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 = 0.06, antiderivative size = 50, normalized size of antiderivative = 1.00, 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 204
Rule 618
Rule 2660
Rule 2735
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*}
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Mathematica [A] time = 0.04, 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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fricas [A] time = 0.78, size = 192, normalized size = 3.84 \[ \left [-\frac {\sqrt {-a^{2} + b^{2}} a \log \left (-\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \relax (x)^{2} - 2 \, a b \sin \relax (x) - a^{2} - b^{2} - 2 \, {\left (a \cos \relax (x) \sin \relax (x) + b \cos \relax (x)\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \relax (x)^{2} - 2 \, a b \sin \relax (x) - a^{2} - b^{2}}\right ) - 2 \, {\left (a^{2} - b^{2}\right )} x}{2 \, {\left (a^{2} b - b^{3}\right )}}, \frac {\sqrt {a^{2} - b^{2}} a \arctan \left (-\frac {a \sin \relax (x) + b}{\sqrt {a^{2} - b^{2}} \cos \relax (x)}\right ) + {\left (a^{2} - b^{2}\right )} x}{a^{2} b - b^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 58, normalized size = 1.16 \[ -\frac {2 \, {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (a) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, x\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )} a}{\sqrt {a^{2} - b^{2}} b} + \frac {x}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 54, normalized size = 1.08 \[ \frac {2 \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{b}-\frac {2 a \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{b \sqrt {a^{2}-b^{2}}} \]
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 = 6.70, size = 101, normalized size = 2.02 \[ \frac {x}{b}-\frac {2\,a\,\mathrm {atanh}\left (\frac {\sin \left (\frac {x}{2}\right )\,a^4-\cos \left (\frac {x}{2}\right )\,a^3\,b-3\,\sin \left (\frac {x}{2}\right )\,a^2\,b^2+\cos \left (\frac {x}{2}\right )\,a\,b^3+2\,\sin \left (\frac {x}{2}\right )\,b^4}{{\left (b^2-a^2\right )}^{3/2}\,\left (2\,b\,\sin \left (\frac {x}{2}\right )+a\,\cos \left (\frac {x}{2}\right )\right )}\right )}{b\,\sqrt {b^2-a^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 80.94, size = 236, normalized size = 4.72 \[ \begin {cases} \tilde {\infty } x & \text {for}\: a = 0 \wedge b = 0 \\\frac {b x \tan {\left (\frac {x}{2} \right )}}{b^{2} \tan {\left (\frac {x}{2} \right )} - b \sqrt {b^{2}}} + \frac {2 b}{b^{2} \tan {\left (\frac {x}{2} \right )} - b \sqrt {b^{2}}} - \frac {x \sqrt {b^{2}}}{b^{2} \tan {\left (\frac {x}{2} \right )} - b \sqrt {b^{2}}} & \text {for}\: a = - \sqrt {b^{2}} \\\frac {b x \tan {\left (\frac {x}{2} \right )}}{b^{2} \tan {\left (\frac {x}{2} \right )} + b \sqrt {b^{2}}} + \frac {2 b}{b^{2} \tan {\left (\frac {x}{2} \right )} + b \sqrt {b^{2}}} + \frac {x \sqrt {b^{2}}}{b^{2} \tan {\left (\frac {x}{2} \right )} + b \sqrt {b^{2}}} & \text {for}\: a = \sqrt {b^{2}} \\- \frac {\cos {\relax (x )}}{a} & \text {for}\: b = 0 \\\frac {x}{b} & \text {for}\: a = 0 \\- \frac {a \log {\left (\tan {\left (\frac {x}{2} \right )} + \frac {b}{a} - \frac {\sqrt {- a^{2} + b^{2}}}{a} \right )}}{b \sqrt {- a^{2} + b^{2}}} + \frac {a \log {\left (\tan {\left (\frac {x}{2} \right )} + \frac {b}{a} + \frac {\sqrt {- a^{2} + b^{2}}}{a} \right )}}{b \sqrt {- a^{2} + b^{2}}} + \frac {x}{b} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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