Optimal. Leaf size=26 \[ x+\frac{2 \sin ^3(x)}{3 (\cos (x)+1)^3}-\frac{2 \sin (x)}{\cos (x)+1} \]
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Rubi [A] time = 0.0711771, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {4392, 2680, 8} \[ x+\frac{2 \sin ^3(x)}{3 (\cos (x)+1)^3}-\frac{2 \sin (x)}{\cos (x)+1} \]
Antiderivative was successfully verified.
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Rule 4392
Rule 2680
Rule 8
Rubi steps
\begin{align*} \int \frac{1}{(\cot (x)+\csc (x))^4} \, dx &=\int \frac{\sin ^4(x)}{(1+\cos (x))^4} \, dx\\ &=\frac{2 \sin ^3(x)}{3 (1+\cos (x))^3}-\int \frac{\sin ^2(x)}{(1+\cos (x))^2} \, dx\\ &=-\frac{2 \sin (x)}{1+\cos (x)}+\frac{2 \sin ^3(x)}{3 (1+\cos (x))^3}+\int 1 \, dx\\ &=x-\frac{2 \sin (x)}{1+\cos (x)}+\frac{2 \sin ^3(x)}{3 (1+\cos (x))^3}\\ \end{align*}
Mathematica [A] time = 0.0140729, size = 30, normalized size = 1.15 \[ x-\frac{8}{3} \tan \left (\frac{x}{2}\right )+\frac{2}{3} \tan \left (\frac{x}{2}\right ) \sec ^2\left (\frac{x}{2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.073, size = 17, normalized size = 0.7 \begin{align*}{\frac{2}{3} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{3}}-2\,\tan \left ( x/2 \right ) +x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47989, size = 47, normalized size = 1.81 \begin{align*} -\frac{2 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{2 \, \sin \left (x\right )^{3}}{3 \,{\left (\cos \left (x\right ) + 1\right )}^{3}} + 2 \, \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.93125, size = 123, normalized size = 4.73 \begin{align*} \frac{3 \, x \cos \left (x\right )^{2} + 6 \, x \cos \left (x\right ) - 4 \,{\left (2 \, \cos \left (x\right ) + 1\right )} \sin \left (x\right ) + 3 \, x}{3 \,{\left (\cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21487, size = 22, normalized size = 0.85 \begin{align*} \frac{2}{3} \, \tan \left (\frac{1}{2} \, x\right )^{3} + x - 2 \, \tan \left (\frac{1}{2} \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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