Optimal. Leaf size=49 \[ -B \tan ^{-1}\left (\frac {B \cos (x)}{\sqrt {A^2+B^2 \sin ^2(x)}}\right )-A \tanh ^{-1}\left (\frac {A \cos (x)}{\sqrt {A^2+B^2 \sin ^2(x)}}\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 57, normalized size of antiderivative = 1.16, number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3186, 402, 217, 203, 377, 206} \[ -B \tan ^{-1}\left (\frac {B \cos (x)}{\sqrt {A^2-B^2 \cos ^2(x)+B^2}}\right )-A \tanh ^{-1}\left (\frac {A \cos (x)}{\sqrt {A^2-B^2 \cos ^2(x)+B^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 217
Rule 377
Rule 402
Rule 3186
Rubi steps
\begin {align*} \int \csc (x) \sqrt {A^2+B^2 \sin ^2(x)} \, dx &=-\operatorname {Subst}\left (\int \frac {\sqrt {A^2+B^2-B^2 x^2}}{1-x^2} \, dx,x,\cos (x)\right )\\ &=-\left (A^2 \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {A^2+B^2-B^2 x^2}} \, dx,x,\cos (x)\right )\right )-B^2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {A^2+B^2-B^2 x^2}} \, dx,x,\cos (x)\right )\\ &=-\left (A^2 \operatorname {Subst}\left (\int \frac {1}{1-A^2 x^2} \, dx,x,\frac {\cos (x)}{\sqrt {A^2+B^2-B^2 \cos ^2(x)}}\right )\right )-B^2 \operatorname {Subst}\left (\int \frac {1}{1+B^2 x^2} \, dx,x,\frac {\cos (x)}{\sqrt {A^2+B^2-B^2 \cos ^2(x)}}\right )\\ &=-B \tan ^{-1}\left (\frac {B \cos (x)}{\sqrt {A^2+B^2-B^2 \cos ^2(x)}}\right )-A \tanh ^{-1}\left (\frac {A \cos (x)}{\sqrt {A^2+B^2-B^2 \cos ^2(x)}}\right )\\ \end {align*}
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Mathematica [B] time = 0.11, size = 99, normalized size = 2.02 \[ \sqrt {-B^2} \log \left (\sqrt {2 A^2-B^2 \cos (2 x)+B^2}+\sqrt {2} \sqrt {-B^2} \cos (x)\right )-\sqrt {A^2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {A^2} \cos (x)}{\sqrt {2 A^2-B^2 \cos (2 x)+B^2}}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.55, size = 244, normalized size = 4.98 \[ \frac {1}{2} \, B \arctan \left (-\frac {{\left (A^{4} + 2 \, A^{2} B^{2} + B^{4}\right )} \cos \relax (x) \sin \relax (x) - 2 \, {\left (2 \, B^{3} \cos \relax (x)^{3} - {\left (A^{2} B + B^{3}\right )} \cos \relax (x)\right )} \sqrt {-B^{2} \cos \relax (x)^{2} + A^{2} + B^{2}}}{4 \, B^{4} \cos \relax (x)^{4} + A^{4} + 2 \, A^{2} B^{2} + B^{4} - {\left (A^{4} + 6 \, A^{2} B^{2} + 5 \, B^{4}\right )} \cos \relax (x)^{2}}\right ) - \frac {1}{2} \, B \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x)}\right ) - \frac {1}{2} \, A \log \left (-B^{2} \cos \relax (x)^{2} + A B \cos \relax (x) \sin \relax (x) + A^{2} + B^{2} + \sqrt {-B^{2} \cos \relax (x)^{2} + A^{2} + B^{2}} {\left (A \cos \relax (x) + B \sin \relax (x)\right )}\right ) + \frac {1}{2} \, A \log \left (-B^{2} \cos \relax (x)^{2} - A B \cos \relax (x) \sin \relax (x) + A^{2} + B^{2} - \sqrt {-B^{2} \cos \relax (x)^{2} + A^{2} + B^{2}} {\left (A \cos \relax (x) - B \sin \relax (x)\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {B^{2} \sin \relax (x)^{2} + A^{2}}}{\sin \relax (x)}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.19, size = 149, normalized size = 3.04 \[ -\frac {\sqrt {\left (B^{2} \left (\sin ^{2}\relax (x )\right )+A^{2}\right ) \left (\cos ^{2}\relax (x )\right )}\, \left (A \,\mathrm {csgn}\relax (A ) \ln \left (-\frac {A^{2} \left (\sin ^{2}\relax (x )\right )-B^{2} \left (\sin ^{2}\relax (x )\right )-2 A^{2}-2 \sqrt {\left (B^{2} \left (\sin ^{2}\relax (x )\right )+A^{2}\right ) \left (\cos ^{2}\relax (x )\right )}\, A \,\mathrm {csgn}\relax (A )}{\sin \relax (x )^{2}}\right )-B \arctan \left (\frac {\left (2 B^{2} \left (\sin ^{2}\relax (x )\right )+A^{2}-B^{2}\right ) \mathrm {csgn}\relax (B )}{2 \sqrt {\left (B^{2} \left (\sin ^{2}\relax (x )\right )+A^{2}\right ) \left (\cos ^{2}\relax (x )\right )}\, B}\right ) \mathrm {csgn}\relax (B )\right )}{2 \sqrt {B^{2} \left (\sin ^{2}\relax (x )\right )+A^{2}}\, \cos \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.36, size = 116, normalized size = 2.37 \[ -B \arcsin \left (\frac {B^{2} \cos \relax (x)}{\sqrt {A^{2} B^{2} + B^{4}}}\right ) - \frac {1}{2} \, A \log \left (B^{2} - \frac {A^{2}}{\cos \relax (x) - 1} - \frac {\sqrt {-B^{2} \cos \relax (x)^{2} + A^{2} + B^{2}} A}{\cos \relax (x) - 1}\right ) + \frac {1}{2} \, A \log \left (-B^{2} + \frac {A^{2}}{\cos \relax (x) + 1} + \frac {\sqrt {-B^{2} \cos \relax (x)^{2} + A^{2} + B^{2}} A}{\cos \relax (x) + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {A^2+B^2\,{\sin \relax (x)}^2}}{\sin \relax (x)} \,d x \]
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 \[ \int \frac {\sqrt {A^{2} + B^{2} \sin ^{2}{\relax (x )}}}{\sin {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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