Optimal. Leaf size=36 \[ -\frac {\csc ^3(x)}{3 a}+\frac {\csc ^2(x)}{2 a}+\frac {\csc (x)}{a}+\frac {\log (\sin (x))}{a} \]
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Rubi [A] time = 0.05, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3879, 75} \[ -\frac {\csc ^3(x)}{3 a}+\frac {\csc ^2(x)}{2 a}+\frac {\csc (x)}{a}+\frac {\log (\sin (x))}{a} \]
Antiderivative was successfully verified.
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Rule 75
Rule 3879
Rubi steps
\begin {align*} \int \frac {\cot ^5(x)}{a+a \csc (x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a-a x)^2 (a+a x)}{x^4} \, dx,x,\sin (x)\right )}{a^4}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a^3}{x^4}-\frac {a^3}{x^3}-\frac {a^3}{x^2}+\frac {a^3}{x}\right ) \, dx,x,\sin (x)\right )}{a^4}\\ &=\frac {\csc (x)}{a}+\frac {\csc ^2(x)}{2 a}-\frac {\csc ^3(x)}{3 a}+\frac {\log (\sin (x))}{a}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 26, normalized size = 0.72 \[ \frac {-\frac {1}{3} \csc ^3(x)+\frac {\csc ^2(x)}{2}+\csc (x)+\log (\sin (x))}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 45, normalized size = 1.25 \[ \frac {6 \, {\left (\cos \relax (x)^{2} - 1\right )} \log \left (\frac {1}{2} \, \sin \relax (x)\right ) \sin \relax (x) + 6 \, \cos \relax (x)^{2} - 3 \, \sin \relax (x) - 4}{6 \, {\left (a \cos \relax (x)^{2} - a\right )} \sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 29, normalized size = 0.81 \[ \frac {\frac {6 \, \sin \relax (x)^{2} + 3 \, \sin \relax (x) - 2}{\sin \relax (x)^{3}} + 6 \, \log \left ({\left | \sin \relax (x) \right |}\right )}{6 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.38, size = 35, normalized size = 0.97 \[ \frac {1}{2 a \sin \relax (x )^{2}}-\frac {1}{3 a \sin \relax (x )^{3}}+\frac {\ln \left (\sin \relax (x )\right )}{a}+\frac {1}{a \sin \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 29, normalized size = 0.81 \[ \frac {\log \left (\sin \relax (x)\right )}{a} + \frac {6 \, \sin \relax (x)^{2} + 3 \, \sin \relax (x) - 2}{6 \, a \sin \relax (x)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.30, size = 80, normalized size = 2.22 \[ \frac {3\,\mathrm {tan}\left (\frac {x}{2}\right )}{8\,a}-\frac {\ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )}{a}+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{8\,a}-\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{24\,a}+\frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{a}+\frac {3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+\mathrm {tan}\left (\frac {x}{2}\right )-\frac {1}{3}}{8\,a\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\cot ^{5}{\relax (x )}}{\csc {\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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