Optimal. Leaf size=100 \[ -\frac {\cot ^5(c+d x)}{5 a^2 d}-\frac {2 \cot ^3(c+d x)}{3 a^2 d}-\frac {\tanh ^{-1}(\cos (c+d x))}{4 a^2 d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{2 a^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{4 a^2 d} \]
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Rubi [A] time = 0.15, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2709, 3767, 8, 3768, 3770} \[ -\frac {\cot ^5(c+d x)}{5 a^2 d}-\frac {2 \cot ^3(c+d x)}{3 a^2 d}-\frac {\tanh ^{-1}(\cos (c+d x))}{4 a^2 d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{2 a^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{4 a^2 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2709
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\cot ^6(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\int \left (-a^4 \csc ^2(c+d x)+2 a^4 \csc ^3(c+d x)-2 a^4 \csc ^5(c+d x)+a^4 \csc ^6(c+d x)\right ) \, dx}{a^6}\\ &=-\frac {\int \csc ^2(c+d x) \, dx}{a^2}+\frac {\int \csc ^6(c+d x) \, dx}{a^2}+\frac {2 \int \csc ^3(c+d x) \, dx}{a^2}-\frac {2 \int \csc ^5(c+d x) \, dx}{a^2}\\ &=-\frac {\cot (c+d x) \csc (c+d x)}{a^2 d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{2 a^2 d}+\frac {\int \csc (c+d x) \, dx}{a^2}-\frac {3 \int \csc ^3(c+d x) \, dx}{2 a^2}+\frac {\operatorname {Subst}(\int 1 \, dx,x,\cot (c+d x))}{a^2 d}-\frac {\operatorname {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{a^2 d}\\ &=-\frac {\tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac {2 \cot ^3(c+d x)}{3 a^2 d}-\frac {\cot ^5(c+d x)}{5 a^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{4 a^2 d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{2 a^2 d}-\frac {3 \int \csc (c+d x) \, dx}{4 a^2}\\ &=-\frac {\tanh ^{-1}(\cos (c+d x))}{4 a^2 d}-\frac {2 \cot ^3(c+d x)}{3 a^2 d}-\frac {\cot ^5(c+d x)}{5 a^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{4 a^2 d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{2 a^2 d}\\ \end {align*}
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Mathematica [A] time = 0.56, size = 189, normalized size = 1.89 \[ -\frac {\csc ^5(c+d x) \left (-180 \sin (2 (c+d x))-30 \sin (4 (c+d x))+200 \cos (c+d x)+20 \cos (3 (c+d x))-28 \cos (5 (c+d x))-150 \sin (c+d x) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+75 \sin (3 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-15 \sin (5 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+150 \sin (c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-75 \sin (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+15 \sin (5 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )}{960 a^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 167, normalized size = 1.67 \[ \frac {56 \, \cos \left (d x + c\right )^{5} - 80 \, \cos \left (d x + c\right )^{3} - 15 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 15 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 30 \, {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} - 2 \, a^{2} d \cos \left (d x + c\right )^{2} + a^{2} d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 157, normalized size = 1.57 \[ \frac {\frac {120 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac {274 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 90 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 25 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}} + \frac {3 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 25 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 90 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{10}}}{480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.66, size = 170, normalized size = 1.70 \[ \frac {\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )}{160 a^{2} d}-\frac {\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )}{32 a^{2} d}+\frac {5 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d \,a^{2}}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d \,a^{2}}+\frac {3}{16 d \,a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{2}}-\frac {1}{160 a^{2} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {1}{32 a^{2} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {5}{96 a^{2} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 195, normalized size = 1.95 \[ -\frac {\frac {\frac {90 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {25 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{2}} - \frac {120 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} - \frac {{\left (\frac {15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {25 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {90 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - 3\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{5}}{a^{2} \sin \left (d x + c\right )^{5}}}{480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.02, size = 149, normalized size = 1.49 \[ \frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96\,a^2\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{32\,a^2\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,a^2\,d}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{4\,a^2\,d}-\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,a^2\,d}+\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {1}{5}\right )}{32\,a^2\,d} \]
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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