Optimal. Leaf size=60 \[ -\frac {3 \cos (c+d x)}{a^3 d}+\frac {\sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac {\tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac {7 x}{2 a^3} \]
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Rubi [A] time = 0.15, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2869, 2757, 3770, 2638, 2635, 8} \[ -\frac {3 \cos (c+d x)}{a^3 d}+\frac {\sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac {\tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac {7 x}{2 a^3} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2638
Rule 2757
Rule 2869
Rule 3770
Rubi steps
\begin {align*} \int \frac {\cos ^5(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\int \csc (c+d x) (a-a \sin (c+d x))^3 \, dx}{a^6}\\ &=\frac {\int \left (-3 a^3+a^3 \csc (c+d x)+3 a^3 \sin (c+d x)-a^3 \sin ^2(c+d x)\right ) \, dx}{a^6}\\ &=-\frac {3 x}{a^3}+\frac {\int \csc (c+d x) \, dx}{a^3}-\frac {\int \sin ^2(c+d x) \, dx}{a^3}+\frac {3 \int \sin (c+d x) \, dx}{a^3}\\ &=-\frac {3 x}{a^3}-\frac {\tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac {3 \cos (c+d x)}{a^3 d}+\frac {\cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {\int 1 \, dx}{2 a^3}\\ &=-\frac {7 x}{2 a^3}-\frac {\tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac {3 \cos (c+d x)}{a^3 d}+\frac {\cos (c+d x) \sin (c+d x)}{2 a^3 d}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 63, normalized size = 1.05 \[ \frac {\sin (2 (c+d x))-12 \cos (c+d x)-2 \left (-2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+7 c+7 d x\right )}{4 a^3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 59, normalized size = 0.98 \[ -\frac {7 \, d x - \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 6 \, \cos \left (d x + c\right ) + \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{2 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 89, normalized size = 1.48 \[ -\frac {\frac {7 \, {\left (d x + c\right )}}{a^{3}} - \frac {2 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} + \frac {2 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{3}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.62, size = 159, normalized size = 2.65 \[ -\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {6 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {6}{a^{3} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {7 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3} d}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 161, normalized size = 2.68 \[ \frac {\frac {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {6 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - 6}{a^{3} + \frac {2 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} - \frac {7 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} + \frac {\log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.44, size = 150, normalized size = 2.50 \[ \frac {7\,\mathrm {atan}\left (\frac {49}{49\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+14}-\frac {14\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{49\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+14}\right )}{a^3\,d}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^3\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+6}{d\,\left (a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+2\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^3\right )} \]
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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