Optimal. Leaf size=65 \[ -\frac {\cot ^3(x)}{3 a^2}-\frac {4 \cot (x)}{a^2}-\frac {13 \cos (x)}{3 a^2 (\sin (x)+1)}-\frac {\cos (x)}{3 a^2 (\sin (x)+1)^2}+\frac {5 \tanh ^{-1}(\cos (x))}{a^2}+\frac {\cot (x) \csc (x)}{a^2} \]
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Rubi [A] time = 0.15, antiderivative size = 71, normalized size of antiderivative = 1.09, number of steps used = 7, 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} \[ -\frac {4 \cot ^3(x)}{a^2}-\frac {12 \cot (x)}{a^2}+\frac {5 \tanh ^{-1}(\cos (x))}{a^2}+\frac {5 \cot (x) \csc (x)}{a^2}+\frac {10 \cot (x) \csc ^2(x)}{3 a^2 (\sin (x)+1)}+\frac {\cot (x) \csc ^2(x)}{3 (a \sin (x)+a)^2} \]
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))^2} \, dx &=\frac {\cot (x) \csc ^2(x)}{3 (a+a \sin (x))^2}+\frac {\int \frac {\csc ^4(x) (6 a-4 a \sin (x))}{a+a \sin (x)} \, dx}{3 a^2}\\ &=\frac {10 \cot (x) \csc ^2(x)}{3 a^2 (1+\sin (x))}+\frac {\cot (x) \csc ^2(x)}{3 (a+a \sin (x))^2}+\frac {\int \csc ^4(x) \left (36 a^2-30 a^2 \sin (x)\right ) \, dx}{3 a^4}\\ &=\frac {10 \cot (x) \csc ^2(x)}{3 a^2 (1+\sin (x))}+\frac {\cot (x) \csc ^2(x)}{3 (a+a \sin (x))^2}-\frac {10 \int \csc ^3(x) \, dx}{a^2}+\frac {12 \int \csc ^4(x) \, dx}{a^2}\\ &=\frac {5 \cot (x) \csc (x)}{a^2}+\frac {10 \cot (x) \csc ^2(x)}{3 a^2 (1+\sin (x))}+\frac {\cot (x) \csc ^2(x)}{3 (a+a \sin (x))^2}-\frac {5 \int \csc (x) \, dx}{a^2}-\frac {12 \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (x)\right )}{a^2}\\ &=\frac {5 \tanh ^{-1}(\cos (x))}{a^2}-\frac {12 \cot (x)}{a^2}-\frac {4 \cot ^3(x)}{a^2}+\frac {5 \cot (x) \csc (x)}{a^2}+\frac {10 \cot (x) \csc ^2(x)}{3 a^2 (1+\sin (x))}+\frac {\cot (x) \csc ^2(x)}{3 (a+a \sin (x))^2}\\ \end {align*}
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Mathematica [B] time = 3.59, size = 238, normalized size = 3.66 \[ \frac {\left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right ) \left (16 \sin \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )^3+208 \sin \left (\frac {x}{2}\right ) \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^2-8 \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )-6 \cos \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )^3-\cos \left (\frac {x}{2}\right ) \left (\cot \left (\frac {x}{2}\right )+1\right )^3+6 \sin \left (\frac {x}{2}\right ) \left (\cot \left (\frac {x}{2}\right )+1\right )^3+120 \log \left (\cos \left (\frac {x}{2}\right )\right ) \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^3-120 \log \left (\sin \left (\frac {x}{2}\right )\right ) \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^3+44 \tan \left (\frac {x}{2}\right ) \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^3-44 \cot \left (\frac {x}{2}\right ) \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^3\right )}{24 a^2 (\sin (x)+1)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 266, normalized size = 4.09 \[ -\frac {48 \, \cos \relax (x)^{5} - 18 \, \cos \relax (x)^{4} - 108 \, \cos \relax (x)^{3} + 22 \, \cos \relax (x)^{2} - 15 \, {\left (\cos \relax (x)^{5} + 2 \, \cos \relax (x)^{4} - 2 \, \cos \relax (x)^{3} - 4 \, \cos \relax (x)^{2} + {\left (\cos \relax (x)^{4} - \cos \relax (x)^{3} - 3 \, \cos \relax (x)^{2} + \cos \relax (x) + 2\right )} \sin \relax (x) + \cos \relax (x) + 2\right )} \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + 15 \, {\left (\cos \relax (x)^{5} + 2 \, \cos \relax (x)^{4} - 2 \, \cos \relax (x)^{3} - 4 \, \cos \relax (x)^{2} + {\left (\cos \relax (x)^{4} - \cos \relax (x)^{3} - 3 \, \cos \relax (x)^{2} + \cos \relax (x) + 2\right )} \sin \relax (x) + \cos \relax (x) + 2\right )} \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) - 2 \, {\left (24 \, \cos \relax (x)^{4} + 33 \, \cos \relax (x)^{3} - 21 \, \cos \relax (x)^{2} - 32 \, \cos \relax (x) - 1\right )} \sin \relax (x) + 62 \, \cos \relax (x) - 2}{6 \, {\left (a^{2} \cos \relax (x)^{5} + 2 \, a^{2} \cos \relax (x)^{4} - 2 \, a^{2} \cos \relax (x)^{3} - 4 \, a^{2} \cos \relax (x)^{2} + a^{2} \cos \relax (x) + 2 \, a^{2} + {\left (a^{2} \cos \relax (x)^{4} - a^{2} \cos \relax (x)^{3} - 3 \, a^{2} \cos \relax (x)^{2} + a^{2} \cos \relax (x) + 2 \, a^{2}\right )} \sin \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 114, normalized size = 1.75 \[ -\frac {5 \, \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{a^{2}} + \frac {110 \, \tan \left (\frac {1}{2} \, x\right )^{6} + 45 \, \tan \left (\frac {1}{2} \, x\right )^{5} - 231 \, \tan \left (\frac {1}{2} \, x\right )^{4} - 232 \, \tan \left (\frac {1}{2} \, x\right )^{3} - 30 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 3 \, \tan \left (\frac {1}{2} \, x\right ) - 1}{24 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} + \tan \left (\frac {1}{2} \, x\right )\right )}^{3} a^{2}} + \frac {a^{4} \tan \left (\frac {1}{2} \, x\right )^{3} - 6 \, a^{4} \tan \left (\frac {1}{2} \, x\right )^{2} + 45 \, a^{4} \tan \left (\frac {1}{2} \, x\right )}{24 \, a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 115, normalized size = 1.77 \[ \frac {\tan ^{3}\left (\frac {x}{2}\right )}{24 a^{2}}-\frac {\tan ^{2}\left (\frac {x}{2}\right )}{4 a^{2}}+\frac {15 \tan \left (\frac {x}{2}\right )}{8 a^{2}}-\frac {1}{24 a^{2} \tan \left (\frac {x}{2}\right )^{3}}+\frac {1}{4 a^{2} \tan \left (\frac {x}{2}\right )^{2}}-\frac {15}{8 a^{2} \tan \left (\frac {x}{2}\right )}-\frac {5 \ln \left (\tan \left (\frac {x}{2}\right )\right )}{a^{2}}-\frac {4}{3 a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {2}{a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {10}{a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.04, size = 178, normalized size = 2.74 \[ \frac {\frac {3 \, \sin \relax (x)}{\cos \relax (x) + 1} - \frac {30 \, \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} - \frac {342 \, \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} - \frac {561 \, \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} - \frac {285 \, \sin \relax (x)^{5}}{{\left (\cos \relax (x) + 1\right )}^{5}} - 1}{24 \, {\left (\frac {a^{2} \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} + \frac {3 \, a^{2} \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} + \frac {3 \, a^{2} \sin \relax (x)^{5}}{{\left (\cos \relax (x) + 1\right )}^{5}} + \frac {a^{2} \sin \relax (x)^{6}}{{\left (\cos \relax (x) + 1\right )}^{6}}\right )}} + \frac {\frac {45 \, \sin \relax (x)}{\cos \relax (x) + 1} - \frac {6 \, \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {\sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}}}{24 \, a^{2}} - \frac {5 \, \log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.40, size = 101, normalized size = 1.55 \[ \frac {15\,\mathrm {tan}\left (\frac {x}{2}\right )}{8\,a^2}-\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{4\,a^2}+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{24\,a^2}-\frac {5\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{a^2}-\frac {\frac {95\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5}{8}+\frac {187\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4}{8}+\frac {57\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{4}+\frac {5\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{4}-\frac {\mathrm {tan}\left (\frac {x}{2}\right )}{8}+\frac {1}{24}}{a^2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^3} \]
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 \[ \frac {\int \frac {\csc ^{4}{\relax (x )}}{\sin ^{2}{\relax (x )} + 2 \sin {\relax (x )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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