Optimal. Leaf size=193 \[ -\frac {b \sin ^5(x)}{5 \left (a^2+b^2\right )}+\frac {b \sin ^3(x)}{3 \left (a^2+b^2\right )}-\frac {a^2 b \sin ^3(x)}{3 \left (a^2+b^2\right )^2}+\frac {a \cos ^5(x)}{5 \left (a^2+b^2\right )}-\frac {a \cos ^3(x)}{3 \left (a^2+b^2\right )}+\frac {a b^2 \cos ^3(x)}{3 \left (a^2+b^2\right )^2}+\frac {a^2 b^3 \sin (x)}{\left (a^2+b^2\right )^3}-\frac {a^3 b^2 \cos (x)}{\left (a^2+b^2\right )^3}+\frac {a^3 b^3 \tanh ^{-1}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}} \]
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Rubi [A] time = 0.36, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, 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 14
Rule 30
Rule 206
Rule 2564
Rule 2565
Rule 2637
Rule 2638
Rule 3074
Rule 3109
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*}
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Mathematica [A] time = 1.48, size = 223, normalized size = 1.16 \[ \frac {3 a^5 \cos (5 x)-30 a^4 b \sin (x)+15 a^4 b \sin (3 x)-3 a^4 b \sin (5 x)+6 a^3 b^2 \cos (5 x)+240 a^2 b^3 \sin (x)+10 a^2 b^3 \sin (3 x)-6 a^2 b^3 \sin (5 x)-30 a \left (a^4+8 a^2 b^2-b^4\right ) \cos (x)-5 a \left (a^4-2 a^2 b^2-3 b^4\right ) \cos (3 x)+3 a b^4 \cos (5 x)+30 b^5 \sin (x)-5 b^5 \sin (3 x)-3 b^5 \sin (5 x)}{240 \left (a^2+b^2\right )^3}-\frac {2 a^3 b^3 \tanh ^{-1}\left (\frac {a \tan \left (\frac {x}{2}\right )-b}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 307, normalized size = 1.59 \[ \frac {15 \, \sqrt {a^{2} + b^{2}} a^{3} b^{3} \log \left (\frac {2 \, a b \cos \relax (x) \sin \relax (x) + {\left (a^{2} - b^{2}\right )} \cos \relax (x)^{2} - 2 \, a^{2} - b^{2} - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \relax (x) - a \sin \relax (x)\right )}}{2 \, a b \cos \relax (x) \sin \relax (x) + {\left (a^{2} - b^{2}\right )} \cos \relax (x)^{2} + b^{2}}\right ) + 6 \, {\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \cos \relax (x)^{5} - 10 \, {\left (a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \cos \relax (x)^{3} - 30 \, {\left (a^{5} b^{2} + a^{3} b^{4}\right )} \cos \relax (x) - 2 \, {\left (3 \, a^{6} b - 11 \, a^{4} b^{3} - 16 \, a^{2} b^{5} - 2 \, b^{7} + 3 \, {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \cos \relax (x)^{4} - {\left (6 \, a^{6} b + 13 \, a^{4} b^{3} + 8 \, a^{2} b^{5} + b^{7}\right )} \cos \relax (x)^{2}\right )} \sin \relax (x)}{30 \, {\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.69, size = 361, normalized size = 1.87 \[ \frac {a^{3} b^{3} \log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, x\right ) - 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, x\right ) - 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, {\left (15 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, x\right )^{9} + 15 \, a b^{4} \tan \left (\frac {1}{2} \, x\right )^{8} + 80 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, x\right )^{7} + 20 \, b^{5} \tan \left (\frac {1}{2} \, x\right )^{7} - 30 \, a^{5} \tan \left (\frac {1}{2} \, x\right )^{6} - 90 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, x\right )^{6} - 48 \, a^{4} b \tan \left (\frac {1}{2} \, x\right )^{5} + 34 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, x\right )^{5} - 8 \, b^{5} \tan \left (\frac {1}{2} \, x\right )^{5} + 10 \, a^{5} \tan \left (\frac {1}{2} \, x\right )^{4} - 50 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, x\right )^{4} + 30 \, a b^{4} \tan \left (\frac {1}{2} \, x\right )^{4} + 80 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, x\right )^{3} + 20 \, b^{5} \tan \left (\frac {1}{2} \, x\right )^{3} - 10 \, a^{5} \tan \left (\frac {1}{2} \, x\right )^{2} - 70 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, x\right )^{2} + 15 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, x\right ) - 2 \, a^{5} - 14 \, a^{3} b^{2} + 3 \, a b^{4}\right )}}{15 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} {\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 305, normalized size = 1.58 \[ -\frac {16 a^{3} b^{3} \arctanh \left (\frac {2 a \tan \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (8 a^{6}+24 a^{4} b^{2}+24 a^{2} b^{4}+8 b^{6}\right ) \sqrt {a^{2}+b^{2}}}-\frac {2 \left (-a^{2} b^{3} \left (\tan ^{9}\left (\frac {x}{2}\right )\right )-a \,b^{4} \left (\tan ^{8}\left (\frac {x}{2}\right )\right )+\left (-\frac {16}{3} a^{2} b^{3}-\frac {4}{3} b^{5}\right ) \left (\tan ^{7}\left (\frac {x}{2}\right )\right )+\left (2 a^{5}+6 a^{3} b^{2}\right ) \left (\tan ^{6}\left (\frac {x}{2}\right )\right )+\left (\frac {16}{5} a^{4} b -\frac {34}{15} a^{2} b^{3}+\frac {8}{15} b^{5}\right ) \left (\tan ^{5}\left (\frac {x}{2}\right )\right )+\left (-\frac {2}{3} a^{5}+\frac {10}{3} a^{3} b^{2}-2 a \,b^{4}\right ) \left (\tan ^{4}\left (\frac {x}{2}\right )\right )+\left (-\frac {16}{3} a^{2} b^{3}-\frac {4}{3} b^{5}\right ) \left (\tan ^{3}\left (\frac {x}{2}\right )\right )+\left (\frac {2}{3} a^{5}+\frac {14}{3} a^{3} b^{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-a^{2} b^{3} \tan \left (\frac {x}{2}\right )+\frac {2 a^{5}}{15}+\frac {14 a^{3} b^{2}}{15}-\frac {a \,b^{4}}{5}\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 521, normalized size = 2.70 \[ \frac {a^{3} b^{3} \log \left (\frac {b - \frac {a \sin \relax (x)}{\cos \relax (x) + 1} + \sqrt {a^{2} + b^{2}}}{b - \frac {a \sin \relax (x)}{\cos \relax (x) + 1} - \sqrt {a^{2} + b^{2}}}\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 (2 \, a^{5} + 14 \, a^{3} b^{2} - 3 \, a b^{4} - \frac {15 \, a^{2} b^{3} \sin \relax (x)}{\cos \relax (x) + 1} - \frac {15 \, a b^{4} \sin \relax (x)^{8}}{{\left (\cos \relax (x) + 1\right )}^{8}} - \frac {15 \, a^{2} b^{3} \sin \relax (x)^{9}}{{\left (\cos \relax (x) + 1\right )}^{9}} + \frac {10 \, {\left (a^{5} + 7 \, a^{3} b^{2}\right )} \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} - \frac {20 \, {\left (4 \, a^{2} b^{3} + b^{5}\right )} \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} - \frac {10 \, {\left (a^{5} - 5 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} + \frac {2 \, {\left (24 \, a^{4} b - 17 \, a^{2} b^{3} + 4 \, b^{5}\right )} \sin \relax (x)^{5}}{{\left (\cos \relax (x) + 1\right )}^{5}} + \frac {30 \, {\left (a^{5} + 3 \, a^{3} b^{2}\right )} \sin \relax (x)^{6}}{{\left (\cos \relax (x) + 1\right )}^{6}} - \frac {20 \, {\left (4 \, a^{2} b^{3} + b^{5}\right )} \sin \relax (x)^{7}}{{\left (\cos \relax (x) + 1\right )}^{7}}\right )}}{15 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6} + \frac {5 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {10 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} + \frac {10 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \sin \relax (x)^{6}}{{\left (\cos \relax (x) + 1\right )}^{6}} + \frac {5 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \sin \relax (x)^{8}}{{\left (\cos \relax (x) + 1\right )}^{8}} + \frac {{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \sin \relax (x)^{10}}{{\left (\cos \relax (x) + 1\right )}^{10}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.64, size = 600, normalized size = 3.11 \[ \frac {\frac {8\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3\,\left (4\,a^2\,b^3+b^5\right )}{3\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}-\frac {4\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\left (a^5+7\,a^3\,b^2\right )}{3\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}-\frac {4\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6\,\left (a^5+3\,a^3\,b^2\right )}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}-\frac {2\,\left (2\,a^5+14\,a^3\,b^2-3\,a\,b^4\right )}{15\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {4\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4\,\left (a^5-5\,a^3\,b^2+3\,a\,b^4\right )}{3\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {8\,b^3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^7\,\left (4\,a^2+b^2\right )}{3\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {2\,a^2\,b^3\,\mathrm {tan}\left (\frac {x}{2}\right )}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}+\frac {2\,a\,b^4\,{\mathrm {tan}\left (\frac {x}{2}\right )}^8}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}+\frac {2\,a^2\,b^3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^9}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}-\frac {4\,b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5\,\left (24\,a^4-17\,a^2\,b^2+4\,b^4\right )}{15\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}}{{\mathrm {tan}\left (\frac {x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+1}+\frac {2\,a^3\,b^3\,\mathrm {atanh}\left (\frac {2\,a^6\,b+2\,b^7+6\,a^2\,b^5+6\,a^4\,b^3-2\,a\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}{2\,{\left (a^2+b^2\right )}^{7/2}}\right )}{{\left (a^2+b^2\right )}^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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