Optimal. Leaf size=176 \[ \frac{b x}{8 \left (a^2+b^2\right )}-\frac{a^2 b x}{2 \left (a^2+b^2\right )^2}+\frac{a^2 b^3 x}{\left (a^2+b^2\right )^3}+\frac{a \sin ^4(x)}{4 \left (a^2+b^2\right )}-\frac{a b^2 \sin ^2(x)}{2 \left (a^2+b^2\right )^2}-\frac{b \sin (x) \cos ^3(x)}{4 \left (a^2+b^2\right )}+\frac{b \sin (x) \cos (x)}{8 \left (a^2+b^2\right )}+\frac{a^2 b \sin (x) \cos (x)}{2 \left (a^2+b^2\right )^2}-\frac{a^3 b^2 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^3} \]
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Rubi [A] time = 0.278453, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {3109, 2568, 2635, 8, 2564, 30, 3097, 3133} \[ \frac{b x}{8 \left (a^2+b^2\right )}-\frac{a^2 b x}{2 \left (a^2+b^2\right )^2}+\frac{a^2 b^3 x}{\left (a^2+b^2\right )^3}+\frac{a \sin ^4(x)}{4 \left (a^2+b^2\right )}-\frac{a b^2 \sin ^2(x)}{2 \left (a^2+b^2\right )^2}-\frac{b \sin (x) \cos ^3(x)}{4 \left (a^2+b^2\right )}+\frac{b \sin (x) \cos (x)}{8 \left (a^2+b^2\right )}+\frac{a^2 b \sin (x) \cos (x)}{2 \left (a^2+b^2\right )^2}-\frac{a^3 b^2 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 3109
Rule 2568
Rule 2635
Rule 8
Rule 2564
Rule 30
Rule 3097
Rule 3133
Rubi steps
\begin{align*} \int \frac{\cos ^2(x) \sin ^3(x)}{a \cos (x)+b \sin (x)} \, dx &=\frac{a \int \cos (x) \sin ^3(x) \, dx}{a^2+b^2}+\frac{b \int \cos ^2(x) \sin ^2(x) \, dx}{a^2+b^2}-\frac{(a b) \int \frac{\cos (x) \sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx}{a^2+b^2}\\ &=-\frac{b \cos ^3(x) \sin (x)}{4 \left (a^2+b^2\right )}-\frac{\left (a^2 b\right ) \int \sin ^2(x) \, dx}{\left (a^2+b^2\right )^2}-\frac{\left (a b^2\right ) \int \cos (x) \sin (x) \, dx}{\left (a^2+b^2\right )^2}+\frac{\left (a^2 b^2\right ) \int \frac{\sin (x)}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^2}+\frac{a \operatorname{Subst}\left (\int x^3 \, dx,x,\sin (x)\right )}{a^2+b^2}+\frac{b \int \cos ^2(x) \, dx}{4 \left (a^2+b^2\right )}\\ &=\frac{a^2 b^3 x}{\left (a^2+b^2\right )^3}+\frac{a^2 b \cos (x) \sin (x)}{2 \left (a^2+b^2\right )^2}+\frac{b \cos (x) \sin (x)}{8 \left (a^2+b^2\right )}-\frac{b \cos ^3(x) \sin (x)}{4 \left (a^2+b^2\right )}+\frac{a \sin ^4(x)}{4 \left (a^2+b^2\right )}-\frac{\left (a^3 b^2\right ) \int \frac{b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^3}-\frac{\left (a^2 b\right ) \int 1 \, dx}{2 \left (a^2+b^2\right )^2}-\frac{\left (a b^2\right ) \operatorname{Subst}(\int x \, dx,x,\sin (x))}{\left (a^2+b^2\right )^2}+\frac{b \int 1 \, dx}{8 \left (a^2+b^2\right )}\\ &=\frac{a^2 b^3 x}{\left (a^2+b^2\right )^3}-\frac{a^2 b x}{2 \left (a^2+b^2\right )^2}+\frac{b x}{8 \left (a^2+b^2\right )}-\frac{a^3 b^2 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^3}+\frac{a^2 b \cos (x) \sin (x)}{2 \left (a^2+b^2\right )^2}+\frac{b \cos (x) \sin (x)}{8 \left (a^2+b^2\right )}-\frac{b \cos ^3(x) \sin (x)}{4 \left (a^2+b^2\right )}-\frac{a b^2 \sin ^2(x)}{2 \left (a^2+b^2\right )^2}+\frac{a \sin ^4(x)}{4 \left (a^2+b^2\right )}\\ \end{align*}
Mathematica [C] time = 0.507061, size = 178, normalized size = 1.01 \[ \frac{-32 i a^3 b^2 x+24 a^2 b^3 x+8 a^2 b^3 \sin (2 x)-2 a^2 b^3 \sin (4 x)+2 a^3 b^2 \cos (4 x)-4 a \left (a^4-b^4\right ) \cos (2 x)+32 i a^3 b^2 \tan ^{-1}(\tan (x))-16 a^3 b^2 \log \left ((a \cos (x)+b \sin (x))^2\right )-12 a^4 b x+8 a^4 b \sin (2 x)-a^4 b \sin (4 x)+a^5 \cos (4 x)+a b^4 \cos (4 x)+4 b^5 x-b^5 \sin (4 x)}{32 \left (a^2+b^2\right )^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.075, size = 363, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.78876, size = 582, normalized size = 3.31 \begin{align*} -\frac{a^{3} b^{2} \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^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac{a^{3} b^{2} \log \left (\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac{{\left (3 \, a^{4} b - 6 \, a^{2} b^{3} - b^{5}\right )} \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{4 \,{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} - \frac{\frac{8 \, a b^{2} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac{16 \, a^{3} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac{8 \, a b^{2} \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} - \frac{{\left (3 \, a^{2} b - b^{3}\right )} \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac{{\left (11 \, a^{2} b + 7 \, b^{3}\right )} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{{\left (11 \, a^{2} b + 7 \, b^{3}\right )} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac{{\left (3 \, a^{2} b - b^{3}\right )} \sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}}}{4 \,{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4} + \frac{4 \,{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{6 \,{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac{4 \,{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (x\right )^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.55469, size = 396, normalized size = 2.25 \begin{align*} -\frac{4 \, a^{3} b^{2} \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 (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (x\right )^{4} + 4 \,{\left (a^{5} + a^{3} b^{2}\right )} \cos \left (x\right )^{2} +{\left (3 \, a^{4} b - 6 \, a^{2} b^{3} - b^{5}\right )} x +{\left (2 \,{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (x\right )^{3} -{\left (5 \, a^{4} b + 6 \, a^{2} b^{3} + b^{5}\right )} \cos \left (x\right )\right )} \sin \left (x\right )}{8 \,{\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.
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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 [A] time = 1.12061, size = 371, normalized size = 2.11 \begin{align*} -\frac{a^{3} b^{3} \log \left ({\left | b \tan \left (x\right ) + a \right |}\right )}{a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}} + \frac{a^{3} b^{2} \log \left (\tan \left (x\right )^{2} + 1\right )}{2 \,{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} - \frac{{\left (3 \, a^{4} b - 6 \, a^{2} b^{3} - b^{5}\right )} x}{8 \,{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} - \frac{6 \, a^{3} b^{2} \tan \left (x\right )^{4} - 5 \, a^{4} b \tan \left (x\right )^{3} - 6 \, a^{2} b^{3} \tan \left (x\right )^{3} - b^{5} \tan \left (x\right )^{3} + 4 \, a^{5} \tan \left (x\right )^{2} + 16 \, a^{3} b^{2} \tan \left (x\right )^{2} - 3 \, a^{4} b \tan \left (x\right ) - 2 \, a^{2} b^{3} \tan \left (x\right ) + b^{5} \tan \left (x\right ) + 2 \, a^{5} + 6 \, a^{3} b^{2} - 2 \, a b^{4}}{8 \,{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}{\left (\tan \left (x\right )^{2} + 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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