \,a^{3}}-\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{3} d}+\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d \,a^{3}}-\frac {23}{16 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {13 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{3} d}-\frac {1}{160 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {1}{2 a^{3} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {3}{64 a^{3} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {17}{96 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 234, normalized size = 2.05 \[ \frac {\frac {\frac {1380 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {480 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {170 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {45 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {6 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {1560 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} + \frac {{\left (\frac {45 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {170 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {480 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {1380 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - 6\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{5}}{a^{3} \sin \left (d x + c\right )^{5}}}{960 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.85, size = 291, normalized size = 2.55 \[ -\frac {6\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+45\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-45\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-170\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+480\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-1380\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+1380\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-480\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+170\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1560\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{960\,a^3\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5} \]
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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