Optimal. Leaf size=42 \[ \frac{2 \cot (x)}{a}-\frac{3 \tanh ^{-1}(\cos (x))}{2 a}-\frac{3 \cot (x) \csc (x)}{2 a}+\frac{\cot (x) \csc (x)}{a \sin (x)+a} \]
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Rubi [A] time = 0.0663447, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462, Rules used = {2768, 2748, 3768, 3770, 3767, 8} \[ \frac{2 \cot (x)}{a}-\frac{3 \tanh ^{-1}(\cos (x))}{2 a}-\frac{3 \cot (x) \csc (x)}{2 a}+\frac{\cot (x) \csc (x)}{a \sin (x)+a} \]
Antiderivative was successfully verified.
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Rule 2768
Rule 2748
Rule 3768
Rule 3770
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \frac{\csc ^3(x)}{a+a \sin (x)} \, dx &=\frac{\cot (x) \csc (x)}{a+a \sin (x)}-\frac{\int \csc ^3(x) (-3 a+2 a \sin (x)) \, dx}{a^2}\\ &=\frac{\cot (x) \csc (x)}{a+a \sin (x)}-\frac{2 \int \csc ^2(x) \, dx}{a}+\frac{3 \int \csc ^3(x) \, dx}{a}\\ &=-\frac{3 \cot (x) \csc (x)}{2 a}+\frac{\cot (x) \csc (x)}{a+a \sin (x)}+\frac{3 \int \csc (x) \, dx}{2 a}+\frac{2 \operatorname{Subst}(\int 1 \, dx,x,\cot (x))}{a}\\ &=-\frac{3 \tanh ^{-1}(\cos (x))}{2 a}+\frac{2 \cot (x)}{a}-\frac{3 \cot (x) \csc (x)}{2 a}+\frac{\cot (x) \csc (x)}{a+a \sin (x)}\\ \end{align*}
Mathematica [A] time = 0.308347, size = 83, normalized size = 1.98 \[ \frac{-4 \tan \left (\frac{x}{2}\right )+4 \cot \left (\frac{x}{2}\right )-\csc ^2\left (\frac{x}{2}\right )+\sec ^2\left (\frac{x}{2}\right )+12 \log \left (\sin \left (\frac{x}{2}\right )\right )-12 \log \left (\cos \left (\frac{x}{2}\right )\right )-\frac{16 \sin \left (\frac{x}{2}\right )}{\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )}}{8 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 67, normalized size = 1.6 \begin{align*}{\frac{1}{8\,a} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}}-{\frac{1}{2\,a}\tan \left ({\frac{x}{2}} \right ) }+2\,{\frac{1}{a \left ( \tan \left ( x/2 \right ) +1 \right ) }}-{\frac{1}{8\,a} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-2}}+{\frac{1}{2\,a} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-1}}+{\frac{3}{2\,a}\ln \left ( \tan \left ({\frac{x}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.73902, size = 131, normalized size = 3.12 \begin{align*} -\frac{\frac{4 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}}{8 \, a} + \frac{\frac{3 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{20 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - 1}{8 \,{\left (\frac{a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{a \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}}\right )}} + \frac{3 \, \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.76625, size = 433, normalized size = 10.31 \begin{align*} \frac{8 \, \cos \left (x\right )^{3} + 6 \, \cos \left (x\right )^{2} - 3 \,{\left (\cos \left (x\right )^{3} + \cos \left (x\right )^{2} +{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right ) - \cos \left (x\right ) - 1\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + 3 \,{\left (\cos \left (x\right )^{3} + \cos \left (x\right )^{2} +{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right ) - \cos \left (x\right ) - 1\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) - 2 \,{\left (4 \, \cos \left (x\right )^{2} + \cos \left (x\right ) - 2\right )} \sin \left (x\right ) - 6 \, \cos \left (x\right ) - 4}{4 \,{\left (a \cos \left (x\right )^{3} + a \cos \left (x\right )^{2} - a \cos \left (x\right ) +{\left (a \cos \left (x\right )^{2} - a\right )} \sin \left (x\right ) - a\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\csc ^{3}{\left (x \right )}}{\sin{\left (x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.12281, size = 99, normalized size = 2.36 \begin{align*} \frac{3 \, \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) \right |}\right )}{2 \, a} + \frac{a \tan \left (\frac{1}{2} \, x\right )^{2} - 4 \, a \tan \left (\frac{1}{2} \, x\right )}{8 \, a^{2}} + \frac{2}{a{\left (\tan \left (\frac{1}{2} \, x\right ) + 1\right )}} - \frac{18 \, \tan \left (\frac{1}{2} \, x\right )^{2} - 4 \, \tan \left (\frac{1}{2} \, x\right ) + 1}{8 \, a \tan \left (\frac{1}{2} \, x\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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