Optimal. Leaf size=54 \[ \frac {B x}{b}-\frac {2 B \sqrt {a^2-b^2} \tan ^{-1}\left (\frac {a \tan \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2}}\right )}{a b} \]
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Rubi [A] time = 0.08, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2735, 2660, 618, 204} \[ \frac {B x}{b}-\frac {2 B \sqrt {a^2-b^2} \tan ^{-1}\left (\frac {a \tan \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2}}\right )}{a b} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 2660
Rule 2735
Rubi steps
\begin {align*} \int \frac {\frac {b B}{a}+B \sin (x)}{a+b \sin (x)} \, dx &=\frac {B x}{b}-\frac {\left (a B-\frac {b^2 B}{a}\right ) \int \frac {1}{a+b \sin (x)} \, dx}{b}\\ &=\frac {B x}{b}-\frac {\left (2 \left (a B-\frac {b^2 B}{a}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{b}\\ &=\frac {B x}{b}+\frac {\left (4 \left (a B-\frac {b^2 B}{a}\right )\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}\\ &=\frac {B x}{b}-\frac {2 \sqrt {a^2-b^2} B \tan ^{-1}\left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{a b}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 52, normalized size = 0.96 \[ \frac {B \left (a x-2 \sqrt {a^2-b^2} \tan ^{-1}\left (\frac {a \tan \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2}}\right )\right )}{a b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 163, normalized size = 3.02 \[ \left [\frac {2 \, B a x + \sqrt {-a^{2} + b^{2}} B \log \left (\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \relax (x)^{2} - 2 \, a b \sin \relax (x) - a^{2} - b^{2} + 2 \, {\left (a \cos \relax (x) \sin \relax (x) + b \cos \relax (x)\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \relax (x)^{2} - 2 \, a b \sin \relax (x) - a^{2} - b^{2}}\right )}{2 \, a b}, \frac {B a x + \sqrt {a^{2} - b^{2}} B \arctan \left (-\frac {a \sin \relax (x) + b}{\sqrt {a^{2} - b^{2}} \cos \relax (x)}\right )}{a b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 73, normalized size = 1.35 \[ \frac {B x}{b} - \frac {2 \, {\left (B a^{2} - B b^{2}\right )} {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (a) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, x\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 99, normalized size = 1.83 \[ \frac {2 B \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{b}-\frac {2 B a \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{b \sqrt {a^{2}-b^{2}}}+\frac {2 B b \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a \sqrt {a^{2}-b^{2}}} \]
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 = 8.12, size = 94, normalized size = 1.74 \[ \frac {2\,B\,\mathrm {atan}\left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )}{b}+\frac {2\,B\,\mathrm {atanh}\left (\frac {-\sin \left (\frac {x}{2}\right )\,a^2+\cos \left (\frac {x}{2}\right )\,a\,b+2\,\sin \left (\frac {x}{2}\right )\,b^2}{\sqrt {b^2-a^2}\,\left (2\,b\,\sin \left (\frac {x}{2}\right )+a\,\cos \left (\frac {x}{2}\right )\right )}\right )\,\sqrt {b^2-a^2}}{a\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 50.12, size = 87, normalized size = 1.61 \[ \begin {cases} \frac {B x}{b} + \frac {B \sqrt {- a^{2} + b^{2}} \log {\left (\tan {\left (\frac {x}{2} \right )} + \frac {b}{a} - \frac {\sqrt {- a^{2} + b^{2}}}{a} \right )}}{a b} - \frac {B \sqrt {- a^{2} + b^{2}} \log {\left (\tan {\left (\frac {x}{2} \right )} + \frac {b}{a} + \frac {\sqrt {- a^{2} + b^{2}}}{a} \right )}}{a b} & \text {for}\: b \neq 0 \\- \frac {B \cos {\relax (x )}}{a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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