Optimal. Leaf size=82 \[ \frac {x \left (2 a^2+b^2\right )}{2 b^3}-\frac {2 a^3 \tan ^{-1}\left (\frac {a \tan \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2}}+\frac {a \cos (x)}{b^2}-\frac {\sin (x) \cos (x)}{2 b} \]
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Rubi [A] time = 0.16, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2793, 3023, 2735, 2660, 618, 204} \[ \frac {x \left (2 a^2+b^2\right )}{2 b^3}-\frac {2 a^3 \tan ^{-1}\left (\frac {a \tan \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2}}+\frac {a \cos (x)}{b^2}-\frac {\sin (x) \cos (x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 2660
Rule 2735
Rule 2793
Rule 3023
Rubi steps
\begin {align*} \int \frac {\sin ^3(x)}{a+b \sin (x)} \, dx &=-\frac {\cos (x) \sin (x)}{2 b}+\frac {\int \frac {a+b \sin (x)-2 a \sin ^2(x)}{a+b \sin (x)} \, dx}{2 b}\\ &=\frac {a \cos (x)}{b^2}-\frac {\cos (x) \sin (x)}{2 b}+\frac {\int \frac {a b+\left (2 a^2+b^2\right ) \sin (x)}{a+b \sin (x)} \, dx}{2 b^2}\\ &=\frac {\left (2 a^2+b^2\right ) x}{2 b^3}+\frac {a \cos (x)}{b^2}-\frac {\cos (x) \sin (x)}{2 b}-\frac {a^3 \int \frac {1}{a+b \sin (x)} \, dx}{b^3}\\ &=\frac {\left (2 a^2+b^2\right ) x}{2 b^3}+\frac {a \cos (x)}{b^2}-\frac {\cos (x) \sin (x)}{2 b}-\frac {\left (2 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{b^3}\\ &=\frac {\left (2 a^2+b^2\right ) x}{2 b^3}+\frac {a \cos (x)}{b^2}-\frac {\cos (x) \sin (x)}{2 b}+\frac {\left (4 a^3\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^3}\\ &=\frac {\left (2 a^2+b^2\right ) x}{2 b^3}-\frac {2 a^3 \tan ^{-1}\left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2}}+\frac {a \cos (x)}{b^2}-\frac {\cos (x) \sin (x)}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 78, normalized size = 0.95 \[ \frac {4 a^2 x-\frac {8 a^3 \tan ^{-1}\left (\frac {a \tan \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+4 a b \cos (x)+2 b^2 x-b^2 \sin (2 x)}{4 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 291, normalized size = 3.55 \[ \left [-\frac {\sqrt {-a^{2} + b^{2}} a^{3} \log \left (-\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \relax (x)^{2} - 2 \, a b \sin \relax (x) - a^{2} - b^{2} - 2 \, {\left (a \cos \relax (x) \sin \relax (x) + b \cos \relax (x)\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \relax (x)^{2} - 2 \, a b \sin \relax (x) - a^{2} - b^{2}}\right ) + {\left (a^{2} b^{2} - b^{4}\right )} \cos \relax (x) \sin \relax (x) - {\left (2 \, a^{4} - a^{2} b^{2} - b^{4}\right )} x - 2 \, {\left (a^{3} b - a b^{3}\right )} \cos \relax (x)}{2 \, {\left (a^{2} b^{3} - b^{5}\right )}}, \frac {2 \, \sqrt {a^{2} - b^{2}} a^{3} \arctan \left (-\frac {a \sin \relax (x) + b}{\sqrt {a^{2} - b^{2}} \cos \relax (x)}\right ) - {\left (a^{2} b^{2} - b^{4}\right )} \cos \relax (x) \sin \relax (x) + {\left (2 \, a^{4} - a^{2} b^{2} - b^{4}\right )} x + 2 \, {\left (a^{3} b - a b^{3}\right )} \cos \relax (x)}{2 \, {\left (a^{2} b^{3} - b^{5}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 112, normalized size = 1.37 \[ -\frac {2 \, {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (a) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, x\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )} a^{3}}{\sqrt {a^{2} - b^{2}} b^{3}} + \frac {{\left (2 \, a^{2} + b^{2}\right )} x}{2 \, b^{3}} + \frac {b \tan \left (\frac {1}{2} \, x\right )^{3} + 2 \, a \tan \left (\frac {1}{2} \, x\right )^{2} - b \tan \left (\frac {1}{2} \, x\right ) + 2 \, a}{{\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 142, normalized size = 1.73 \[ \frac {\tan ^{3}\left (\frac {x}{2}\right )}{b \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{2}}+\frac {2 \left (\tan ^{2}\left (\frac {x}{2}\right )\right ) a}{b^{2} \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{2}}-\frac {\tan \left (\frac {x}{2}\right )}{b \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{2}}+\frac {2 a}{b^{2} \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{2}}+\frac {2 \arctan \left (\tan \left (\frac {x}{2}\right )\right ) a^{2}}{b^{3}}-\frac {2 a^{3} \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{3} \sqrt {a^{2}-b^{2}}}+\frac {x}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.08, size = 1004, normalized size = 12.24 \[ \frac {\frac {2\,a}{b^2}-\frac {\mathrm {tan}\left (\frac {x}{2}\right )}{b}+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{b}+\frac {2\,a\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{b^2}}{{\mathrm {tan}\left (\frac {x}{2}\right )}^4+2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+1}-\frac {\mathrm {atan}\left (\frac {40\,a^3\,\mathrm {tan}\left (\frac {x}{2}\right )}{8\,a\,b^2+40\,a^3+\frac {48\,a^5}{b^2}}+\frac {48\,a^5\,\mathrm {tan}\left (\frac {x}{2}\right )}{48\,a^5+40\,a^3\,b^2+8\,a\,b^4}+\frac {8\,a\,b\,\mathrm {tan}\left (\frac {x}{2}\right )}{8\,a\,b+\frac {40\,a^3}{b}+\frac {48\,a^5}{b^3}}\right )\,\left (a^2\,2{}\mathrm {i}+b^2\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{b^3}+\frac {a^3\,\mathrm {atan}\left (\frac {\frac {a^3\,\left (\frac {8\,\left (4\,a^6\,b^2+4\,a^4\,b^4+a^2\,b^6\right )}{b^5}+\frac {8\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (-8\,a^7\,b^2+4\,a^5\,b^4+7\,a^3\,b^6+2\,a\,b^8\right )}{b^6}+\frac {a^3\,\left (64\,a^4\,\mathrm {tan}\left (\frac {x}{2}\right )+\frac {8\,\left (2\,a^3\,b^6+2\,a\,b^8\right )}{b^5}+\frac {a^3\,\left (32\,a^2\,b^3+\frac {8\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (12\,a\,b^{10}-8\,a^3\,b^8\right )}{b^6}\right )}{b^3\,\sqrt {b^2-a^2}}\right )}{b^3\,\sqrt {b^2-a^2}}\right )\,1{}\mathrm {i}}{b^3\,\sqrt {b^2-a^2}}+\frac {a^3\,\left (\frac {8\,\left (4\,a^6\,b^2+4\,a^4\,b^4+a^2\,b^6\right )}{b^5}+\frac {8\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (-8\,a^7\,b^2+4\,a^5\,b^4+7\,a^3\,b^6+2\,a\,b^8\right )}{b^6}-\frac {a^3\,\left (64\,a^4\,\mathrm {tan}\left (\frac {x}{2}\right )+\frac {8\,\left (2\,a^3\,b^6+2\,a\,b^8\right )}{b^5}-\frac {a^3\,\left (32\,a^2\,b^3+\frac {8\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (12\,a\,b^{10}-8\,a^3\,b^8\right )}{b^6}\right )}{b^3\,\sqrt {b^2-a^2}}\right )}{b^3\,\sqrt {b^2-a^2}}\right )\,1{}\mathrm {i}}{b^3\,\sqrt {b^2-a^2}}}{\frac {16\,\left (2\,a^7+a^5\,b^2\right )}{b^5}+\frac {16\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (8\,a^8+8\,a^6\,b^2+2\,a^4\,b^4\right )}{b^6}+\frac {a^3\,\left (\frac {8\,\left (4\,a^6\,b^2+4\,a^4\,b^4+a^2\,b^6\right )}{b^5}+\frac {8\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (-8\,a^7\,b^2+4\,a^5\,b^4+7\,a^3\,b^6+2\,a\,b^8\right )}{b^6}+\frac {a^3\,\left (64\,a^4\,\mathrm {tan}\left (\frac {x}{2}\right )+\frac {8\,\left (2\,a^3\,b^6+2\,a\,b^8\right )}{b^5}+\frac {a^3\,\left (32\,a^2\,b^3+\frac {8\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (12\,a\,b^{10}-8\,a^3\,b^8\right )}{b^6}\right )}{b^3\,\sqrt {b^2-a^2}}\right )}{b^3\,\sqrt {b^2-a^2}}\right )}{b^3\,\sqrt {b^2-a^2}}-\frac {a^3\,\left (\frac {8\,\left (4\,a^6\,b^2+4\,a^4\,b^4+a^2\,b^6\right )}{b^5}+\frac {8\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (-8\,a^7\,b^2+4\,a^5\,b^4+7\,a^3\,b^6+2\,a\,b^8\right )}{b^6}-\frac {a^3\,\left (64\,a^4\,\mathrm {tan}\left (\frac {x}{2}\right )+\frac {8\,\left (2\,a^3\,b^6+2\,a\,b^8\right )}{b^5}-\frac {a^3\,\left (32\,a^2\,b^3+\frac {8\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (12\,a\,b^{10}-8\,a^3\,b^8\right )}{b^6}\right )}{b^3\,\sqrt {b^2-a^2}}\right )}{b^3\,\sqrt {b^2-a^2}}\right )}{b^3\,\sqrt {b^2-a^2}}}\right )\,2{}\mathrm {i}}{b^3\,\sqrt {b^2-a^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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