Optimal. Leaf size=112 \[ \frac{a \sin ^3(x)}{3 \left (a^2+b^2\right )}-\frac{a b^2 \sin (x)}{\left (a^2+b^2\right )^2}-\frac{b \cos ^3(x)}{3 \left (a^2+b^2\right )}+\frac{a^2 b \cos (x)}{\left (a^2+b^2\right )^2}-\frac{a^2 b^2 \tanh ^{-1}\left (\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.196009, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {3109, 2565, 30, 2564, 2637, 2638, 3074, 206} \[ \frac{a \sin ^3(x)}{3 \left (a^2+b^2\right )}-\frac{a b^2 \sin (x)}{\left (a^2+b^2\right )^2}-\frac{b \cos ^3(x)}{3 \left (a^2+b^2\right )}+\frac{a^2 b \cos (x)}{\left (a^2+b^2\right )^2}-\frac{a^2 b^2 \tanh ^{-1}\left (\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3109
Rule 2565
Rule 30
Rule 2564
Rule 2637
Rule 2638
Rule 3074
Rule 206
Rubi steps
\begin{align*} \int \frac{\cos ^2(x) \sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx &=\frac{a \int \cos (x) \sin ^2(x) \, dx}{a^2+b^2}+\frac{b \int \cos ^2(x) \sin (x) \, dx}{a^2+b^2}-\frac{(a b) \int \frac{\cos (x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx}{a^2+b^2}\\ &=-\frac{\left (a^2 b\right ) \int \sin (x) \, dx}{\left (a^2+b^2\right )^2}-\frac{\left (a b^2\right ) \int \cos (x) \, dx}{\left (a^2+b^2\right )^2}+\frac{\left (a^2 b^2\right ) \int \frac{1}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^2}+\frac{a \operatorname{Subst}\left (\int x^2 \, dx,x,\sin (x)\right )}{a^2+b^2}-\frac{b \operatorname{Subst}\left (\int x^2 \, dx,x,\cos (x)\right )}{a^2+b^2}\\ &=\frac{a^2 b \cos (x)}{\left (a^2+b^2\right )^2}-\frac{b \cos ^3(x)}{3 \left (a^2+b^2\right )}-\frac{a b^2 \sin (x)}{\left (a^2+b^2\right )^2}+\frac{a \sin ^3(x)}{3 \left (a^2+b^2\right )}-\frac{\left (a^2 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{a^2+b^2-x^2} \, dx,x,b \cos (x)-a \sin (x)\right )}{\left (a^2+b^2\right )^2}\\ &=-\frac{a^2 b^2 \tanh ^{-1}\left (\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}+\frac{a^2 b \cos (x)}{\left (a^2+b^2\right )^2}-\frac{b \cos ^3(x)}{3 \left (a^2+b^2\right )}-\frac{a b^2 \sin (x)}{\left (a^2+b^2\right )^2}+\frac{a \sin ^3(x)}{3 \left (a^2+b^2\right )}\\ \end{align*}
Mathematica [A] time = 0.655489, size = 115, normalized size = 1.03 \[ \frac{2 a^2 b^2 \tanh ^{-1}\left (\frac{a \tan \left (\frac{x}{2}\right )-b}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}-\frac{\left (3 b^3-9 a^2 b\right ) \cos (x)+b \left (a^2+b^2\right ) \cos (3 x)+2 a \sin (x) \left (\left (a^2+b^2\right ) \cos (2 x)-a^2+5 b^2\right )}{12 \left (a^2+b^2\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.091, size = 168, normalized size = 1.5 \begin{align*} 2\,{\frac{-a{b}^{2} \left ( \tan \left ( x/2 \right ) \right ) ^{5}-{b}^{3} \left ( \tan \left ( x/2 \right ) \right ) ^{4}+ \left ( 4/3\,{a}^{3}-2/3\,a{b}^{2} \right ) \left ( \tan \left ( x/2 \right ) \right ) ^{3}+2\,{a}^{2}b \left ( \tan \left ( x/2 \right ) \right ) ^{2}-a{b}^{2}\tan \left ( x/2 \right ) +2/3\,{a}^{2}b-1/3\,{b}^{3}}{ \left ({a}^{4}+2\,{a}^{2}{b}^{2}+{b}^{4} \right ) \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{3}}}+8\,{\frac{{a}^{2}{b}^{2}}{ \left ( 4\,{a}^{4}+8\,{a}^{2}{b}^{2}+4\,{b}^{4} \right ) \sqrt{{a}^{2}+{b}^{2}}}{\it Artanh} \left ( 1/2\,{\frac{2\,a\tan \left ( x/2 \right ) -2\,b}{\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 [B] time = 0.535051, size = 498, normalized size = 4.45 \begin{align*} \frac{3 \, \sqrt{a^{2} + b^{2}} a^{2} b^{2} \log \left (-\frac{2 \, a b \cos \left (x\right ) \sin \left (x\right ) +{\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} - 2 \, a^{2} - b^{2} + 2 \, \sqrt{a^{2} + b^{2}}{\left (b \cos \left (x\right ) - a \sin \left (x\right )\right )}}{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 (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (x\right )^{3} + 6 \,{\left (a^{4} b + a^{2} b^{3}\right )} \cos \left (x\right ) + 2 \,{\left (a^{5} - a^{3} b^{2} - 2 \, a b^{4} -{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )}{6 \,{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\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 = 1.22947, size = 259, normalized size = 2.31 \begin{align*} -\frac{a^{2} b^{2} \log \left (\frac{{\left | 2 \, a \tan \left (\frac{1}{2} \, x\right ) - 2 \, b - 2 \, \sqrt{a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac{1}{2} \, x\right ) - 2 \, b + 2 \, \sqrt{a^{2} + b^{2}} \right |}}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt{a^{2} + b^{2}}} - \frac{2 \,{\left (3 \, a b^{2} \tan \left (\frac{1}{2} \, x\right )^{5} + 3 \, b^{3} \tan \left (\frac{1}{2} \, x\right )^{4} - 4 \, a^{3} \tan \left (\frac{1}{2} \, x\right )^{3} + 2 \, a b^{2} \tan \left (\frac{1}{2} \, x\right )^{3} - 6 \, a^{2} b \tan \left (\frac{1}{2} \, x\right )^{2} + 3 \, a b^{2} \tan \left (\frac{1}{2} \, x\right ) - 2 \, a^{2} b + b^{3}\right )}}{3 \,{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}{\left (\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]