Optimal. Leaf size=36 \[ -\frac {\cos (c+d x)}{a^2 d}-\frac {\tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac {2 x}{a^2} \]
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Rubi [A] time = 0.13, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2869, 2746, 2735, 3770} \[ -\frac {\cos (c+d x)}{a^2 d}-\frac {\tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac {2 x}{a^2} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2746
Rule 2869
Rule 3770
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\int \csc (c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4}\\ &=-\frac {\cos (c+d x)}{a^2 d}+\frac {\int \csc (c+d x) \left (a^2-2 a^2 \sin (c+d x)\right ) \, dx}{a^4}\\ &=-\frac {2 x}{a^2}-\frac {\cos (c+d x)}{a^2 d}+\frac {\int \csc (c+d x) \, dx}{a^2}\\ &=-\frac {2 x}{a^2}-\frac {\tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac {\cos (c+d x)}{a^2 d}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 46, normalized size = 1.28 \[ -\frac {\cos (c+d x)-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+2 c+2 d x}{a^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 45, normalized size = 1.25 \[ -\frac {4 \, d x + 2 \, \cos \left (d x + c\right ) + \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{2 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 52, normalized size = 1.44 \[ -\frac {\frac {2 \, {\left (d x + c\right )}}{a^{2}} - \frac {\log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{2}} + \frac {2}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.51, size = 60, normalized size = 1.67 \[ -\frac {2}{a^{2} d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {4 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2}}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 82, normalized size = 2.28 \[ -\frac {\frac {2}{a^{2} + \frac {a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}} + \frac {4 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} - \frac {\log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.72, size = 97, normalized size = 2.69 \[ \frac {4\,\mathrm {atan}\left (\frac {16}{16\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+8}-\frac {8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+8}\right )}{a^2\,d}-\frac {2}{d\,\left (a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^2\right )}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\cos ^{4}{\left (c + d x \right )} \csc {\left (c + d x \right )}}{\sin ^{2}{\left (c + d x \right )} + 2 \sin {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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