Optimal. Leaf size=28 \[ \frac{4}{1-\cos (x)}-\frac{2}{(1-\cos (x))^2}+\log (1-\cos (x)) \]
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Rubi [A] time = 0.0510641, antiderivative size = 28, 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} \[ \frac{4}{1-\cos (x)}-\frac{2}{(1-\cos (x))^2}+\log (1-\cos (x)) \]
Antiderivative was successfully verified.
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Rule 4392
Rule 2667
Rule 43
Rubi steps
\begin{align*} \int (\cot (x)+\csc (x))^5 \, dx &=\int (1+\cos (x))^5 \csc ^5(x) \, dx\\ &=-\operatorname{Subst}\left (\int \frac{(1+x)^2}{(1-x)^3} \, dx,x,\cos (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{1}{1-x}-\frac{4}{(-1+x)^3}-\frac{4}{(-1+x)^2}\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 = 0.0734023, size = 32, normalized size = 1.14 \[ -\frac{1}{2} \csc ^4\left (\frac{x}{2}\right )+2 \csc ^2\left (\frac{x}{2}\right )+2 \log \left (\sin \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.048, size = 105, normalized size = 3.8 \begin{align*} -{\frac{ \left ( \cot \left ( x \right ) \right ) ^{4}}{4}}+{\frac{ \left ( \cot \left ( x \right ) \right ) ^{2}}{2}}+\ln \left ( \sin \left ( x \right ) \right ) -{\frac{5\, \left ( \cos \left ( x \right ) \right ) ^{5}}{4\, \left ( \sin \left ( x \right ) \right ) ^{4}}}+{\frac{5\, \left ( \cos \left ( x \right ) \right ) ^{5}}{8\, \left ( \sin \left ( x \right ) \right ) ^{2}}}+{\frac{5\, \left ( \cos \left ( x \right ) \right ) ^{3}}{8}}+{\frac{5\,\cos \left ( x \right ) }{8}}+\ln \left ( \csc \left ( x \right ) -\cot \left ( x \right ) \right ) -{\frac{5\, \left ( \cos \left ( x \right ) \right ) ^{4}}{2\, \left ( \sin \left ( x \right ) \right ) ^{4}}}-{\frac{5\, \left ( \cos \left ( x \right ) \right ) ^{3}}{2\, \left ( \sin \left ( x \right ) \right ) ^{4}}}-{\frac{5\, \left ( \cos \left ( x \right ) \right ) ^{3}}{4\, \left ( \sin \left ( x \right ) \right ) ^{2}}}-{\frac{5}{4\, \left ( \sin \left ( x \right ) \right ) ^{4}}}+ \left ( -{\frac{ \left ( \csc \left ( x \right ) \right ) ^{3}}{4}}-{\frac{3\,\csc \left ( x \right ) }{8}} \right ) \cot \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.01307, size = 169, normalized size = 6.04 \begin{align*} -\frac{5}{2} \, \cot \left (x\right )^{4} - \frac{5 \,{\left (5 \, \cos \left (x\right )^{3} - 3 \, \cos \left (x\right )\right )}}{8 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )}} + \frac{3 \, \cos \left (x\right )^{3} - 5 \, \cos \left (x\right )}{8 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )}} - \frac{5 \,{\left (\cos \left (x\right )^{3} + \cos \left (x\right )\right )}}{4 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )}} + \frac{4 \, \sin \left (x\right )^{2} - 1}{4 \, \sin \left (x\right )^{4}} - \frac{5}{4 \, \sin \left (x\right )^{4}} + \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 = 2.03381, size = 126, normalized size = 4.5 \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 = 1.14025, size = 30, normalized size = 1.07 \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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