Optimal. Leaf size=81 \[ -\frac {\left (a^2+b^2\right )^2 \log (a+b \cot (x))}{b^5}+\frac {a \left (a^2+2 b^2\right ) \cot (x)}{b^4}-\frac {\left (a^2+2 b^2\right ) \cot ^2(x)}{2 b^3}+\frac {a \cot ^3(x)}{3 b^2}-\frac {\cot ^4(x)}{4 b} \]
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Rubi [A] time = 0.10, antiderivative size = 81, 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 = {3506, 697} \[ -\frac {\left (a^2+2 b^2\right ) \cot ^2(x)}{2 b^3}+\frac {a \left (a^2+2 b^2\right ) \cot (x)}{b^4}-\frac {\left (a^2+b^2\right )^2 \log (a+b \cot (x))}{b^5}+\frac {a \cot ^3(x)}{3 b^2}-\frac {\cot ^4(x)}{4 b} \]
Antiderivative was successfully verified.
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Rule 697
Rule 3506
Rubi steps
\begin {align*} \int \frac {\csc ^6(x)}{a+b \cot (x)} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\left (1+\frac {x^2}{b^2}\right )^2}{a+x} \, dx,x,b \cot (x)\right )}{b}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {a \left (-a^2-2 b^2\right )}{b^4}+\frac {\left (a^2+2 b^2\right ) x}{b^4}-\frac {a x^2}{b^4}+\frac {x^3}{b^4}+\frac {\left (a^2+b^2\right )^2}{b^4 (a+x)}\right ) \, dx,x,b \cot (x)\right )}{b}\\ &=\frac {a \left (a^2+2 b^2\right ) \cot (x)}{b^4}-\frac {\left (a^2+2 b^2\right ) \cot ^2(x)}{2 b^3}+\frac {a \cot ^3(x)}{3 b^2}-\frac {\cot ^4(x)}{4 b}-\frac {\left (a^2+b^2\right )^2 \log (a+b \cot (x))}{b^5}\\ \end {align*}
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Mathematica [A] time = 0.41, size = 85, normalized size = 1.05 \[ \frac {-6 b^2 \left (a^2+b^2\right ) \csc ^2(x)+4 a b \cot (x) \left (3 a^2+b^2 \csc ^2(x)+5 b^2\right )+12 \left (a^2+b^2\right )^2 (\log (\sin (x))-\log (a \sin (x)+b \cos (x)))-3 b^4 \csc ^4(x)}{12 b^5} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.70, size = 248, normalized size = 3.06 \[ -\frac {6 \, a^{2} b^{2} + 9 \, b^{4} - 6 \, {\left (a^{2} b^{2} + b^{4}\right )} \cos \relax (x)^{2} + 6 \, {\left ({\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cos \relax (x)^{4} + a^{4} + 2 \, a^{2} b^{2} + b^{4} - 2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cos \relax (x)^{2}\right )} \log \left (2 \, a b \cos \relax (x) \sin \relax (x) - {\left (a^{2} - b^{2}\right )} \cos \relax (x)^{2} + a^{2}\right ) - 6 \, {\left ({\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cos \relax (x)^{4} + a^{4} + 2 \, a^{2} b^{2} + b^{4} - 2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cos \relax (x)^{2}\right )} \log \left (-\frac {1}{4} \, \cos \relax (x)^{2} + \frac {1}{4}\right ) + 4 \, {\left ({\left (3 \, a^{3} b + 5 \, a b^{3}\right )} \cos \relax (x)^{3} - 3 \, {\left (a^{3} b + 2 \, a b^{3}\right )} \cos \relax (x)\right )} \sin \relax (x)}{12 \, {\left (b^{5} \cos \relax (x)^{4} - 2 \, b^{5} \cos \relax (x)^{2} + b^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.32, size = 151, normalized size = 1.86 \[ \frac {{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \log \left ({\left | \tan \relax (x) \right |}\right )}{b^{5}} - \frac {{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \log \left ({\left | a \tan \relax (x) + b \right |}\right )}{a b^{5}} - \frac {25 \, a^{4} \tan \relax (x)^{4} + 50 \, a^{2} b^{2} \tan \relax (x)^{4} + 25 \, b^{4} \tan \relax (x)^{4} - 12 \, a^{3} b \tan \relax (x)^{3} - 24 \, a b^{3} \tan \relax (x)^{3} + 6 \, a^{2} b^{2} \tan \relax (x)^{2} + 12 \, b^{4} \tan \relax (x)^{2} - 4 \, a b^{3} \tan \relax (x) + 3 \, b^{4}}{12 \, b^{5} \tan \relax (x)^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 133, normalized size = 1.64 \[ -\frac {\ln \left (a \tan \relax (x )+b \right ) a^{4}}{b^{5}}-\frac {2 \ln \left (a \tan \relax (x )+b \right ) a^{2}}{b^{3}}-\frac {\ln \left (a \tan \relax (x )+b \right )}{b}-\frac {1}{4 b \tan \relax (x )^{4}}-\frac {a^{2}}{2 b^{3} \tan \relax (x )^{2}}-\frac {1}{b \tan \relax (x )^{2}}+\frac {\ln \left (\tan \relax (x )\right ) a^{4}}{b^{5}}+\frac {2 \ln \left (\tan \relax (x )\right ) a^{2}}{b^{3}}+\frac {\ln \left (\tan \relax (x )\right )}{b}+\frac {a}{3 b^{2} \tan \relax (x )^{3}}+\frac {a^{3}}{b^{4} \tan \relax (x )}+\frac {2 a}{b^{2} \tan \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 106, normalized size = 1.31 \[ -\frac {{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (a \tan \relax (x) + b\right )}{b^{5}} + \frac {{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \relax (x)\right )}{b^{5}} + \frac {4 \, a b^{2} \tan \relax (x) + 12 \, {\left (a^{3} + 2 \, a b^{2}\right )} \tan \relax (x)^{3} - 3 \, b^{3} - 6 \, {\left (a^{2} b + 2 \, b^{3}\right )} \tan \relax (x)^{2}}{12 \, b^{4} \tan \relax (x)^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.38, size = 110, normalized size = 1.36 \[ -\frac {\frac {1}{4\,b}-\frac {a\,\mathrm {tan}\relax (x)}{3\,b^2}+\frac {{\mathrm {tan}\relax (x)}^2\,\left (a^2+2\,b^2\right )}{2\,b^3}-\frac {a\,{\mathrm {tan}\relax (x)}^3\,\left (a^2+2\,b^2\right )}{b^4}}{{\mathrm {tan}\relax (x)}^4}-\frac {2\,\mathrm {atanh}\left (\frac {\left (b+2\,a\,\mathrm {tan}\relax (x)\right )\,{\left (a^2+b^2\right )}^2}{b\,\left (a^4+2\,a^2\,b^2+b^4\right )}\right )\,{\left (a^2+b^2\right )}^2}{b^5} \]
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 \[ \int \frac {\csc ^{6}{\relax (x )}}{a + b \cot {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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