Optimal. Leaf size=81 \[ -\frac{\left (a^2+2 b^2\right ) \cot ^2(x)}{2 b^3}+\frac{a \left (a^2+2 b^2\right ) \cot (x)}{b^4}-\frac{\left (a^2+b^2\right )^2 \log (a+b \cot (x))}{b^5}+\frac{a \cot ^3(x)}{3 b^2}-\frac{\cot ^4(x)}{4 b} \]
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Rubi [A] time = 0.0996165, antiderivative size = 81, normalized size of antiderivative = 1., 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 3506
Rule 697
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*}
Mathematica [A] time = 0.374664, 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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Maple [A] time = 0.056, size = 133, normalized size = 1.6 \begin{align*} -{\frac{1}{4\,b \left ( \tan \left ( x \right ) \right ) ^{4}}}-{\frac{{a}^{2}}{2\,{b}^{3} \left ( \tan \left ( x \right ) \right ) ^{2}}}-{\frac{1}{b \left ( \tan \left ( x \right ) \right ) ^{2}}}+{\frac{\ln \left ( \tan \left ( x \right ) \right ){a}^{4}}{{b}^{5}}}+2\,{\frac{\ln \left ( \tan \left ( x \right ) \right ){a}^{2}}{{b}^{3}}}+{\frac{\ln \left ( \tan \left ( x \right ) \right ) }{b}}+{\frac{a}{3\,{b}^{2} \left ( \tan \left ( x \right ) \right ) ^{3}}}+{\frac{{a}^{3}}{{b}^{4}\tan \left ( x \right ) }}+2\,{\frac{a}{{b}^{2}\tan \left ( x \right ) }}-{\frac{\ln \left ( a\tan \left ( x \right ) +b \right ){a}^{4}}{{b}^{5}}}-2\,{\frac{\ln \left ( a\tan \left ( x \right ) +b \right ){a}^{2}}{{b}^{3}}}-{\frac{\ln \left ( a\tan \left ( x \right ) +b \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.21936, size = 143, normalized size = 1.77 \begin{align*} -\frac{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (a \tan \left (x\right ) + b\right )}{b^{5}} + \frac{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (x\right )\right )}{b^{5}} + \frac{4 \, a b^{2} \tan \left (x\right ) + 12 \,{\left (a^{3} + 2 \, a b^{2}\right )} \tan \left (x\right )^{3} - 3 \, b^{3} - 6 \,{\left (a^{2} b + 2 \, b^{3}\right )} \tan \left (x\right )^{2}}{12 \, b^{4} \tan \left (x\right )^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.88844, size = 599, normalized size = 7.4 \begin{align*} -\frac{6 \, a^{2} b^{2} + 9 \, b^{4} - 6 \,{\left (a^{2} b^{2} + b^{4}\right )} \cos \left (x\right )^{2} + 6 \,{\left ({\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (x\right )^{4} + a^{4} + 2 \, a^{2} b^{2} + b^{4} - 2 \,{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (x\right )^{2}\right )} \log \left (2 \, a b \cos \left (x\right ) \sin \left (x\right ) -{\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + a^{2}\right ) - 6 \,{\left ({\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (x\right )^{4} + a^{4} + 2 \, a^{2} b^{2} + b^{4} - 2 \,{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (x\right )^{2}\right )} \log \left (-\frac{1}{4} \, \cos \left (x\right )^{2} + \frac{1}{4}\right ) + 4 \,{\left ({\left (3 \, a^{3} b + 5 \, a b^{3}\right )} \cos \left (x\right )^{3} - 3 \,{\left (a^{3} b + 2 \, a b^{3}\right )} \cos \left (x\right )\right )} \sin \left (x\right )}{12 \,{\left (b^{5} \cos \left (x\right )^{4} - 2 \, b^{5} \cos \left (x\right )^{2} + b^{5}\right )}} \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 [B] time = 1.34744, size = 204, normalized size = 2.52 \begin{align*} \frac{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \log \left ({\left | \tan \left (x\right ) \right |}\right )}{b^{5}} - \frac{{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \log \left ({\left | a \tan \left (x\right ) + b \right |}\right )}{a b^{5}} - \frac{25 \, a^{4} \tan \left (x\right )^{4} + 50 \, a^{2} b^{2} \tan \left (x\right )^{4} + 25 \, b^{4} \tan \left (x\right )^{4} - 12 \, a^{3} b \tan \left (x\right )^{3} - 24 \, a b^{3} \tan \left (x\right )^{3} + 6 \, a^{2} b^{2} \tan \left (x\right )^{2} + 12 \, b^{4} \tan \left (x\right )^{2} - 4 \, a b^{3} \tan \left (x\right ) + 3 \, b^{4}}{12 \, b^{5} \tan \left (x\right )^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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