Optimal. Leaf size=66 \[ -\frac {2 b \tan ^{-1}\left (\frac {a \tan \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}-\frac {a \cos (x)}{\left (a^2-b^2\right ) (a+b \sin (x))} \]
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Rubi [A] time = 0.06, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {2754, 12, 2660, 618, 204} \[ -\frac {2 b \tan ^{-1}\left (\frac {a \tan \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}-\frac {a \cos (x)}{\left (a^2-b^2\right ) (a+b \sin (x))} \]
Antiderivative was successfully verified.
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Rule 12
Rule 204
Rule 618
Rule 2660
Rule 2754
Rubi steps
\begin {align*} \int \frac {\sin (x)}{(a+b \sin (x))^2} \, dx &=-\frac {a \cos (x)}{\left (a^2-b^2\right ) (a+b \sin (x))}+\frac {\int \frac {b}{a+b \sin (x)} \, dx}{-a^2+b^2}\\ &=-\frac {a \cos (x)}{\left (a^2-b^2\right ) (a+b \sin (x))}-\frac {b \int \frac {1}{a+b \sin (x)} \, dx}{a^2-b^2}\\ &=-\frac {a \cos (x)}{\left (a^2-b^2\right ) (a+b \sin (x))}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{a^2-b^2}\\ &=-\frac {a \cos (x)}{\left (a^2-b^2\right ) (a+b \sin (x))}+\frac {(4 b) \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 )}{a^2-b^2}\\ &=-\frac {2 b \tan ^{-1}\left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}-\frac {a \cos (x)}{\left (a^2-b^2\right ) (a+b \sin (x))}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 67, normalized size = 1.02 \[ -\frac {2 b \tan ^{-1}\left (\frac {a \tan \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}-\frac {a \cos (x)}{(a-b) (a+b) (a+b \sin (x))} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 266, normalized size = 4.03 \[ \left [\frac {{\left (b^{2} \sin \relax (x) + a b\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \relax (x)^{2} - 2 \, a b \sin \relax (x) - a^{2} - b^{2} + 2 \, {\left (a \cos \relax (x) \sin \relax (x) + b \cos \relax (x)\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \relax (x)^{2} - 2 \, a b \sin \relax (x) - a^{2} - b^{2}}\right ) - 2 \, {\left (a^{3} - a b^{2}\right )} \cos \relax (x)}{2 \, {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4} + {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \sin \relax (x)\right )}}, \frac {{\left (b^{2} \sin \relax (x) + a b\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \relax (x) + b}{\sqrt {a^{2} - b^{2}} \cos \relax (x)}\right ) - {\left (a^{3} - a b^{2}\right )} \cos \relax (x)}{a^{5} - 2 \, a^{3} b^{2} + a b^{4} + {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \sin \relax (x)}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 90, normalized size = 1.36 \[ -\frac {2 \, {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (a) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, x\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )} b}{{\left (a^{2} - b^{2}\right )}^{\frac {3}{2}}} - \frac {2 \, {\left (b \tan \left (\frac {1}{2} \, x\right ) + a\right )}}{{\left (a \tan \left (\frac {1}{2} \, x\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, x\right ) + a\right )} {\left (a^{2} - b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 99, normalized size = 1.50 \[ \frac {-8 \tan \left (\frac {x}{2}\right ) b -8 a}{\left (4 a^{2}-4 b^{2}\right ) \left (\left (\tan ^{2}\left (\frac {x}{2}\right )\right ) a +2 \tan \left (\frac {x}{2}\right ) b +a \right )}-\frac {8 b \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (4 a^{2}-4 b^{2}\right ) \sqrt {a^{2}-b^{2}}} \]
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 = 6.49, size = 123, normalized size = 1.86 \[ -\frac {\frac {2\,a}{a^2-b^2}+\frac {2\,b\,\mathrm {tan}\left (\frac {x}{2}\right )}{a^2-b^2}}{a\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {x}{2}\right )+a}-\frac {2\,b\,\mathrm {atan}\left (\frac {\left (a^2-b^2\right )\,\left (\frac {2\,b^2}{{\left (a+b\right )}^{3/2}\,{\left (a-b\right )}^{3/2}}+\frac {2\,a\,b\,\mathrm {tan}\left (\frac {x}{2}\right )}{{\left (a+b\right )}^{3/2}\,{\left (a-b\right )}^{3/2}}\right )}{2\,b}\right )}{{\left (a+b\right )}^{3/2}\,{\left (a-b\right )}^{3/2}} \]
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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