Optimal. Leaf size=193 \[ -\frac{b \sin ^5(x)}{5 \left (a^2+b^2\right )}+\frac{b \sin ^3(x)}{3 \left (a^2+b^2\right )}-\frac{a^2 b \sin ^3(x)}{3 \left (a^2+b^2\right )^2}+\frac{a^2 b^3 \sin (x)}{\left (a^2+b^2\right )^3}+\frac{a \cos ^5(x)}{5 \left (a^2+b^2\right )}-\frac{a \cos ^3(x)}{3 \left (a^2+b^2\right )}+\frac{a b^2 \cos ^3(x)}{3 \left (a^2+b^2\right )^2}-\frac{a^3 b^2 \cos (x)}{\left (a^2+b^2\right )^3}+\frac{a^3 b^3 \tanh ^{-1}\left (\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}} \]
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Rubi [A] time = 0.358671, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {3109, 2564, 14, 2565, 30, 2637, 2638, 3074, 206} \[ -\frac{b \sin ^5(x)}{5 \left (a^2+b^2\right )}+\frac{b \sin ^3(x)}{3 \left (a^2+b^2\right )}-\frac{a^2 b \sin ^3(x)}{3 \left (a^2+b^2\right )^2}+\frac{a^2 b^3 \sin (x)}{\left (a^2+b^2\right )^3}+\frac{a \cos ^5(x)}{5 \left (a^2+b^2\right )}-\frac{a \cos ^3(x)}{3 \left (a^2+b^2\right )}+\frac{a b^2 \cos ^3(x)}{3 \left (a^2+b^2\right )^2}-\frac{a^3 b^2 \cos (x)}{\left (a^2+b^2\right )^3}+\frac{a^3 b^3 \tanh ^{-1}\left (\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 3109
Rule 2564
Rule 14
Rule 2565
Rule 30
Rule 2637
Rule 2638
Rule 3074
Rule 206
Rubi steps
\begin{align*} \int \frac{\cos ^3(x) \sin ^3(x)}{a \cos (x)+b \sin (x)} \, dx &=\frac{a \int \cos ^2(x) \sin ^3(x) \, dx}{a^2+b^2}+\frac{b \int \cos ^3(x) \sin ^2(x) \, dx}{a^2+b^2}-\frac{(a b) \int \frac{\cos ^2(x) \sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx}{a^2+b^2}\\ &=-\frac{\left (a^2 b\right ) \int \cos (x) \sin ^2(x) \, dx}{\left (a^2+b^2\right )^2}-\frac{\left (a b^2\right ) \int \cos ^2(x) \sin (x) \, dx}{\left (a^2+b^2\right )^2}+\frac{\left (a^2 b^2\right ) \int \frac{\cos (x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^2}-\frac{a \operatorname{Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,\cos (x)\right )}{a^2+b^2}+\frac{b \operatorname{Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,\sin (x)\right )}{a^2+b^2}\\ &=\frac{\left (a^3 b^2\right ) \int \sin (x) \, dx}{\left (a^2+b^2\right )^3}+\frac{\left (a^2 b^3\right ) \int \cos (x) \, dx}{\left (a^2+b^2\right )^3}-\frac{\left (a^3 b^3\right ) \int \frac{1}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^3}-\frac{\left (a^2 b\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,\sin (x)\right )}{\left (a^2+b^2\right )^2}+\frac{\left (a b^2\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,\cos (x)\right )}{\left (a^2+b^2\right )^2}-\frac{a \operatorname{Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,\cos (x)\right )}{a^2+b^2}+\frac{b \operatorname{Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,\sin (x)\right )}{a^2+b^2}\\ &=-\frac{a^3 b^2 \cos (x)}{\left (a^2+b^2\right )^3}+\frac{a b^2 \cos ^3(x)}{3 \left (a^2+b^2\right )^2}-\frac{a \cos ^3(x)}{3 \left (a^2+b^2\right )}+\frac{a \cos ^5(x)}{5 \left (a^2+b^2\right )}+\frac{a^2 b^3 \sin (x)}{\left (a^2+b^2\right )^3}-\frac{a^2 b \sin ^3(x)}{3 \left (a^2+b^2\right )^2}+\frac{b \sin ^3(x)}{3 \left (a^2+b^2\right )}-\frac{b \sin ^5(x)}{5 \left (a^2+b^2\right )}+\frac{\left (a^3 b^3\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 )^3}\\ &=\frac{a^3 b^3 \tanh ^{-1}\left (\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}-\frac{a^3 b^2 \cos (x)}{\left (a^2+b^2\right )^3}+\frac{a b^2 \cos ^3(x)}{3 \left (a^2+b^2\right )^2}-\frac{a \cos ^3(x)}{3 \left (a^2+b^2\right )}+\frac{a \cos ^5(x)}{5 \left (a^2+b^2\right )}+\frac{a^2 b^3 \sin (x)}{\left (a^2+b^2\right )^3}-\frac{a^2 b \sin ^3(x)}{3 \left (a^2+b^2\right )^2}+\frac{b \sin ^3(x)}{3 \left (a^2+b^2\right )}-\frac{b \sin ^5(x)}{5 \left (a^2+b^2\right )}\\ \end{align*}
Mathematica [A] time = 1.55243, size = 223, normalized size = 1.16 \[ \frac{240 a^2 b^3 \sin (x)+10 a^2 b^3 \sin (3 x)-6 a^2 b^3 \sin (5 x)+6 a^3 b^2 \cos (5 x)-30 a \left (8 a^2 b^2+a^4-b^4\right ) \cos (x)-5 a \left (-2 a^2 b^2+a^4-3 b^4\right ) \cos (3 x)-30 a^4 b \sin (x)+15 a^4 b \sin (3 x)-3 a^4 b \sin (5 x)+3 a^5 \cos (5 x)+3 a b^4 \cos (5 x)+30 b^5 \sin (x)-5 b^5 \sin (3 x)-3 b^5 \sin (5 x)}{240 \left (a^2+b^2\right )^3}-\frac{2 a^3 b^3 \tanh ^{-1}\left (\frac{a \tan \left (\frac{x}{2}\right )-b}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.111, size = 305, normalized size = 1.6 \begin{align*} -2\,{\frac{1}{ \left ({a}^{4}+2\,{a}^{2}{b}^{2}+{b}^{4} \right ) \left ({a}^{2}+{b}^{2} \right ) \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{5}} \left ( -{a}^{2}{b}^{3} \left ( \tan \left ( x/2 \right ) \right ) ^{9}-a{b}^{4} \left ( \tan \left ( x/2 \right ) \right ) ^{8}+ \left ( -16/3\,{a}^{2}{b}^{3}-4/3\,{b}^{5} \right ) \left ( \tan \left ( x/2 \right ) \right ) ^{7}+ \left ( 2\,{a}^{5}+6\,{a}^{3}{b}^{2} \right ) \left ( \tan \left ( x/2 \right ) \right ) ^{6}+ \left ({\frac{16\,{a}^{4}b}{5}}-{\frac{34\,{a}^{2}{b}^{3}}{15}}+{\frac{8\,{b}^{5}}{15}} \right ) \left ( \tan \left ( x/2 \right ) \right ) ^{5}+ \left ( -2/3\,{a}^{5}+10/3\,{a}^{3}{b}^{2}-2\,a{b}^{4} \right ) \left ( \tan \left ( x/2 \right ) \right ) ^{4}+ \left ( -16/3\,{a}^{2}{b}^{3}-4/3\,{b}^{5} \right ) \left ( \tan \left ( x/2 \right ) \right ) ^{3}+ \left ( 2/3\,{a}^{5}+14/3\,{a}^{3}{b}^{2} \right ) \left ( \tan \left ( x/2 \right ) \right ) ^{2}-{a}^{2}{b}^{3}\tan \left ( x/2 \right ) +2/15\,{a}^{5}+{\frac{14\,{a}^{3}{b}^{2}}{15}}-1/5\,a{b}^{4} \right ) }-16\,{\frac{{a}^{3}{b}^{3}}{ \left ( 8\,{a}^{6}+24\,{a}^{4}{b}^{2}+24\,{a}^{2}{b}^{4}+8\,{b}^{6} \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.
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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 [A] time = 0.583998, size = 706, normalized size = 3.66 \begin{align*} \frac{15 \, \sqrt{a^{2} + b^{2}} a^{3} b^{3} \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 ) + 6 \,{\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (x\right )^{5} - 10 \,{\left (a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \cos \left (x\right )^{3} - 30 \,{\left (a^{5} b^{2} + a^{3} b^{4}\right )} \cos \left (x\right ) - 2 \,{\left (3 \, a^{6} b - 11 \, a^{4} b^{3} - 16 \, a^{2} b^{5} - 2 \, b^{7} + 3 \,{\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \cos \left (x\right )^{4} -{\left (6 \, a^{6} b + 13 \, a^{4} b^{3} + 8 \, a^{2} b^{5} + b^{7}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )}{30 \,{\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\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 [B] time = 1.23626, size = 487, normalized size = 2.52 \begin{align*} \frac{a^{3} b^{3} \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^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \sqrt{a^{2} + b^{2}}} + \frac{2 \,{\left (15 \, a^{2} b^{3} \tan \left (\frac{1}{2} \, x\right )^{9} + 15 \, a b^{4} \tan \left (\frac{1}{2} \, x\right )^{8} + 80 \, a^{2} b^{3} \tan \left (\frac{1}{2} \, x\right )^{7} + 20 \, b^{5} \tan \left (\frac{1}{2} \, x\right )^{7} - 30 \, a^{5} \tan \left (\frac{1}{2} \, x\right )^{6} - 90 \, a^{3} b^{2} \tan \left (\frac{1}{2} \, x\right )^{6} - 48 \, a^{4} b \tan \left (\frac{1}{2} \, x\right )^{5} + 34 \, a^{2} b^{3} \tan \left (\frac{1}{2} \, x\right )^{5} - 8 \, b^{5} \tan \left (\frac{1}{2} \, x\right )^{5} + 10 \, a^{5} \tan \left (\frac{1}{2} \, x\right )^{4} - 50 \, a^{3} b^{2} \tan \left (\frac{1}{2} \, x\right )^{4} + 30 \, a b^{4} \tan \left (\frac{1}{2} \, x\right )^{4} + 80 \, a^{2} b^{3} \tan \left (\frac{1}{2} \, x\right )^{3} + 20 \, b^{5} \tan \left (\frac{1}{2} \, x\right )^{3} - 10 \, a^{5} \tan \left (\frac{1}{2} \, x\right )^{2} - 70 \, a^{3} b^{2} \tan \left (\frac{1}{2} \, x\right )^{2} + 15 \, a^{2} b^{3} \tan \left (\frac{1}{2} \, x\right ) - 2 \, a^{5} - 14 \, a^{3} b^{2} + 3 \, a b^{4}\right )}}{15 \,{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}{\left (\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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