Optimal. Leaf size=54 \[ -\frac {\cot (c+d x)}{a^3 d}-\frac {4 \cos (c+d x)}{a^3 d (\sin (c+d x)+1)}+\frac {3 \tanh ^{-1}(\cos (c+d x))}{a^3 d} \]
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Rubi [A] time = 0.24, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2875, 2872, 3770, 3767, 8, 2648} \[ -\frac {\cot (c+d x)}{a^3 d}-\frac {4 \cos (c+d x)}{a^3 d (\sin (c+d x)+1)}+\frac {3 \tanh ^{-1}(\cos (c+d x))}{a^3 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2648
Rule 2872
Rule 2875
Rule 3767
Rule 3770
Rubi steps
\begin {align*} \int \frac {\cos ^2(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\int \csc ^2(c+d x) \sec ^2(c+d x) (a-a \sin (c+d x))^3 \, dx}{a^6}\\ &=\frac {\int \left (-3 a \csc (c+d x)+a \csc ^2(c+d x)+\frac {4 a}{1+\sin (c+d x)}\right ) \, dx}{a^4}\\ &=\frac {\int \csc ^2(c+d x) \, dx}{a^3}-\frac {3 \int \csc (c+d x) \, dx}{a^3}+\frac {4 \int \frac {1}{1+\sin (c+d x)} \, dx}{a^3}\\ &=\frac {3 \tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac {4 \cos (c+d x)}{a^3 d (1+\sin (c+d x))}-\frac {\operatorname {Subst}(\int 1 \, dx,x,\cot (c+d x))}{a^3 d}\\ &=\frac {3 \tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac {\cot (c+d x)}{a^3 d}-\frac {4 \cos (c+d x)}{a^3 d (1+\sin (c+d x))}\\ \end {align*}
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Mathematica [B] time = 0.64, size = 156, normalized size = 2.89 \[ -\frac {\tan \left (\frac {1}{2} (c+d x)\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^5 \left (\cos \left (\frac {1}{2} (c+d x)\right ) \left (\cot ^2\left (\frac {1}{2} (c+d x)\right )+6 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-6 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+\cot \left (\frac {1}{2} (c+d x)\right ) \left (6 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-6 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+1\right )-17\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^3 d (\sin (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 165, normalized size = 3.06 \[ \frac {10 \, \cos \left (d x + c\right )^{2} + 3 \, {\left (\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right ) - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 3 \, {\left (\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right ) - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2 \, {\left (5 \, \cos \left (d x + c\right ) + 4\right )} \sin \left (d x + c\right ) + 2 \, \cos \left (d x + c\right ) - 8}{2 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d - {\left (a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 90, normalized size = 1.67 \[ -\frac {\frac {6 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac {\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{3}} - \frac {3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 14 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} a^{3}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.59, size = 77, normalized size = 1.43 \[ \frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d \,a^{3}}-\frac {1}{2 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3}}-\frac {8}{d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.46, size = 116, normalized size = 2.15 \[ -\frac {\frac {\frac {17 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1}{\frac {a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}} + \frac {6 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} - \frac {\sin \left (d x + c\right )}{a^{3} {\left (\cos \left (d x + c\right ) + 1\right )}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.67, size = 87, normalized size = 1.61 \[ \frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a^3\,d}-\frac {17\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1}{d\,\left (2\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}-\frac {3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^3\,d} \]
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 \[ \frac {\int \frac {\cos ^{4}{\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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