Optimal. Leaf size=20 \[ -\frac{2}{1-\cos (x)}-\log (1-\cos (x)) \]
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Rubi [A] time = 0.0480181, antiderivative size = 20, 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} \[ -\frac{2}{1-\cos (x)}-\log (1-\cos (x)) \]
Antiderivative was successfully verified.
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Rule 4392
Rule 2667
Rule 43
Rubi steps
\begin{align*} \int (\cot (x)+\csc (x))^3 \, dx &=\int (1+\cos (x))^3 \csc ^3(x) \, dx\\ &=-\operatorname{Subst}\left (\int \frac{1+x}{(1-x)^2} \, dx,x,\cos (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{2}{(-1+x)^2}+\frac{1}{-1+x}\right ) \, dx,x,\cos (x)\right )\\ &=-\frac{2}{1-\cos (x)}-\log (1-\cos (x))\\ \end{align*}
Mathematica [A] time = 0.0394034, size = 20, normalized size = 1. \[ -\csc ^2\left (\frac{x}{2}\right )-2 \log \left (\sin \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.037, size = 49, normalized size = 2.5 \begin{align*} -{\frac{ \left ( \cot \left ( x \right ) \right ) ^{2}}{2}}-\ln \left ( \sin \left ( x \right ) \right ) -{\frac{3\, \left ( \cos \left ( x \right ) \right ) ^{3}}{2\, \left ( \sin \left ( x \right ) \right ) ^{2}}}-{\frac{3\,\cos \left ( x \right ) }{2}}-\ln \left ( \csc \left ( x \right ) -\cot \left ( x \right ) \right ) -{\frac{3}{2\, \left ( \sin \left ( x \right ) \right ) ^{2}}}-{\frac{\cot \left ( x \right ) \csc \left ( x \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.95826, size = 62, normalized size = 3.1 \begin{align*} -\frac{3}{2} \, \cot \left (x\right )^{2} + \frac{2 \, \cos \left (x\right )}{\cos \left (x\right )^{2} - 1} - \frac{1}{2 \, \sin \left (x\right )^{2}} - \frac{1}{2} \, \log \left (\sin \left (x\right )^{2}\right ) + \frac{1}{2} \, \log \left (\cos \left (x\right ) + 1\right ) - \frac{1}{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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Fricas [A] time = 2.05848, size = 77, normalized size = 3.85 \begin{align*} -\frac{{\left (\cos \left (x\right ) - 1\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) - 2}{\cos \left (x\right ) - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 82.1944, size = 44, normalized size = 2.2 \begin{align*} - \frac{\log{\left (\cos{\left (x \right )} - 1 \right )}}{2} + \frac{\log{\left (\cos{\left (x \right )} + 1 \right )}}{2} + \frac{\log{\left (\csc ^{2}{\left (x \right )} \right )}}{2} - 2 \csc ^{2}{\left (x \right )} + \frac{4 \cos{\left (x \right )}}{2 \cos ^{2}{\left (x \right )} - 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1422, size = 24, normalized size = 1.2 \begin{align*} \frac{2}{\cos \left (x\right ) - 1} - \log \left (-\cos \left (x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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