Optimal. Leaf size=38 \[ -\frac {\tanh ^{-1}\left (\frac {\sinh (x) (a \coth (x)+b)}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}} \]
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Rubi [A] time = 0.04, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3509, 206} \[ -\frac {\tanh ^{-1}\left (\frac {\sinh (x) (a \coth (x)+b)}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 3509
Rubi steps
\begin {align*} \int \frac {\text {csch}(x)}{a+b \coth (x)} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{a^2-b^2-x^2} \, dx,x,i (-i b-i a \coth (x)) \sinh (x)\right )\\ &=-\frac {i \tan ^{-1}\left (\frac {(-i b-i a \coth (x)) \sinh (x)}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 46, normalized size = 1.21 \[ \frac {2 \tan ^{-1}\left (\frac {a+b \tanh \left (\frac {x}{2}\right )}{\sqrt {b-a} \sqrt {a+b}}\right )}{\sqrt {b-a} \sqrt {a+b}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 147, normalized size = 3.87 \[ \left [\frac {\log \left (\frac {{\left (a + b\right )} \cosh \relax (x)^{2} + 2 \, {\left (a + b\right )} \cosh \relax (x) \sinh \relax (x) + {\left (a + b\right )} \sinh \relax (x)^{2} - 2 \, \sqrt {a^{2} - b^{2}} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} + a - b}{{\left (a + b\right )} \cosh \relax (x)^{2} + 2 \, {\left (a + b\right )} \cosh \relax (x) \sinh \relax (x) + {\left (a + b\right )} \sinh \relax (x)^{2} - a + b}\right )}{\sqrt {a^{2} - b^{2}}}, \frac {2 \, \sqrt {-a^{2} + b^{2}} \arctan \left (\frac {\sqrt {-a^{2} + b^{2}}}{{\left (a + b\right )} \cosh \relax (x) + {\left (a + b\right )} \sinh \relax (x)}\right )}{a^{2} - b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 35, normalized size = 0.92 \[ \frac {2 \, \arctan \left (\frac {a e^{x} + b e^{x}}{\sqrt {-a^{2} + b^{2}}}\right )}{\sqrt {-a^{2} + b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 39, normalized size = 1.03 \[ \frac {2 \arctan \left (\frac {2 \tanh \left (\frac {x}{2}\right ) b +2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}} \]
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 = 0.17, size = 35, normalized size = 0.92 \[ -\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {b^2-a^2}}{a-b}\right )}{\sqrt {b^2-a^2}} \]
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 \[ \int \frac {\operatorname {csch}{\relax (x )}}{a + b \coth {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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