Optimal. Leaf size=24 \[ -\frac{4}{\cos (x)+1}+\frac{2}{(\cos (x)+1)^2}-\log (\cos (x)+1) \]
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Rubi [A] time = 0.0502716, antiderivative size = 24, 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, 2667, 43} \[ -\frac{4}{\cos (x)+1}+\frac{2}{(\cos (x)+1)^2}-\log (\cos (x)+1) \]
Antiderivative was successfully verified.
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Rule 4392
Rule 2667
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{(\cot (x)+\csc (x))^5} \, dx &=\int \frac{\sin ^5(x)}{(1+\cos (x))^5} \, dx\\ &=-\operatorname{Subst}\left (\int \frac{(1-x)^2}{(1+x)^3} \, dx,x,\cos (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{4}{(1+x)^3}-\frac{4}{(1+x)^2}+\frac{1}{1+x}\right ) \, dx,x,\cos (x)\right )\\ &=\frac{2}{(1+\cos (x))^2}-\frac{4}{1+\cos (x)}-\log (1+\cos (x))\\ \end{align*}
Mathematica [A] time = 0.0134744, size = 32, normalized size = 1.33 \[ \frac{1}{2} \sec ^4\left (\frac{x}{2}\right )-2 \sec ^2\left (\frac{x}{2}\right )-2 \log \left (\cos \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.08, size = 25, normalized size = 1. \begin{align*} 2\, \left ( 1+\cos \left ( x \right ) \right ) ^{-2}-4\, \left ( 1+\cos \left ( x \right ) \right ) ^{-1}-\ln \left ( 1+\cos \left ( x \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.56519, size = 53, normalized size = 2.21 \begin{align*} -\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{\sin \left (x\right )^{4}}{2 \,{\left (\cos \left (x\right ) + 1\right )}^{4}} + \log \left (\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.1921, size = 126, normalized size = 5.25 \begin{align*} -\frac{{\left (\cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + 4 \, \cos \left (x\right ) + 2}{\cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1} \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 = 1.13402, size = 30, normalized size = 1.25 \begin{align*} -\frac{2 \,{\left (2 \, \cos \left (x\right ) + 1\right )}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \log \left (\cos \left (x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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