Optimal. Leaf size=37 \[ -\frac{4 \cot ^3(x)}{15 a}-\frac{4 \cot (x)}{5 a}+\frac{\csc ^3(x)}{5 (a \cos (x)+a)} \]
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Rubi [A] time = 0.0484188, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2672, 3767} \[ -\frac{4 \cot ^3(x)}{15 a}-\frac{4 \cot (x)}{5 a}+\frac{\csc ^3(x)}{5 (a \cos (x)+a)} \]
Antiderivative was successfully verified.
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Rule 2672
Rule 3767
Rubi steps
\begin{align*} \int \frac{\csc ^4(x)}{a+a \cos (x)} \, dx &=\frac{\csc ^3(x)}{5 (a+a \cos (x))}+\frac{4 \int \csc ^4(x) \, dx}{5 a}\\ &=\frac{\csc ^3(x)}{5 (a+a \cos (x))}-\frac{4 \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (x)\right )}{5 a}\\ &=-\frac{4 \cot (x)}{5 a}-\frac{4 \cot ^3(x)}{15 a}+\frac{\csc ^3(x)}{5 (a+a \cos (x))}\\ \end{align*}
Mathematica [A] time = 0.0565487, size = 38, normalized size = 1.03 \[ \frac{(-6 \cos (x)-2 \cos (2 x)+2 \cos (3 x)+\cos (4 x)) \csc ^3(x)}{15 a (\cos (x)+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 45, normalized size = 1.2 \begin{align*}{\frac{1}{16\,a} \left ({\frac{1}{5} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{5}}+{\frac{4}{3} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{3}}+6\,\tan \left ( x/2 \right ) -4\, \left ( \tan \left ( x/2 \right ) \right ) ^{-1}-{\frac{1}{3} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.07647, size = 95, normalized size = 2.57 \begin{align*} \frac{\frac{90 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{20 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{3 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}}}{240 \, a} - \frac{{\left (\frac{12 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )}{\left (\cos \left (x\right ) + 1\right )}^{3}}{48 \, a \sin \left (x\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54182, size = 153, normalized size = 4.14 \begin{align*} -\frac{8 \, \cos \left (x\right )^{4} + 8 \, \cos \left (x\right )^{3} - 12 \, \cos \left (x\right )^{2} - 12 \, \cos \left (x\right ) + 3}{15 \,{\left (a \cos \left (x\right )^{3} + a \cos \left (x\right )^{2} - a \cos \left (x\right ) - a\right )} \sin \left (x\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 ^{4}{\left (x \right )}}{\cos{\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 = 1.14274, size = 80, normalized size = 2.16 \begin{align*} -\frac{12 \, \tan \left (\frac{1}{2} \, x\right )^{2} + 1}{48 \, a \tan \left (\frac{1}{2} \, x\right )^{3}} + \frac{3 \, a^{4} \tan \left (\frac{1}{2} \, x\right )^{5} + 20 \, a^{4} \tan \left (\frac{1}{2} \, x\right )^{3} + 90 \, a^{4} \tan \left (\frac{1}{2} \, x\right )}{240 \, a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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