Optimal. Leaf size=72 \[ \frac {\left (a^2-b^2\right )^2 \log (a+b \csc (x))}{a b^4}-\frac {\left (a^2-2 b^2\right ) \csc (x)}{b^3}+\frac {a \csc ^2(x)}{2 b^2}+\frac {\log (\sin (x))}{a}-\frac {\csc ^3(x)}{3 b} \]
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Rubi [A] time = 0.08, antiderivative size = 72, 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 = {3885, 894} \[ -\frac {\left (a^2-2 b^2\right ) \csc (x)}{b^3}+\frac {\left (a^2-b^2\right )^2 \log (a+b \csc (x))}{a b^4}+\frac {a \csc ^2(x)}{2 b^2}+\frac {\log (\sin (x))}{a}-\frac {\csc ^3(x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 894
Rule 3885
Rubi steps
\begin {align*} \int \frac {\cot ^5(x)}{a+b \csc (x)} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\left (b^2-x^2\right )^2}{x (a+x)} \, dx,x,b \csc (x)\right )}{b^4}\\ &=-\frac {\operatorname {Subst}\left (\int \left (a^2 \left (1-\frac {2 b^2}{a^2}\right )+\frac {b^4}{a x}-a x+x^2-\frac {\left (a^2-b^2\right )^2}{a (a+x)}\right ) \, dx,x,b \csc (x)\right )}{b^4}\\ &=-\frac {\left (a^2-2 b^2\right ) \csc (x)}{b^3}+\frac {a \csc ^2(x)}{2 b^2}-\frac {\csc ^3(x)}{3 b}+\frac {\left (a^2-b^2\right )^2 \log (a+b \csc (x))}{a b^4}+\frac {\log (\sin (x))}{a}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 85, normalized size = 1.18 \[ \frac {3 a^2 b^2 \csc ^2(x)-6 a b \left (a^2-2 b^2\right ) \csc (x)-6 a^2 \left (a^2-2 b^2\right ) \log (\sin (x))+6 \left (a^2-b^2\right )^2 \log (a \sin (x)+b)-2 a b^3 \csc ^3(x)}{6 a b^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 151, normalized size = 2.10 \[ -\frac {3 \, a^{2} b^{2} \sin \relax (x) - 6 \, a^{3} b + 10 \, a b^{3} + 6 \, {\left (a^{3} b - 2 \, a b^{3}\right )} \cos \relax (x)^{2} + 6 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4} - {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cos \relax (x)^{2}\right )} \log \left (a \sin \relax (x) + b\right ) \sin \relax (x) - 6 \, {\left (a^{4} - 2 \, a^{2} b^{2} - {\left (a^{4} - 2 \, a^{2} b^{2}\right )} \cos \relax (x)^{2}\right )} \log \left (\frac {1}{2} \, \sin \relax (x)\right ) \sin \relax (x)}{6 \, {\left (a b^{4} \cos \relax (x)^{2} - a b^{4}\right )} \sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 90, normalized size = 1.25 \[ -\frac {{\left (a^{3} - 2 \, a b^{2}\right )} \log \left ({\left | \sin \relax (x) \right |}\right )}{b^{4}} + \frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \log \left ({\left | a \sin \relax (x) + b \right |}\right )}{a b^{4}} + \frac {3 \, a b^{2} \sin \relax (x) - 2 \, b^{3} - 6 \, {\left (a^{2} b - 2 \, b^{3}\right )} \sin \relax (x)^{2}}{6 \, b^{4} \sin \relax (x)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 100, normalized size = 1.39 \[ \frac {a^{3} \ln \left (b +a \sin \relax (x )\right )}{b^{4}}-\frac {2 a \ln \left (b +a \sin \relax (x )\right )}{b^{2}}+\frac {\ln \left (b +a \sin \relax (x )\right )}{a}-\frac {1}{3 b \sin \relax (x )^{3}}-\frac {a^{2}}{b^{3} \sin \relax (x )}+\frac {2}{b \sin \relax (x )}+\frac {a}{2 b^{2} \sin \relax (x )^{2}}-\frac {a^{3} \ln \left (\sin \relax (x )\right )}{b^{4}}+\frac {2 a \ln \left (\sin \relax (x )\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 84, normalized size = 1.17 \[ -\frac {{\left (a^{3} - 2 \, a b^{2}\right )} \log \left (\sin \relax (x)\right )}{b^{4}} + \frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (a \sin \relax (x) + b\right )}{a b^{4}} + \frac {3 \, a b \sin \relax (x) - 6 \, {\left (a^{2} - 2 \, b^{2}\right )} \sin \relax (x)^{2} - 2 \, b^{2}}{6 \, b^{3} \sin \relax (x)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.57, size = 157, normalized size = 2.18 \[ \mathrm {tan}\left (\frac {x}{2}\right )\,\left (\frac {7}{8\,b}-\frac {a^2}{2\,b^3}\right )-\frac {\ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )}{a}-\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{24\,b}+\frac {a\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{8\,b^2}-\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\left (4\,a^2-7\,b^2\right )+\frac {b^2}{3}-a\,b\,\mathrm {tan}\left (\frac {x}{2}\right )}{8\,b^3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3}+\frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )\,\left (2\,a\,b^2-a^3\right )}{b^4}+\frac {\ln \left (b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+2\,a\,\mathrm {tan}\left (\frac {x}{2}\right )+b\right )\,{\left (a^2-b^2\right )}^2}{a\,b^4} \]
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 \[ \int \frac {\cot ^{5}{\relax (x )}}{a + b \csc {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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