Optimal. Leaf size=103 \[ -\frac {136 \cot ^3(x)}{15 a^3}-\frac {136 \cot (x)}{5 a^3}+\frac {23 \tanh ^{-1}(\cos (x))}{2 a^3}+\frac {23 \cot (x) \csc (x)}{2 a^3}+\frac {23 \cot (x) \csc ^2(x)}{3 \left (a^3 \sin (x)+a^3\right )}+\frac {13 \cot (x) \csc ^2(x)}{15 a (a \sin (x)+a)^2}+\frac {\cot (x) \csc ^2(x)}{5 (a \sin (x)+a)^3} \]
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Rubi [A] time = 0.24, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2766, 2978, 2748, 3767, 3768, 3770} \[ -\frac {136 \cot ^3(x)}{15 a^3}-\frac {136 \cot (x)}{5 a^3}+\frac {23 \tanh ^{-1}(\cos (x))}{2 a^3}+\frac {23 \cot (x) \csc (x)}{2 a^3}+\frac {23 \cot (x) \csc ^2(x)}{3 \left (a^3 \sin (x)+a^3\right )}+\frac {13 \cot (x) \csc ^2(x)}{15 a (a \sin (x)+a)^2}+\frac {\cot (x) \csc ^2(x)}{5 (a \sin (x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 2748
Rule 2766
Rule 2978
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\csc ^4(x)}{(a+a \sin (x))^3} \, dx &=\frac {\cot (x) \csc ^2(x)}{5 (a+a \sin (x))^3}+\frac {\int \frac {\csc ^4(x) (8 a-5 a \sin (x))}{(a+a \sin (x))^2} \, dx}{5 a^2}\\ &=\frac {\cot (x) \csc ^2(x)}{5 (a+a \sin (x))^3}+\frac {13 \cot (x) \csc ^2(x)}{15 a (a+a \sin (x))^2}+\frac {\int \frac {\csc ^4(x) \left (63 a^2-52 a^2 \sin (x)\right )}{a+a \sin (x)} \, dx}{15 a^4}\\ &=\frac {\cot (x) \csc ^2(x)}{5 (a+a \sin (x))^3}+\frac {13 \cot (x) \csc ^2(x)}{15 a (a+a \sin (x))^2}+\frac {23 \cot (x) \csc ^2(x)}{3 \left (a^3+a^3 \sin (x)\right )}+\frac {\int \csc ^4(x) \left (408 a^3-345 a^3 \sin (x)\right ) \, dx}{15 a^6}\\ &=\frac {\cot (x) \csc ^2(x)}{5 (a+a \sin (x))^3}+\frac {13 \cot (x) \csc ^2(x)}{15 a (a+a \sin (x))^2}+\frac {23 \cot (x) \csc ^2(x)}{3 \left (a^3+a^3 \sin (x)\right )}-\frac {23 \int \csc ^3(x) \, dx}{a^3}+\frac {136 \int \csc ^4(x) \, dx}{5 a^3}\\ &=\frac {23 \cot (x) \csc (x)}{2 a^3}+\frac {\cot (x) \csc ^2(x)}{5 (a+a \sin (x))^3}+\frac {13 \cot (x) \csc ^2(x)}{15 a (a+a \sin (x))^2}+\frac {23 \cot (x) \csc ^2(x)}{3 \left (a^3+a^3 \sin (x)\right )}-\frac {23 \int \csc (x) \, dx}{2 a^3}-\frac {136 \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (x)\right )}{5 a^3}\\ &=\frac {23 \tanh ^{-1}(\cos (x))}{2 a^3}-\frac {136 \cot (x)}{5 a^3}-\frac {136 \cot ^3(x)}{15 a^3}+\frac {23 \cot (x) \csc (x)}{2 a^3}+\frac {\cot (x) \csc ^2(x)}{5 (a+a \sin (x))^3}+\frac {13 \cot (x) \csc ^2(x)}{15 a (a+a \sin (x))^2}+\frac {23 \cot (x) \csc ^2(x)}{3 \left (a^3+a^3 \sin (x)\right )}\\ \end {align*}
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Mathematica [B] time = 0.93, size = 299, normalized size = 2.90 \[ \frac {\left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right ) \left (48 \sin \left (\frac {x}{2}\right )-45 \cos ^3\left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )^5+2752 \sin \left (\frac {x}{2}\right ) \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^4-176 \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^3+352 \sin \left (\frac {x}{2}\right ) \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^2-24 \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )+45 \sin ^3\left (\frac {x}{2}\right ) \left (\cot \left (\frac {x}{2}\right )+1\right )^5+5 \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )^5+1380 \log \left (\cos \left (\frac {x}{2}\right )\right ) \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^5-1380 \log \left (\sin \left (\frac {x}{2}\right )\right ) \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^5+400 \tan \left (\frac {x}{2}\right ) \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^5-5 \sin ^2\left (\frac {x}{2}\right ) \cos \left (\frac {x}{2}\right ) \left (\cot \left (\frac {x}{2}\right )+1\right )^5-400 \cot \left (\frac {x}{2}\right ) \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^5\right )}{120 a^3 (\sin (x)+1)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 333, normalized size = 3.23 \[ \frac {1088 \, \cos \relax (x)^{6} + 2574 \, \cos \relax (x)^{5} - 2428 \, \cos \relax (x)^{4} - 5338 \, \cos \relax (x)^{3} + 1372 \, \cos \relax (x)^{2} + 345 \, {\left (\cos \relax (x)^{6} - 2 \, \cos \relax (x)^{5} - 6 \, \cos \relax (x)^{4} + 4 \, \cos \relax (x)^{3} + 9 \, \cos \relax (x)^{2} - {\left (\cos \relax (x)^{5} + 3 \, \cos \relax (x)^{4} - 3 \, \cos \relax (x)^{3} - 7 \, \cos \relax (x)^{2} + 2 \, \cos \relax (x) + 4\right )} \sin \relax (x) - 2 \, \cos \relax (x) - 4\right )} \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) - 345 \, {\left (\cos \relax (x)^{6} - 2 \, \cos \relax (x)^{5} - 6 \, \cos \relax (x)^{4} + 4 \, \cos \relax (x)^{3} + 9 \, \cos \relax (x)^{2} - {\left (\cos \relax (x)^{5} + 3 \, \cos \relax (x)^{4} - 3 \, \cos \relax (x)^{3} - 7 \, \cos \relax (x)^{2} + 2 \, \cos \relax (x) + 4\right )} \sin \relax (x) - 2 \, \cos \relax (x) - 4\right )} \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + 2 \, {\left (544 \, \cos \relax (x)^{5} - 743 \, \cos \relax (x)^{4} - 1957 \, \cos \relax (x)^{3} + 712 \, \cos \relax (x)^{2} + 1398 \, \cos \relax (x) + 6\right )} \sin \relax (x) + 2784 \, \cos \relax (x) - 12}{60 \, {\left (a^{3} \cos \relax (x)^{6} - 2 \, a^{3} \cos \relax (x)^{5} - 6 \, a^{3} \cos \relax (x)^{4} + 4 \, a^{3} \cos \relax (x)^{3} + 9 \, a^{3} \cos \relax (x)^{2} - 2 \, a^{3} \cos \relax (x) - 4 \, a^{3} - {\left (a^{3} \cos \relax (x)^{5} + 3 \, a^{3} \cos \relax (x)^{4} - 3 \, a^{3} \cos \relax (x)^{3} - 7 \, a^{3} \cos \relax (x)^{2} + 2 \, a^{3} \cos \relax (x) + 4 \, a^{3}\right )} \sin \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 128, normalized size = 1.24 \[ -\frac {23 \, \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{2 \, a^{3}} + \frac {506 \, \tan \left (\frac {1}{2} \, x\right )^{3} - 81 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 9 \, \tan \left (\frac {1}{2} \, x\right ) - 1}{24 \, a^{3} \tan \left (\frac {1}{2} \, x\right )^{3}} - \frac {2 \, {\left (225 \, \tan \left (\frac {1}{2} \, x\right )^{4} + 810 \, \tan \left (\frac {1}{2} \, x\right )^{3} + 1160 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 760 \, \tan \left (\frac {1}{2} \, x\right ) + 197\right )}}{15 \, a^{3} {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{5}} + \frac {a^{6} \tan \left (\frac {1}{2} \, x\right )^{3} - 9 \, a^{6} \tan \left (\frac {1}{2} \, x\right )^{2} + 81 \, a^{6} \tan \left (\frac {1}{2} \, x\right )}{24 \, a^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 141, normalized size = 1.37 \[ \frac {\tan ^{3}\left (\frac {x}{2}\right )}{24 a^{3}}-\frac {3 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{8 a^{3}}+\frac {27 \tan \left (\frac {x}{2}\right )}{8 a^{3}}-\frac {1}{24 a^{3} \tan \left (\frac {x}{2}\right )^{3}}+\frac {3}{8 a^{3} \tan \left (\frac {x}{2}\right )^{2}}-\frac {27}{8 a^{3} \tan \left (\frac {x}{2}\right )}-\frac {23 \ln \left (\tan \left (\frac {x}{2}\right )\right )}{2 a^{3}}-\frac {8}{5 a^{3} \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}+\frac {4}{a^{3} \left (\tan \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {32}{3 a^{3} \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {12}{a^{3} \left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {30}{a^{3} \left (\tan \left (\frac {x}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.67, size = 232, normalized size = 2.25 \[ \frac {\frac {20 \, \sin \relax (x)}{\cos \relax (x) + 1} - \frac {230 \, \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} - \frac {4777 \, \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} - \frac {15785 \, \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} - \frac {22390 \, \sin \relax (x)^{5}}{{\left (\cos \relax (x) + 1\right )}^{5}} - \frac {14940 \, \sin \relax (x)^{6}}{{\left (\cos \relax (x) + 1\right )}^{6}} - \frac {4005 \, \sin \relax (x)^{7}}{{\left (\cos \relax (x) + 1\right )}^{7}} - 5}{120 \, {\left (\frac {a^{3} \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} + \frac {10 \, a^{3} \sin \relax (x)^{5}}{{\left (\cos \relax (x) + 1\right )}^{5}} + \frac {10 \, a^{3} \sin \relax (x)^{6}}{{\left (\cos \relax (x) + 1\right )}^{6}} + \frac {5 \, a^{3} \sin \relax (x)^{7}}{{\left (\cos \relax (x) + 1\right )}^{7}} + \frac {a^{3} \sin \relax (x)^{8}}{{\left (\cos \relax (x) + 1\right )}^{8}}\right )}} + \frac {\frac {81 \, \sin \relax (x)}{\cos \relax (x) + 1} - \frac {9 \, \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {\sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}}}{24 \, a^{3}} - \frac {23 \, \log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right )}{2 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.69, size = 117, normalized size = 1.14 \[ \frac {27\,\mathrm {tan}\left (\frac {x}{2}\right )}{8\,a^3}-\frac {3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{8\,a^3}+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{24\,a^3}-\frac {23\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{2\,a^3}-\frac {\frac {267\,{\mathrm {tan}\left (\frac {x}{2}\right )}^7}{8}+\frac {249\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6}{2}+\frac {2239\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5}{12}+\frac {3157\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4}{24}+\frac {4777\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{120}+\frac {23\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{12}-\frac {\mathrm {tan}\left (\frac {x}{2}\right )}{6}+\frac {1}{24}}{a^3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^5} \]
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 \[ \frac {\int \frac {\csc ^{4}{\relax (x )}}{\sin ^{3}{\relax (x )} + 3 \sin ^{2}{\relax (x )} + 3 \sin {\relax (x )} + 1}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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