Optimal. Leaf size=110 \[ -\frac{b x \left (a^2+2 b^2\right )}{2 a^4}-\frac{\left (2 a^2+3 b^2\right ) \cos (x)}{3 a^3}-\frac{2 b^4 \tanh ^{-1}\left (\frac{a+b \tan \left (\frac{x}{2}\right )}{\sqrt{a^2-b^2}}\right )}{a^4 \sqrt{a^2-b^2}}+\frac{b \sin (x) \cos (x)}{2 a^2}-\frac{\sin ^2(x) \cos (x)}{3 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.398245, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {3853, 4104, 3919, 3831, 2660, 618, 206} \[ -\frac{b x \left (a^2+2 b^2\right )}{2 a^4}-\frac{\left (2 a^2+3 b^2\right ) \cos (x)}{3 a^3}-\frac{2 b^4 \tanh ^{-1}\left (\frac{a+b \tan \left (\frac{x}{2}\right )}{\sqrt{a^2-b^2}}\right )}{a^4 \sqrt{a^2-b^2}}+\frac{b \sin (x) \cos (x)}{2 a^2}-\frac{\sin ^2(x) \cos (x)}{3 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3853
Rule 4104
Rule 3919
Rule 3831
Rule 2660
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{\sin ^3(x)}{a+b \csc (x)} \, dx &=-\frac{\cos (x) \sin ^2(x)}{3 a}+\frac{\int \frac{\left (-3 b+2 a \csc (x)+2 b \csc ^2(x)\right ) \sin ^2(x)}{a+b \csc (x)} \, dx}{3 a}\\ &=\frac{b \cos (x) \sin (x)}{2 a^2}-\frac{\cos (x) \sin ^2(x)}{3 a}-\frac{\int \frac{\left (-2 \left (2 a^2+3 b^2\right )-a b \csc (x)+3 b^2 \csc ^2(x)\right ) \sin (x)}{a+b \csc (x)} \, dx}{6 a^2}\\ &=-\frac{\left (2 a^2+3 b^2\right ) \cos (x)}{3 a^3}+\frac{b \cos (x) \sin (x)}{2 a^2}-\frac{\cos (x) \sin ^2(x)}{3 a}+\frac{\int \frac{-3 b \left (a^2+2 b^2\right )-3 a b^2 \csc (x)}{a+b \csc (x)} \, dx}{6 a^3}\\ &=-\frac{b \left (a^2+2 b^2\right ) x}{2 a^4}-\frac{\left (2 a^2+3 b^2\right ) \cos (x)}{3 a^3}+\frac{b \cos (x) \sin (x)}{2 a^2}-\frac{\cos (x) \sin ^2(x)}{3 a}+\frac{b^4 \int \frac{\csc (x)}{a+b \csc (x)} \, dx}{a^4}\\ &=-\frac{b \left (a^2+2 b^2\right ) x}{2 a^4}-\frac{\left (2 a^2+3 b^2\right ) \cos (x)}{3 a^3}+\frac{b \cos (x) \sin (x)}{2 a^2}-\frac{\cos (x) \sin ^2(x)}{3 a}+\frac{b^3 \int \frac{1}{1+\frac{a \sin (x)}{b}} \, dx}{a^4}\\ &=-\frac{b \left (a^2+2 b^2\right ) x}{2 a^4}-\frac{\left (2 a^2+3 b^2\right ) \cos (x)}{3 a^3}+\frac{b \cos (x) \sin (x)}{2 a^2}-\frac{\cos (x) \sin ^2(x)}{3 a}+\frac{\left (2 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+\frac{2 a x}{b}+x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )}{a^4}\\ &=-\frac{b \left (a^2+2 b^2\right ) x}{2 a^4}-\frac{\left (2 a^2+3 b^2\right ) \cos (x)}{3 a^3}+\frac{b \cos (x) \sin (x)}{2 a^2}-\frac{\cos (x) \sin ^2(x)}{3 a}-\frac{\left (4 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left (1-\frac{a^2}{b^2}\right )-x^2} \, dx,x,\frac{2 a}{b}+2 \tan \left (\frac{x}{2}\right )\right )}{a^4}\\ &=-\frac{b \left (a^2+2 b^2\right ) x}{2 a^4}-\frac{2 b^4 \tanh ^{-1}\left (\frac{b \left (\frac{a}{b}+\tan \left (\frac{x}{2}\right )\right )}{\sqrt{a^2-b^2}}\right )}{a^4 \sqrt{a^2-b^2}}-\frac{\left (2 a^2+3 b^2\right ) \cos (x)}{3 a^3}+\frac{b \cos (x) \sin (x)}{2 a^2}-\frac{\cos (x) \sin ^2(x)}{3 a}\\ \end{align*}
Mathematica [A] time = 0.237856, size = 98, normalized size = 0.89 \[ \frac{-6 b x \left (a^2+2 b^2\right )-3 a \left (3 a^2+4 b^2\right ) \cos (x)+\frac{24 b^4 \tan ^{-1}\left (\frac{a+b \tan \left (\frac{x}{2}\right )}{\sqrt{b^2-a^2}}\right )}{\sqrt{b^2-a^2}}+3 a^2 b \sin (2 x)+a^3 \cos (3 x)}{12 a^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.066, size = 213, normalized size = 1.9 \begin{align*} -{\frac{b}{{a}^{2}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{5} \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-3}}-2\,{\frac{{b}^{2} \left ( \tan \left ( x/2 \right ) \right ) ^{4}}{{a}^{3} \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{3}}}-4\,{\frac{ \left ( \tan \left ( x/2 \right ) \right ) ^{2}}{a \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{3}}}-4\,{\frac{ \left ( \tan \left ( x/2 \right ) \right ) ^{2}{b}^{2}}{{a}^{3} \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{3}}}+{\frac{b}{{a}^{2}}\tan \left ({\frac{x}{2}} \right ) \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-3}}-{\frac{4}{3\,a} \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-3}}-2\,{\frac{{b}^{2}}{{a}^{3} \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{3}}}-{\frac{b}{{a}^{2}}\arctan \left ( \tan \left ({\frac{x}{2}} \right ) \right ) }-2\,{\frac{\arctan \left ( \tan \left ( x/2 \right ) \right ){b}^{3}}{{a}^{4}}}+2\,{\frac{{b}^{4}}{{a}^{4}\sqrt{-{a}^{2}+{b}^{2}}}\arctan \left ( 1/2\,{\frac{2\,b\tan \left ( x/2 \right ) +2\,a}{\sqrt{-{a}^{2}+{b}^{2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.552073, size = 744, normalized size = 6.76 \begin{align*} \left [\frac{3 \, \sqrt{a^{2} - b^{2}} b^{4} \log \left (-\frac{{\left (a^{2} - 2 \, b^{2}\right )} \cos \left (x\right )^{2} + 2 \, a b \sin \left (x\right ) + a^{2} + b^{2} - 2 \,{\left (b \cos \left (x\right ) \sin \left (x\right ) + a \cos \left (x\right )\right )} \sqrt{a^{2} - b^{2}}}{a^{2} \cos \left (x\right )^{2} - 2 \, a b \sin \left (x\right ) - a^{2} - b^{2}}\right ) + 2 \,{\left (a^{5} - a^{3} b^{2}\right )} \cos \left (x\right )^{3} + 3 \,{\left (a^{4} b - a^{2} b^{3}\right )} \cos \left (x\right ) \sin \left (x\right ) - 3 \,{\left (a^{4} b + a^{2} b^{3} - 2 \, b^{5}\right )} x - 6 \,{\left (a^{5} - a b^{4}\right )} \cos \left (x\right )}{6 \,{\left (a^{6} - a^{4} b^{2}\right )}}, -\frac{6 \, \sqrt{-a^{2} + b^{2}} b^{4} \arctan \left (-\frac{\sqrt{-a^{2} + b^{2}}{\left (b \sin \left (x\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \cos \left (x\right )}\right ) - 2 \,{\left (a^{5} - a^{3} b^{2}\right )} \cos \left (x\right )^{3} - 3 \,{\left (a^{4} b - a^{2} b^{3}\right )} \cos \left (x\right ) \sin \left (x\right ) + 3 \,{\left (a^{4} b + a^{2} b^{3} - 2 \, b^{5}\right )} x + 6 \,{\left (a^{5} - a b^{4}\right )} \cos \left (x\right )}{6 \,{\left (a^{6} - a^{4} b^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.4371, size = 201, normalized size = 1.83 \begin{align*} \frac{2 \,{\left (\pi \left \lfloor \frac{x}{2 \, \pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (b\right ) + \arctan \left (\frac{b \tan \left (\frac{1}{2} \, x\right ) + a}{\sqrt{-a^{2} + b^{2}}}\right )\right )} b^{4}}{\sqrt{-a^{2} + b^{2}} a^{4}} - \frac{{\left (a^{2} b + 2 \, b^{3}\right )} x}{2 \, a^{4}} - \frac{3 \, a b \tan \left (\frac{1}{2} \, x\right )^{5} + 6 \, b^{2} \tan \left (\frac{1}{2} \, x\right )^{4} + 12 \, a^{2} \tan \left (\frac{1}{2} \, x\right )^{2} + 12 \, b^{2} \tan \left (\frac{1}{2} \, x\right )^{2} - 3 \, a b \tan \left (\frac{1}{2} \, x\right ) + 4 \, a^{2} + 6 \, b^{2}}{3 \,{\left (\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right )}^{3} a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]