Optimal. Leaf size=117 \[ \frac{\left (a^2+b^2\right )^2}{2 a^3 b^2 (a+b \tan (x))^2}-\frac{\left (a^2-3 b^2\right ) \left (a^2+b^2\right )}{a^4 b^2 (a+b \tan (x))}+\frac{2 \left (a^2+3 b^2\right ) \log (\tan (x))}{a^5}-\frac{2 \left (a^2+3 b^2\right ) \log (a+b \tan (x))}{a^5}+\frac{3 b \cot (x)}{a^4}-\frac{\cot ^2(x)}{2 a^3} \]
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Rubi [A] time = 0.136647, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3087, 894} \[ \frac{\left (a^2+b^2\right )^2}{2 a^3 b^2 (a+b \tan (x))^2}-\frac{\left (a^2-3 b^2\right ) \left (a^2+b^2\right )}{a^4 b^2 (a+b \tan (x))}+\frac{2 \left (a^2+3 b^2\right ) \log (\tan (x))}{a^5}-\frac{2 \left (a^2+3 b^2\right ) \log (a+b \tan (x))}{a^5}+\frac{3 b \cot (x)}{a^4}-\frac{\cot ^2(x)}{2 a^3} \]
Antiderivative was successfully verified.
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Rule 3087
Rule 894
Rubi steps
\begin{align*} \int \frac{\csc ^3(x)}{(a \cos (x)+b \sin (x))^3} \, dx &=\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^2}{x^3 (a+b x)^3} \, dx,x,\tan (x)\right )\\ &=\operatorname{Subst}\left (\int \left (\frac{1}{a^3 x^3}-\frac{3 b}{a^4 x^2}+\frac{2 \left (a^2+3 b^2\right )}{a^5 x}-\frac{\left (a^2+b^2\right )^2}{a^3 b (a+b x)^3}+\frac{a^4-2 a^2 b^2-3 b^4}{a^4 b (a+b x)^2}-\frac{2 b \left (a^2+3 b^2\right )}{a^5 (a+b x)}\right ) \, dx,x,\tan (x)\right )\\ &=\frac{3 b \cot (x)}{a^4}-\frac{\cot ^2(x)}{2 a^3}+\frac{2 \left (a^2+3 b^2\right ) \log (\tan (x))}{a^5}-\frac{2 \left (a^2+3 b^2\right ) \log (a+b \tan (x))}{a^5}+\frac{\left (a^2+b^2\right )^2}{2 a^3 b^2 (a+b \tan (x))^2}-\frac{\left (a^2-3 b^2\right ) \left (a^2+b^2\right )}{a^4 b^2 (a+b \tan (x))}\\ \end{align*}
Mathematica [A] time = 0.760654, size = 208, normalized size = 1.78 \[ \frac{2 b^2 \left (2 \left (a^2+3 b^2\right ) \log (\sin (x))-2 \left (a^2+3 b^2\right ) \log (a \cos (x)+b \sin (x))-3 \left (a^2+b^2\right )\right )+\cot ^2(x) \left (4 a^2 \left (\left (a^2+3 b^2\right ) \log (\sin (x))-\left (a^2+3 b^2\right ) \log (a \cos (x)+b \sin (x))+3 b^2\right )-a^4 \csc ^2(x)\right )-2 a b \cot (x) \left (-4 \left (a^2+3 b^2\right ) \log (\sin (x))+4 a^2 \log (a \cos (x)+b \sin (x))+a^2 \csc ^2(x)+3 a^2+12 b^2 \log (a \cos (x)+b \sin (x))\right )+6 a^3 b \cot ^3(x)+a^4 \csc ^2(x)}{2 a^5 (a \cot (x)+b)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.143, size = 151, normalized size = 1.3 \begin{align*}{\frac{a}{2\,{b}^{2} \left ( a+b\tan \left ( x \right ) \right ) ^{2}}}+{\frac{1}{a \left ( a+b\tan \left ( x \right ) \right ) ^{2}}}+{\frac{{b}^{2}}{2\,{a}^{3} \left ( a+b\tan \left ( x \right ) \right ) ^{2}}}-{\frac{1}{{b}^{2} \left ( a+b\tan \left ( x \right ) \right ) }}+2\,{\frac{1}{{a}^{2} \left ( a+b\tan \left ( x \right ) \right ) }}+3\,{\frac{{b}^{2}}{{a}^{4} \left ( a+b\tan \left ( x \right ) \right ) }}-2\,{\frac{\ln \left ( a+b\tan \left ( x \right ) \right ) }{{a}^{3}}}-6\,{\frac{\ln \left ( a+b\tan \left ( x \right ) \right ){b}^{2}}{{a}^{5}}}-{\frac{1}{2\,{a}^{3} \left ( \tan \left ( x \right ) \right ) ^{2}}}+2\,{\frac{\ln \left ( \tan \left ( x \right ) \right ) }{{a}^{3}}}+6\,{\frac{\ln \left ( \tan \left ( x \right ) \right ){b}^{2}}{{a}^{5}}}+3\,{\frac{b}{{a}^{4}\tan \left ( x \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.26935, size = 416, normalized size = 3.56 \begin{align*} -\frac{a^{4} - \frac{8 \, a^{3} b \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac{2 \,{\left (a^{4} + 22 \, a^{2} b^{2}\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{4 \,{\left (21 \, a^{3} b + 4 \, a b^{3}\right )} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - \frac{{\left (15 \, a^{4} - 144 \, a^{2} b^{2} - 112 \, b^{4}\right )} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} - \frac{4 \,{\left (19 \, a^{3} b + 16 \, a b^{3}\right )} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}}}{8 \,{\left (\frac{a^{7} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{4 \, a^{6} b \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - \frac{4 \, a^{6} b \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac{a^{7} \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} - \frac{2 \,{\left (a^{7} - 2 \, a^{5} b^{2}\right )} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}}\right )}} - \frac{\frac{12 \, b \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}}{8 \, a^{4}} - \frac{2 \,{\left (a^{2} + 3 \, b^{2}\right )} \log \left (-a - \frac{2 \, b \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right )}{a^{5}} + \frac{2 \,{\left (a^{2} + 3 \, b^{2}\right )} \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.592269, size = 887, normalized size = 7.58 \begin{align*} -\frac{24 \, a^{2} b^{2} \cos \left (x\right )^{4} - a^{4} + 6 \, a^{2} b^{2} + 2 \,{\left (a^{4} - 15 \, a^{2} b^{2}\right )} \cos \left (x\right )^{2} - 2 \,{\left ({\left (a^{4} + 2 \, a^{2} b^{2} - 3 \, b^{4}\right )} \cos \left (x\right )^{4} - a^{2} b^{2} - 3 \, b^{4} -{\left (a^{4} + a^{2} b^{2} - 6 \, b^{4}\right )} \cos \left (x\right )^{2} + 2 \,{\left ({\left (a^{3} b + 3 \, a b^{3}\right )} \cos \left (x\right )^{3} -{\left (a^{3} b + 3 \, a b^{3}\right )} \cos \left (x\right )\right )} \sin \left (x\right )\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} + b^{2}\right ) + 2 \,{\left ({\left (a^{4} + 2 \, a^{2} b^{2} - 3 \, b^{4}\right )} \cos \left (x\right )^{4} - a^{2} b^{2} - 3 \, b^{4} -{\left (a^{4} + a^{2} b^{2} - 6 \, b^{4}\right )} \cos \left (x\right )^{2} + 2 \,{\left ({\left (a^{3} b + 3 \, a b^{3}\right )} \cos \left (x\right )^{3} -{\left (a^{3} b + 3 \, a b^{3}\right )} \cos \left (x\right )\right )} \sin \left (x\right )\right )} \log \left (-\frac{1}{4} \, \cos \left (x\right )^{2} + \frac{1}{4}\right ) - 4 \,{\left (3 \,{\left (a^{3} b - a b^{3}\right )} \cos \left (x\right )^{3} -{\left (2 \, a^{3} b - 3 \, a b^{3}\right )} \cos \left (x\right )\right )} \sin \left (x\right )}{2 \,{\left (a^{5} b^{2} -{\left (a^{7} - a^{5} b^{2}\right )} \cos \left (x\right )^{4} +{\left (a^{7} - 2 \, a^{5} b^{2}\right )} \cos \left (x\right )^{2} - 2 \,{\left (a^{6} b \cos \left (x\right )^{3} - a^{6} b \cos \left (x\right )\right )} \sin \left (x\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc ^{3}{\left (x \right )}}{\left (a \cos{\left (x \right )} + b \sin{\left (x \right )}\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19542, size = 197, normalized size = 1.68 \begin{align*} \frac{2 \,{\left (a^{2} + 3 \, b^{2}\right )} \log \left ({\left | \tan \left (x\right ) \right |}\right )}{a^{5}} - \frac{2 \,{\left (a^{2} b + 3 \, b^{3}\right )} \log \left ({\left | b \tan \left (x\right ) + a \right |}\right )}{a^{5} b} - \frac{2 \, a^{4} b \tan \left (x\right )^{3} - 4 \, a^{2} b^{3} \tan \left (x\right )^{3} - 12 \, b^{5} \tan \left (x\right )^{3} + a^{5} \tan \left (x\right )^{2} - 6 \, a^{3} b^{2} \tan \left (x\right )^{2} - 18 \, a b^{4} \tan \left (x\right )^{2} - 4 \, a^{2} b^{3} \tan \left (x\right ) + a^{3} b^{2}}{2 \,{\left (b \tan \left (x\right )^{2} + a \tan \left (x\right )\right )}^{2} a^{4} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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