Optimal. Leaf size=50 \[ \frac{x}{b}-\frac{2 a \tan ^{-1}\left (\frac{a \tan \left (\frac{x}{2}\right )+b}{\sqrt{a^2-b^2}}\right )}{b \sqrt{a^2-b^2}} \]
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Rubi [A] time = 0.0552835, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {2735, 2660, 618, 204} \[ \frac{x}{b}-\frac{2 a \tan ^{-1}\left (\frac{a \tan \left (\frac{x}{2}\right )+b}{\sqrt{a^2-b^2}}\right )}{b \sqrt{a^2-b^2}} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2660
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{\sin (x)}{a+b \sin (x)} \, dx &=\frac{x}{b}-\frac{a \int \frac{1}{a+b \sin (x)} \, dx}{b}\\ &=\frac{x}{b}-\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )}{b}\\ &=\frac{x}{b}+\frac{(4 a) \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}\\ &=\frac{x}{b}-\frac{2 a \tan ^{-1}\left (\frac{b+a \tan \left (\frac{x}{2}\right )}{\sqrt{a^2-b^2}}\right )}{b \sqrt{a^2-b^2}}\\ \end{align*}
Mathematica [A] time = 0.0398464, size = 47, normalized size = 0.94 \[ \frac{x-\frac{2 a \tan ^{-1}\left (\frac{a \tan \left (\frac{x}{2}\right )+b}{\sqrt{a^2-b^2}}\right )}{\sqrt{a^2-b^2}}}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 54, normalized size = 1.1 \begin{align*} 2\,{\frac{\arctan \left ( \tan \left ( x/2 \right ) \right ) }{b}}-2\,{\frac{a}{b\sqrt{{a}^{2}-{b}^{2}}}\arctan \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 = 1.80871, size = 423, normalized size = 8.46 \begin{align*} \left [-\frac{\sqrt{-a^{2} + b^{2}} a \log \left (-\frac{{\left (2 \, a^{2} - b^{2}\right )} \cos \left (x\right )^{2} - 2 \, a b \sin \left (x\right ) - a^{2} - b^{2} - 2 \,{\left (a \cos \left (x\right ) \sin \left (x\right ) + b \cos \left (x\right )\right )} \sqrt{-a^{2} + b^{2}}}{b^{2} \cos \left (x\right )^{2} - 2 \, a b \sin \left (x\right ) - a^{2} - b^{2}}\right ) - 2 \,{\left (a^{2} - b^{2}\right )} x}{2 \,{\left (a^{2} b - b^{3}\right )}}, \frac{\sqrt{a^{2} - b^{2}} a \arctan \left (-\frac{a \sin \left (x\right ) + b}{\sqrt{a^{2} - b^{2}} \cos \left (x\right )}\right ) +{\left (a^{2} - b^{2}\right )} x}{a^{2} b - b^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 99.1242, size = 202, normalized size = 4.04 \begin{align*} \begin{cases} \tilde{\infty } x & \text{for}\: a = 0 \wedge b = 0 \\\frac{x}{b} & \text{for}\: a = 0 \\\frac{x \tan{\left (\frac{x}{2} \right )}}{b \tan{\left (\frac{x}{2} \right )} - b} - \frac{x}{b \tan{\left (\frac{x}{2} \right )} - b} + \frac{2 \tan{\left (\frac{x}{2} \right )}}{b \tan{\left (\frac{x}{2} \right )} - b} & \text{for}\: a = - b \\\frac{x \tan{\left (\frac{x}{2} \right )}}{b \tan{\left (\frac{x}{2} \right )} + b} + \frac{x}{b \tan{\left (\frac{x}{2} \right )} + b} - \frac{2 \tan{\left (\frac{x}{2} \right )}}{b \tan{\left (\frac{x}{2} \right )} + b} & \text{for}\: a = b \\- \frac{\cos{\left (x \right )}}{a} & \text{for}\: b = 0 \\\frac{a^{2} x}{a^{2} b - b^{3}} + \frac{a \sqrt{- a^{2} + b^{2}} \log{\left (\tan{\left (\frac{x}{2} \right )} + \frac{b}{a} - \frac{\sqrt{- a^{2} + b^{2}}}{a} \right )}}{a^{2} b - b^{3}} - \frac{a \sqrt{- a^{2} + b^{2}} \log{\left (\tan{\left (\frac{x}{2} \right )} + \frac{b}{a} + \frac{\sqrt{- a^{2} + b^{2}}}{a} \right )}}{a^{2} b - b^{3}} - \frac{b^{2} x}{a^{2} b - b^{3}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.85884, size = 78, normalized size = 1.56 \begin{align*} -\frac{2 \,{\left (\pi \left \lfloor \frac{x}{2 \, \pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (a\right ) + \arctan \left (\frac{a \tan \left (\frac{1}{2} \, x\right ) + b}{\sqrt{a^{2} - b^{2}}}\right )\right )} a}{\sqrt{a^{2} - b^{2}} b} + \frac{x}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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