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 = 0.05, antiderivative size = 24, normalized size of antiderivative = 1.00, 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 43
Rule 2667
Rule 4392
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 1.99, size = 38, normalized size = 1.58 \[ -\frac {{\left (\cos \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right )} \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + 4 \, \cos \relax (x) + 2}{\cos \relax (x)^{2} + 2 \, \cos \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 22, normalized size = 0.92 \[ -\frac {2 \, {\left (2 \, \cos \relax (x) + 1\right )}}{{\left (\cos \relax (x) + 1\right )}^{2}} - \log \left (\cos \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 25, normalized size = 1.04 \[ \frac {2}{\left (1+\cos \relax (x )\right )^{2}}-\frac {4}{1+\cos \relax (x )}-\ln \left (1+\cos \relax (x )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 39, normalized size = 1.62 \[ -\frac {\sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {\sin \relax (x)^{4}}{2 \, {\left (\cos \relax (x) + 1\right )}^{4}} + \log \left (\frac {\sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.41, size = 26, normalized size = 1.08 \[ \ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )-{\mathrm {tan}\left (\frac {x}{2}\right )}^2+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^4}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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