Optimal. Leaf size=129 \[ \frac{b x \left (3 a^3-a b^2\right )}{\left (a^2+b^2\right )^3}-\frac{\left (a^2-b^2\right ) \sin ^2(x)}{2 \left (a^2+b^2\right )^2}-\frac{a^2 b \sin (x)}{\left (a^2+b^2\right )^2 (a \cos (x)+b \sin (x))}-\frac{a b \sin (x) \cos (x)}{\left (a^2+b^2\right )^2}-\frac{a^2 \left (a^2-3 b^2\right ) \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^3} \]
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Rubi [A] time = 0.506473, antiderivative size = 198, normalized size of antiderivative = 1.53, number of steps used = 17, number of rules used = 13, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.722, Rules used = {3111, 3109, 2564, 30, 2635, 8, 3097, 3133, 3099, 3085, 3483, 3531, 3530} \[ \frac{a^3 b x}{\left (a^2+b^2\right )^3}+\frac{a b x}{\left (a^2+b^2\right )^2}-\frac{a b^3 x}{\left (a^2+b^2\right )^3}+\frac{a b x \left (a^2-b^2\right )}{\left (a^2+b^2\right )^3}-\frac{a^2 \sin ^2(x)}{2 \left (a^2+b^2\right )^2}+\frac{b^2 \sin ^2(x)}{2 \left (a^2+b^2\right )^2}-\frac{a^2 b}{\left (a^2+b^2\right )^2 (a \cot (x)+b)}-\frac{a b \sin (x) \cos (x)}{\left (a^2+b^2\right )^2}-\frac{a^4 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^3}+\frac{3 a^2 b^2 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 3111
Rule 3109
Rule 2564
Rule 30
Rule 2635
Rule 8
Rule 3097
Rule 3133
Rule 3099
Rule 3085
Rule 3483
Rule 3531
Rule 3530
Rubi steps
\begin{align*} \int \frac{\cos (x) \sin ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx &=\frac{a \int \frac{\sin ^3(x)}{a \cos (x)+b \sin (x)} \, dx}{a^2+b^2}+\frac{b \int \frac{\cos (x) \sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx}{a^2+b^2}-\frac{(a b) \int \frac{\sin ^2(x)}{(a \cos (x)+b \sin (x))^2} \, dx}{a^2+b^2}\\ &=-\frac{a^2 \sin ^2(x)}{2 \left (a^2+b^2\right )^2}+\frac{a^3 \int \frac{\sin (x)}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^2}+2 \frac{(a b) \int \sin ^2(x) \, dx}{\left (a^2+b^2\right )^2}+\frac{b^2 \int \cos (x) \sin (x) \, dx}{\left (a^2+b^2\right )^2}-\frac{\left (a b^2\right ) \int \frac{\sin (x)}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^2}-\frac{(a b) \int \frac{1}{(b+a \cot (x))^2} \, dx}{a^2+b^2}\\ &=\frac{a^3 b x}{\left (a^2+b^2\right )^3}-\frac{a b^3 x}{\left (a^2+b^2\right )^3}-\frac{a^2 b}{\left (a^2+b^2\right )^2 (b+a \cot (x))}-\frac{a^2 \sin ^2(x)}{2 \left (a^2+b^2\right )^2}-\frac{a^4 \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^2\right ) \int \frac{b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^3}+2 \left (-\frac{a b \cos (x) \sin (x)}{2 \left (a^2+b^2\right )^2}+\frac{(a b) \int 1 \, dx}{2 \left (a^2+b^2\right )^2}\right )-\frac{(a b) \int \frac{b-a \cot (x)}{b+a \cot (x)} \, dx}{\left (a^2+b^2\right )^2}+\frac{b^2 \operatorname{Subst}(\int x \, dx,x,\sin (x))}{\left (a^2+b^2\right )^2}\\ &=\frac{a^3 b x}{\left (a^2+b^2\right )^3}-\frac{a b^3 x}{\left (a^2+b^2\right )^3}+\frac{a b \left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^3}-\frac{a^2 b}{\left (a^2+b^2\right )^2 (b+a \cot (x))}-\frac{a^4 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^3}+\frac{a^2 b^2 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^3}-\frac{a^2 \sin ^2(x)}{2 \left (a^2+b^2\right )^2}+\frac{b^2 \sin ^2(x)}{2 \left (a^2+b^2\right )^2}+2 \left (\frac{a b x}{2 \left (a^2+b^2\right )^2}-\frac{a b \cos (x) \sin (x)}{2 \left (a^2+b^2\right )^2}\right )+\frac{\left (2 a^2 b^2\right ) \int \frac{-a+b \cot (x)}{b+a \cot (x)} \, dx}{\left (a^2+b^2\right )^3}\\ &=\frac{a^3 b x}{\left (a^2+b^2\right )^3}-\frac{a b^3 x}{\left (a^2+b^2\right )^3}+\frac{a b \left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^3}-\frac{a^2 b}{\left (a^2+b^2\right )^2 (b+a \cot (x))}-\frac{a^4 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^3}+\frac{3 a^2 b^2 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^3}-\frac{a^2 \sin ^2(x)}{2 \left (a^2+b^2\right )^2}+\frac{b^2 \sin ^2(x)}{2 \left (a^2+b^2\right )^2}+2 \left (\frac{a b x}{2 \left (a^2+b^2\right )^2}-\frac{a b \cos (x) \sin (x)}{2 \left (a^2+b^2\right )^2}\right )\\ \end{align*}
Mathematica [C] time = 1.43766, size = 226, normalized size = 1.75 \[ \frac{a \cos (x) \left (\left (a^4-b^4\right ) \cos (2 x)+2 a \left (-b \left (a^2+b^2\right ) \sin (2 x)-a \left (a^2-3 b^2\right ) \log \left ((a \cos (x)+b \sin (x))^2\right )+2 x (-b+i a)^3\right )\right )-b \sin (x) \left (\left (b^4-a^4\right ) \cos (2 x)+2 a \left (2 \left (-3 a^2 b x+a^3 (1+i x)+a b^2 (1-3 i x)+b^3 x\right )+b \left (a^2+b^2\right ) \sin (2 x)+a \left (a^2-3 b^2\right ) \log \left ((a \cos (x)+b \sin (x))^2\right )\right )\right )+4 i a^2 \left (a^2-3 b^2\right ) \tan ^{-1}(\tan (x)) (a \cos (x)+b \sin (x))}{4 \left (a^2+b^2\right )^3 (a \cos (x)+b \sin (x))} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.101, size = 243, normalized size = 1.9 \begin{align*}{\frac{{a}^{3}}{ \left ({a}^{2}+{b}^{2} \right ) ^{2} \left ( a+b\tan \left ( x \right ) \right ) }}-{\frac{{a}^{4}\ln \left ( a+b\tan \left ( x \right ) \right ) }{ \left ({a}^{2}+{b}^{2} \right ) ^{3}}}+3\,{\frac{{a}^{2}\ln \left ( a+b\tan \left ( x \right ) \right ){b}^{2}}{ \left ({a}^{2}+{b}^{2} \right ) ^{3}}}-{\frac{\tan \left ( x \right ){a}^{3}b}{ \left ({a}^{2}+{b}^{2} \right ) ^{3} \left ( \left ( \tan \left ( x \right ) \right ) ^{2}+1 \right ) }}-{\frac{a\tan \left ( x \right ){b}^{3}}{ \left ({a}^{2}+{b}^{2} \right ) ^{3} \left ( \left ( \tan \left ( x \right ) \right ) ^{2}+1 \right ) }}+{\frac{{a}^{4}}{2\, \left ({a}^{2}+{b}^{2} \right ) ^{3} \left ( \left ( \tan \left ( x \right ) \right ) ^{2}+1 \right ) }}-{\frac{{b}^{4}}{2\, \left ({a}^{2}+{b}^{2} \right ) ^{3} \left ( \left ( \tan \left ( x \right ) \right ) ^{2}+1 \right ) }}+{\frac{\ln \left ( \left ( \tan \left ( x \right ) \right ) ^{2}+1 \right ){a}^{4}}{2\, \left ({a}^{2}+{b}^{2} \right ) ^{3}}}-{\frac{3\,\ln \left ( \left ( \tan \left ( x \right ) \right ) ^{2}+1 \right ){a}^{2}{b}^{2}}{2\, \left ({a}^{2}+{b}^{2} \right ) ^{3}}}+3\,{\frac{\arctan \left ( \tan \left ( x \right ) \right ){a}^{3}b}{ \left ({a}^{2}+{b}^{2} \right ) ^{3}}}-{\frac{a\arctan \left ( \tan \left ( x \right ) \right ){b}^{3}}{ \left ({a}^{2}+{b}^{2} \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.70432, size = 350, normalized size = 2.71 \begin{align*} \frac{{\left (3 \, a^{3} b - a b^{3}\right )} x}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac{{\left (a^{4} - 3 \, a^{2} b^{2}\right )} \log \left (b \tan \left (x\right ) + a\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac{{\left (a^{4} - 3 \, a^{2} b^{2}\right )} \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{3 \, a^{3} - a b^{2} + 2 \,{\left (a^{3} - a b^{2}\right )} \tan \left (x\right )^{2} -{\left (a^{2} b + b^{3}\right )} \tan \left (x\right )}{2 \,{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4} +{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \tan \left (x\right )^{3} +{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \tan \left (x\right )^{2} +{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \tan \left (x\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.559608, size = 531, normalized size = 4.12 \begin{align*} \frac{2 \,{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (x\right )^{3} -{\left (a^{5} + 3 \, a b^{4} - 4 \,{\left (3 \, a^{4} b - a^{2} b^{3}\right )} x\right )} \cos \left (x\right ) - 2 \,{\left ({\left (a^{5} - 3 \, a^{3} b^{2}\right )} \cos \left (x\right ) +{\left (a^{4} b - 3 \, a^{2} b^{3}\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 ) -{\left (5 \, a^{4} b - b^{5} + 2 \,{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (x\right )^{2} - 4 \,{\left (3 \, a^{3} b^{2} - a b^{4}\right )} x\right )} \sin \left (x\right )}{4 \,{\left ({\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (x\right ) +{\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \sin \left (x\right )\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 = 1.12032, size = 301, normalized size = 2.33 \begin{align*} \frac{{\left (3 \, a^{3} b - a b^{3}\right )} x}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac{{\left (a^{4} - 3 \, a^{2} b^{2}\right )} \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 (a^{4} b - 3 \, a^{2} b^{3}\right )} \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{2 \, a^{3} \tan \left (x\right )^{2} - 2 \, a b^{2} \tan \left (x\right )^{2} - a^{2} b \tan \left (x\right ) - b^{3} \tan \left (x\right ) + 3 \, a^{3} - a b^{2}}{2 \,{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}{\left (b \tan \left (x\right )^{3} + a \tan \left (x\right )^{2} + b \tan \left (x\right ) + a\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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