Optimal. Leaf size=107 \[ \frac{-b \left (a^2+b^2\right ) \sin (2 x)+a \left (a^2+b^2\right ) \cos (2 x)+3 a \left (a^2-b^2\right )}{2 \left (a^2+b^2\right )^2 (a \cos (x)+b \sin (x))}+\frac{6 a^2 b \tanh ^{-1}\left (\frac{a \tan \left (\frac{x}{2}\right )-b}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}} \]
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Rubi [B] time = 1.16922, antiderivative size = 283, normalized size of antiderivative = 2.64, number of steps used = 19, number of rules used = 11, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.688, Rules used = {4401, 2637, 2638, 6742, 639, 203, 638, 618, 206, 3100, 3074} \[ \frac{3 a^3 \sin (x)}{b^3 \left (a^2+b^2\right )}+\frac{3 a^2 \cos (x)}{b^2 \left (a^2+b^2\right )}+\frac{2 a^2 \left (a+b \tan \left (\frac{x}{2}\right )\right )}{\left (a^2+b^2\right )^2 \left (-a \tan ^2\left (\frac{x}{2}\right )+a+2 b \tan \left (\frac{x}{2}\right )\right )}-\frac{2 a^3 \cos ^2\left (\frac{x}{2}\right ) \left (\left (a^2-b^2\right ) \tan \left (\frac{x}{2}\right )+2 a b\right )}{b^3 \left (a^2+b^2\right )^2}+\frac{2 a^2 \left (3 a^2+b^2\right ) \tanh ^{-1}\left (\frac{b-a \tan \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{5/2}}-\frac{2 a^2 b \tanh ^{-1}\left (\frac{b-a \tan \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}-\frac{3 a^2 \tanh ^{-1}\left (\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2}}-\frac{2 a \sin (x)}{b^3}-\frac{\cos (x)}{b^2} \]
Antiderivative was successfully verified.
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Rule 4401
Rule 2637
Rule 2638
Rule 6742
Rule 639
Rule 203
Rule 638
Rule 618
Rule 206
Rule 3100
Rule 3074
Rubi steps
\begin{align*} \int \frac{\sin ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx &=\int \left (-\frac{2 a \cos (x)}{b^3}+\frac{\sin (x)}{b^2}-\frac{a^3 \cos ^3(x)}{b^3 (a \cos (x)+b \sin (x))^2}+\frac{3 a^2 \cos ^2(x)}{b^3 (a \cos (x)+b \sin (x))}\right ) \, dx\\ &=-\frac{(2 a) \int \cos (x) \, dx}{b^3}+\frac{\left (3 a^2\right ) \int \frac{\cos ^2(x)}{a \cos (x)+b \sin (x)} \, dx}{b^3}-\frac{a^3 \int \frac{\cos ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx}{b^3}+\frac{\int \sin (x) \, dx}{b^2}\\ &=-\frac{\cos (x)}{b^2}+\frac{3 a^2 \cos (x)}{b^2 \left (a^2+b^2\right )}-\frac{2 a \sin (x)}{b^3}-\frac{\left (2 a^3\right ) \operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^3}{\left (1+x^2\right )^2 \left (a+2 b x-a x^2\right )^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )}{b^3}+\frac{\left (3 a^3\right ) \int \cos (x) \, dx}{b^3 \left (a^2+b^2\right )}+\frac{\left (3 a^2\right ) \int \frac{1}{a \cos (x)+b \sin (x)} \, dx}{b \left (a^2+b^2\right )}\\ &=-\frac{\cos (x)}{b^2}+\frac{3 a^2 \cos (x)}{b^2 \left (a^2+b^2\right )}-\frac{2 a \sin (x)}{b^3}+\frac{3 a^3 \sin (x)}{b^3 \left (a^2+b^2\right )}-\frac{\left (2 a^3\right ) \operatorname{Subst}\left (\int \left (\frac{2 \left (a^2-b^2-2 a b x\right )}{\left (a^2+b^2\right )^2 \left (1+x^2\right )^2}+\frac{-a^2+b^2}{\left (a^2+b^2\right )^2 \left (1+x^2\right )}-\frac{2 b^3 x}{a \left (a^2+b^2\right ) \left (-a-2 b x+a x^2\right )^2}-\frac{b^2 \left (3 a^2+b^2\right )}{a \left (a^2+b^2\right )^2 \left (-a-2 b x+a x^2\right )}\right ) \, dx,x,\tan \left (\frac{x}{2}\right )\right )}{b^3}-\frac{\left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{a^2+b^2-x^2} \, dx,x,b \cos (x)-a \sin (x)\right )}{b \left (a^2+b^2\right )}\\ &=-\frac{3 a^2 \tanh ^{-1}\left (\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2}}-\frac{\cos (x)}{b^2}+\frac{3 a^2 \cos (x)}{b^2 \left (a^2+b^2\right )}-\frac{2 a \sin (x)}{b^3}+\frac{3 a^3 \sin (x)}{b^3 \left (a^2+b^2\right )}-\frac{\left (4 a^3\right ) \operatorname{Subst}\left (\int \frac{a^2-b^2-2 a b x}{\left (1+x^2\right )^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )}{b^3 \left (a^2+b^2\right )^2}+\frac{\left (2 a^3 \left (a^2-b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )}{b^3 \left (a^2+b^2\right )^2}+\frac{\left (4 a^2\right ) \operatorname{Subst}\left (\int \frac{x}{\left (-a-2 b x+a x^2\right )^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )}{a^2+b^2}+\frac{\left (2 a^2 \left (3 a^2+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-a-2 b x+a x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )}{b \left (a^2+b^2\right )^2}\\ &=\frac{a^3 \left (a^2-b^2\right ) x}{b^3 \left (a^2+b^2\right )^2}-\frac{3 a^2 \tanh ^{-1}\left (\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2}}-\frac{\cos (x)}{b^2}+\frac{3 a^2 \cos (x)}{b^2 \left (a^2+b^2\right )}-\frac{2 a \sin (x)}{b^3}+\frac{3 a^3 \sin (x)}{b^3 \left (a^2+b^2\right )}-\frac{2 a^3 \cos ^2\left (\frac{x}{2}\right ) \left (2 a b+\left (a^2-b^2\right ) \tan \left (\frac{x}{2}\right )\right )}{b^3 \left (a^2+b^2\right )^2}+\frac{2 a^2 \left (a+b \tan \left (\frac{x}{2}\right )\right )}{\left (a^2+b^2\right )^2 \left (a+2 b \tan \left (\frac{x}{2}\right )-a \tan ^2\left (\frac{x}{2}\right )\right )}-\frac{\left (2 a^2 b\right ) \operatorname{Subst}\left (\int \frac{1}{-a-2 b x+a x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )}{\left (a^2+b^2\right )^2}-\frac{\left (2 a^3 \left (a^2-b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )}{b^3 \left (a^2+b^2\right )^2}-\frac{\left (4 a^2 \left (3 a^2+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 \left (a^2+b^2\right )-x^2} \, dx,x,-2 b+2 a \tan \left (\frac{x}{2}\right )\right )}{b \left (a^2+b^2\right )^2}\\ &=-\frac{3 a^2 \tanh ^{-1}\left (\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2}}+\frac{2 a^2 \left (3 a^2+b^2\right ) \tanh ^{-1}\left (\frac{b-a \tan \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{5/2}}-\frac{\cos (x)}{b^2}+\frac{3 a^2 \cos (x)}{b^2 \left (a^2+b^2\right )}-\frac{2 a \sin (x)}{b^3}+\frac{3 a^3 \sin (x)}{b^3 \left (a^2+b^2\right )}-\frac{2 a^3 \cos ^2\left (\frac{x}{2}\right ) \left (2 a b+\left (a^2-b^2\right ) \tan \left (\frac{x}{2}\right )\right )}{b^3 \left (a^2+b^2\right )^2}+\frac{2 a^2 \left (a+b \tan \left (\frac{x}{2}\right )\right )}{\left (a^2+b^2\right )^2 \left (a+2 b \tan \left (\frac{x}{2}\right )-a \tan ^2\left (\frac{x}{2}\right )\right )}+\frac{\left (4 a^2 b\right ) \operatorname{Subst}\left (\int \frac{1}{4 \left (a^2+b^2\right )-x^2} \, dx,x,-2 b+2 a \tan \left (\frac{x}{2}\right )\right )}{\left (a^2+b^2\right )^2}\\ &=-\frac{3 a^2 \tanh ^{-1}\left (\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2}}-\frac{2 a^2 b \tanh ^{-1}\left (\frac{b-a \tan \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}+\frac{2 a^2 \left (3 a^2+b^2\right ) \tanh ^{-1}\left (\frac{b-a \tan \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{5/2}}-\frac{\cos (x)}{b^2}+\frac{3 a^2 \cos (x)}{b^2 \left (a^2+b^2\right )}-\frac{2 a \sin (x)}{b^3}+\frac{3 a^3 \sin (x)}{b^3 \left (a^2+b^2\right )}-\frac{2 a^3 \cos ^2\left (\frac{x}{2}\right ) \left (2 a b+\left (a^2-b^2\right ) \tan \left (\frac{x}{2}\right )\right )}{b^3 \left (a^2+b^2\right )^2}+\frac{2 a^2 \left (a+b \tan \left (\frac{x}{2}\right )\right )}{\left (a^2+b^2\right )^2 \left (a+2 b \tan \left (\frac{x}{2}\right )-a \tan ^2\left (\frac{x}{2}\right )\right )}\\ \end{align*}
Mathematica [A] time = 0.425441, size = 107, normalized size = 1. \[ \frac{-b \left (a^2+b^2\right ) \sin (2 x)+a \left (a^2+b^2\right ) \cos (2 x)+3 a \left (a^2-b^2\right )}{2 \left (a^2+b^2\right )^2 (a \cos (x)+b \sin (x))}+\frac{6 a^2 b \tanh ^{-1}\left (\frac{a \tan \left (\frac{x}{2}\right )-b}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.107, size = 141, normalized size = 1.3 \begin{align*} 4\,{\frac{-ab\tan \left ( x/2 \right ) +1/2\,{a}^{2}-1/2\,{b}^{2}}{ \left ({a}^{4}+2\,{a}^{2}{b}^{2}+{b}^{4} \right ) \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}+1 \right ) }}-4\,{\frac{{a}^{2}}{{a}^{4}+2\,{a}^{2}{b}^{2}+{b}^{4}} \left ({\frac{1/2\,b\tan \left ( x/2 \right ) +a/2}{ \left ( \tan \left ( x/2 \right ) \right ) ^{2}a-2\,b\tan \left ( x/2 \right ) -a}}-3/2\,{\frac{b}{\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 ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [B] time = 0.546829, size = 570, normalized size = 5.33 \begin{align*} \frac{2 \, a^{5} - 2 \, a^{3} b^{2} - 4 \, a b^{4} + 2 \,{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (x\right )^{2} - 2 \,{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (x\right ) \sin \left (x\right ) + 3 \,{\left (a^{3} b \cos \left (x\right ) + a^{2} b^{2} \sin \left (x\right )\right )} \sqrt{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 ({\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 = 1.20356, size = 251, normalized size = 2.35 \begin{align*} -\frac{3 \, a^{2} b \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^{2} b \tan \left (\frac{1}{2} \, x\right )^{3} - 3 \, a b^{2} \tan \left (\frac{1}{2} \, x\right )^{2} + a^{2} b \tan \left (\frac{1}{2} \, x\right ) - 2 \, b^{3} \tan \left (\frac{1}{2} \, x\right ) + 2 \, a^{3} - a b^{2}\right )}}{{\left (a \tan \left (\frac{1}{2} \, x\right )^{4} - 2 \, b \tan \left (\frac{1}{2} \, x\right )^{3} - 2 \, b \tan \left (\frac{1}{2} \, x\right ) - a\right )}{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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