Optimal. Leaf size=55 \[ -\frac {4 \cot ^3(x)}{3 a}-\frac {4 \cot (x)}{a}+\frac {3 \tanh ^{-1}(\cos (x))}{2 a}+\frac {3 \cot (x) \csc (x)}{2 a}+\frac {\cot (x) \csc ^2(x)}{a \sin (x)+a} \]
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Rubi [A] time = 0.07, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2768, 2748, 3767, 3768, 3770} \[ -\frac {4 \cot ^3(x)}{3 a}-\frac {4 \cot (x)}{a}+\frac {3 \tanh ^{-1}(\cos (x))}{2 a}+\frac {3 \cot (x) \csc (x)}{2 a}+\frac {\cot (x) \csc ^2(x)}{a \sin (x)+a} \]
Antiderivative was successfully verified.
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Rule 2748
Rule 2768
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\csc ^4(x)}{a+a \sin (x)} \, dx &=\frac {\cot (x) \csc ^2(x)}{a+a \sin (x)}-\frac {\int \csc ^4(x) (-4 a+3 a \sin (x)) \, dx}{a^2}\\ &=\frac {\cot (x) \csc ^2(x)}{a+a \sin (x)}-\frac {3 \int \csc ^3(x) \, dx}{a}+\frac {4 \int \csc ^4(x) \, dx}{a}\\ &=\frac {3 \cot (x) \csc (x)}{2 a}+\frac {\cot (x) \csc ^2(x)}{a+a \sin (x)}-\frac {3 \int \csc (x) \, dx}{2 a}-\frac {4 \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (x)\right )}{a}\\ &=\frac {3 \tanh ^{-1}(\cos (x))}{2 a}-\frac {4 \cot (x)}{a}-\frac {4 \cot ^3(x)}{3 a}+\frac {3 \cot (x) \csc (x)}{2 a}+\frac {\cot (x) \csc ^2(x)}{a+a \sin (x)}\\ \end {align*}
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Mathematica [B] time = 0.81, size = 113, normalized size = 2.05 \[ \frac {20 \tan \left (\frac {x}{2}\right )-20 \cot \left (\frac {x}{2}\right )+3 \csc ^2\left (\frac {x}{2}\right )-3 \sec ^2\left (\frac {x}{2}\right )-36 \log \left (\sin \left (\frac {x}{2}\right )\right )+36 \log \left (\cos \left (\frac {x}{2}\right )\right )+\frac {48 \sin \left (\frac {x}{2}\right )}{\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )}-\frac {1}{2} \sin (x) \csc ^4\left (\frac {x}{2}\right )+8 \sin ^4\left (\frac {x}{2}\right ) \csc ^3(x)}{24 a} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 168, normalized size = 3.05 \[ \frac {32 \, \cos \relax (x)^{4} + 14 \, \cos \relax (x)^{3} - 48 \, \cos \relax (x)^{2} + 9 \, {\left (\cos \relax (x)^{4} - 2 \, \cos \relax (x)^{2} - {\left (\cos \relax (x)^{3} + \cos \relax (x)^{2} - \cos \relax (x) - 1\right )} \sin \relax (x) + 1\right )} \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) - 9 \, {\left (\cos \relax (x)^{4} - 2 \, \cos \relax (x)^{2} - {\left (\cos \relax (x)^{3} + \cos \relax (x)^{2} - \cos \relax (x) - 1\right )} \sin \relax (x) + 1\right )} \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + 2 \, {\left (16 \, \cos \relax (x)^{3} + 9 \, \cos \relax (x)^{2} - 15 \, \cos \relax (x) - 6\right )} \sin \relax (x) - 18 \, \cos \relax (x) + 12}{12 \, {\left (a \cos \relax (x)^{4} - 2 \, a \cos \relax (x)^{2} - {\left (a \cos \relax (x)^{3} + a \cos \relax (x)^{2} - a \cos \relax (x) - a\right )} \sin \relax (x) + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 96, normalized size = 1.75 \[ -\frac {3 \, \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{2 \, a} + \frac {a^{2} \tan \left (\frac {1}{2} \, x\right )^{3} - 3 \, a^{2} \tan \left (\frac {1}{2} \, x\right )^{2} + 21 \, a^{2} \tan \left (\frac {1}{2} \, x\right )}{24 \, a^{3}} - \frac {2}{a {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}} + \frac {66 \, \tan \left (\frac {1}{2} \, x\right )^{3} - 21 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 3 \, \tan \left (\frac {1}{2} \, x\right ) - 1}{24 \, a \tan \left (\frac {1}{2} \, x\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 89, normalized size = 1.62 \[ \frac {\tan ^{3}\left (\frac {x}{2}\right )}{24 a}-\frac {\tan ^{2}\left (\frac {x}{2}\right )}{8 a}+\frac {7 \tan \left (\frac {x}{2}\right )}{8 a}-\frac {1}{24 a \tan \left (\frac {x}{2}\right )^{3}}+\frac {1}{8 a \tan \left (\frac {x}{2}\right )^{2}}-\frac {7}{8 a \tan \left (\frac {x}{2}\right )}-\frac {3 \ln \left (\tan \left (\frac {x}{2}\right )\right )}{2 a}-\frac {2}{a \left (\tan \left (\frac {x}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.57, size = 120, normalized size = 2.18 \[ \frac {\frac {21 \, \sin \relax (x)}{\cos \relax (x) + 1} - \frac {3 \, \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {\sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}}}{24 \, a} + \frac {\frac {2 \, \sin \relax (x)}{\cos \relax (x) + 1} - \frac {18 \, \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} - \frac {69 \, \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} - 1}{24 \, {\left (\frac {a \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} + \frac {a \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}}\right )}} - \frac {3 \, \log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right )}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.50, size = 89, normalized size = 1.62 \[ \frac {7\,\mathrm {tan}\left (\frac {x}{2}\right )}{8\,a}-\frac {23\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+6\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2-\frac {2\,\mathrm {tan}\left (\frac {x}{2}\right )}{3}+\frac {1}{3}}{8\,a\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+8\,a\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3}-\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{8\,a}+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{24\,a}-\frac {3\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{2\,a} \]
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 \[ \frac {\int \frac {\csc ^{4}{\relax (x )}}{\sin {\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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