Optimal. Leaf size=106 \[ \frac {3 \cot (c+d x)}{a^3 d}+\frac {17 \cos (c+d x)}{3 a^3 d (\sin (c+d x)+1)}+\frac {2 \cos (c+d x)}{3 a^3 d (\sin (c+d x)+1)^2}-\frac {11 \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a^3 d} \]
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Rubi [A] time = 0.27, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2874, 2966, 3770, 3767, 8, 3768, 2650, 2648} \[ \frac {3 \cot (c+d x)}{a^3 d}+\frac {17 \cos (c+d x)}{3 a^3 d (\sin (c+d x)+1)}+\frac {2 \cos (c+d x)}{3 a^3 d (\sin (c+d x)+1)^2}-\frac {11 \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a^3 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2648
Rule 2650
Rule 2874
Rule 2966
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\cot ^2(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\int \frac {\csc ^3(c+d x) (a-a \sin (c+d x))}{(a+a \sin (c+d x))^2} \, dx}{a^2}\\ &=\frac {\int \left (\frac {5 \csc (c+d x)}{a}-\frac {3 \csc ^2(c+d x)}{a}+\frac {\csc ^3(c+d x)}{a}-\frac {2}{a (1+\sin (c+d x))^2}-\frac {5}{a (1+\sin (c+d x))}\right ) \, dx}{a^2}\\ &=\frac {\int \csc ^3(c+d x) \, dx}{a^3}-\frac {2 \int \frac {1}{(1+\sin (c+d x))^2} \, dx}{a^3}-\frac {3 \int \csc ^2(c+d x) \, dx}{a^3}+\frac {5 \int \csc (c+d x) \, dx}{a^3}-\frac {5 \int \frac {1}{1+\sin (c+d x)} \, dx}{a^3}\\ &=-\frac {5 \tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac {2 \cos (c+d x)}{3 a^3 d (1+\sin (c+d x))^2}+\frac {5 \cos (c+d x)}{a^3 d (1+\sin (c+d x))}+\frac {\int \csc (c+d x) \, dx}{2 a^3}-\frac {2 \int \frac {1}{1+\sin (c+d x)} \, dx}{3 a^3}+\frac {3 \operatorname {Subst}(\int 1 \, dx,x,\cot (c+d x))}{a^3 d}\\ &=-\frac {11 \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}+\frac {3 \cot (c+d x)}{a^3 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac {2 \cos (c+d x)}{3 a^3 d (1+\sin (c+d x))^2}+\frac {17 \cos (c+d x)}{3 a^3 d (1+\sin (c+d x))}\\ \end {align*}
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Mathematica [B] time = 5.95, size = 308, normalized size = 2.91 \[ \frac {\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^3 \left (-32 \sin \left (\frac {1}{2} (c+d x)\right )-272 \sin \left (\frac {1}{2} (c+d x)\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2+16 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+3 \cos \left (\frac {1}{2} (c+d x)\right ) \left (\tan \left (\frac {1}{2} (c+d x)\right )+1\right )^3-3 \sin \left (\frac {1}{2} (c+d x)\right ) \left (\cot \left (\frac {1}{2} (c+d x)\right )+1\right )^3-132 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^3+132 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^3-36 \tan \left (\frac {1}{2} (c+d x)\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^3+36 \cot \left (\frac {1}{2} (c+d x)\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^3\right )}{24 a^3 d (\sin (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 365, normalized size = 3.44 \[ -\frac {104 \, \cos \left (d x + c\right )^{4} + 142 \, \cos \left (d x + c\right )^{3} - 90 \, \cos \left (d x + c\right )^{2} + 33 \, {\left (\cos \left (d x + c\right )^{4} - \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{3} + 2 \, \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) + \cos \left (d x + c\right ) + 2\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 33 \, {\left (\cos \left (d x + c\right )^{4} - \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{3} + 2 \, \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) + \cos \left (d x + c\right ) + 2\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2 \, {\left (52 \, \cos \left (d x + c\right )^{3} - 19 \, \cos \left (d x + c\right )^{2} - 64 \, \cos \left (d x + c\right ) + 4\right )} \sin \left (d x + c\right ) - 136 \, \cos \left (d x + c\right ) - 8}{12 \, {\left (a^{3} d \cos \left (d x + c\right )^{4} - a^{3} d \cos \left (d x + c\right )^{3} - 3 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d \cos \left (d x + c\right ) + 2 \, a^{3} d - {\left (a^{3} d \cos \left (d x + c\right )^{3} + 2 \, a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d \cos \left (d x + c\right ) - 2 \, a^{3} d\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 143, normalized size = 1.35 \[ \frac {\frac {132 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac {3 \, {\left (66 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} + \frac {3 \, {\left (a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a^{6}} + \frac {16 \, {\left (21 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 36 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 19\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{3}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.69, size = 157, normalized size = 1.48 \[ \frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{3}}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d \,a^{3}}-\frac {1}{8 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {3}{2 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {11 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{3}}+\frac {8}{3 d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {4}{d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {14}{d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 247, normalized size = 2.33 \[ \frac {\frac {\frac {27 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {403 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {681 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {372 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - 3}{\frac {a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}} - \frac {3 \, {\left (\frac {12 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}}{a^{3}} + \frac {132 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.66, size = 178, normalized size = 1.68 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a^3\,d}+\frac {11\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,a^3\,d}+\frac {62\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {227\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}+\frac {403\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{6}+\frac {9\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}-\frac {1}{2}}{d\,\left (4\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+12\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+12\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+4\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}-\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a^3\,d} \]
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 \[ \frac {\int \frac {\cos ^{2}{\left (c + d x \right )} \csc ^{3}{\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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