Optimal. Leaf size=61 \[ \frac {2 (a A-b B) \tan ^{-1}\left (\frac {a \tan \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2}}-\frac {B \tanh ^{-1}(\cos (x))}{a} \]
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Rubi [A] time = 0.13, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2828, 3001, 3770, 2660, 618, 204} \[ \frac {2 (a A-b B) \tan ^{-1}\left (\frac {a \tan \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2}}-\frac {B \tanh ^{-1}(\cos (x))}{a} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 2660
Rule 2828
Rule 3001
Rule 3770
Rubi steps
\begin {align*} \int \frac {A+B \csc (x)}{a+b \sin (x)} \, dx &=\int \frac {\csc (x) (B+A \sin (x))}{a+b \sin (x)} \, dx\\ &=\frac {B \int \csc (x) \, dx}{a}+\frac {(a A-b B) \int \frac {1}{a+b \sin (x)} \, dx}{a}\\ &=-\frac {B \tanh ^{-1}(\cos (x))}{a}+\frac {(2 (a A-b B)) \operatorname {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{a}\\ &=-\frac {B \tanh ^{-1}(\cos (x))}{a}-\frac {(4 (a A-b 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}\\ &=\frac {2 (a A-b B) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2}}-\frac {B \tanh ^{-1}(\cos (x))}{a}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 72, normalized size = 1.18 \[ \frac {\frac {2 (a A-b B) \tan ^{-1}\left (\frac {a \tan \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+B \left (\log \left (\sin \left (\frac {x}{2}\right )\right )-\log \left (\cos \left (\frac {x}{2}\right )\right )\right )}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.04, size = 265, normalized size = 4.34 \[ \left [\frac {{\left (A a - B b\right )} \sqrt {-a^{2} + b^{2}} \log \left (-\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \relax (x)^{2} - 2 \, a b \sin \relax (x) - a^{2} - b^{2} - 2 \, {\left (a \cos \relax (x) \sin \relax (x) + b \cos \relax (x)\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \relax (x)^{2} - 2 \, a b \sin \relax (x) - a^{2} - b^{2}}\right ) - {\left (B a^{2} - B b^{2}\right )} \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + {\left (B a^{2} - B b^{2}\right )} \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right )}{2 \, {\left (a^{3} - a b^{2}\right )}}, -\frac {2 \, {\left (A a - B b\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \relax (x) + b}{\sqrt {a^{2} - b^{2}} \cos \relax (x)}\right ) + {\left (B a^{2} - B b^{2}\right )} \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) - {\left (B a^{2} - B b^{2}\right )} \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right )}{2 \, {\left (a^{3} - a b^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 71, normalized size = 1.16 \[ \frac {B \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{a} + \frac {2 \, {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (a) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, x\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )} {\left (A a - B b\right )}}{\sqrt {a^{2} - b^{2}} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 94, normalized size = 1.54 \[ \frac {B \ln \left (\tan \left (\frac {x}{2}\right )\right )}{a}+\frac {2 A \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}-\frac {2 \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right ) B b}{a \sqrt {a^{2}-b^{2}}} \]
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 = 15.62, size = 134, normalized size = 2.20 \[ \frac {B\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{a}-\frac {\ln \left (b+a\,\mathrm {tan}\left (\frac {x}{2}\right )+\sqrt {b^2-a^2}\right )\,\sqrt {-\left (a+b\right )\,\left (a-b\right )}\,\left (A\,a-B\,b\right )}{a\,b^2-a^3}-\frac {\ln \left (b+a\,\mathrm {tan}\left (\frac {x}{2}\right )-\sqrt {b^2-a^2}\right )\,\left (A\,a\,\sqrt {b^2-a^2}-B\,b\,\sqrt {b^2-a^2}\right )}{a\,\left (a^2-b^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {A + B \csc {\relax (x )}}{a + b \sin {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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