Optimal. Leaf size=52 \[ -\frac {b \tan ^{-1}\left (\frac {a \sinh (x)+b \cosh (x)}{\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2}}-\frac {\tanh ^{-1}(\cosh (x))}{a} \]
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Rubi [A] time = 0.14, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {3518, 3110, 3770, 3074, 206} \[ -\frac {b \tan ^{-1}\left (\frac {a \sinh (x)+b \cosh (x)}{\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2}}-\frac {\tanh ^{-1}(\cosh (x))}{a} \]
Antiderivative was successfully verified.
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Rule 206
Rule 3074
Rule 3110
Rule 3518
Rule 3770
Rubi steps
\begin {align*} \int \frac {\text {csch}(x)}{a+b \tanh (x)} \, dx &=\int \frac {\coth (x)}{a \cosh (x)+b \sinh (x)} \, dx\\ &=i \int \left (-\frac {i \text {csch}(x)}{a}+\frac {i b}{a (a \cosh (x)+b \sinh (x))}\right ) \, dx\\ &=\frac {\int \text {csch}(x) \, dx}{a}-\frac {b \int \frac {1}{a \cosh (x)+b \sinh (x)} \, dx}{a}\\ &=-\frac {\tanh ^{-1}(\cosh (x))}{a}-\frac {(i b) \operatorname {Subst}\left (\int \frac {1}{a^2-b^2-x^2} \, dx,x,-i b \cosh (x)-i a \sinh (x)\right )}{a}\\ &=-\frac {b \tan ^{-1}\left (\frac {b \cosh (x)+a \sinh (x)}{\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2}}-\frac {\tanh ^{-1}(\cosh (x))}{a}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 59, normalized size = 1.13 \[ \frac {\log \left (\tanh \left (\frac {x}{2}\right )\right )-\frac {2 b \tan ^{-1}\left (\frac {a \tanh \left (\frac {x}{2}\right )+b}{\sqrt {a-b} \sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b}}}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 237, normalized size = 4.56 \[ \left [-\frac {\sqrt {-a^{2} + b^{2}} b \log \left (\frac {{\left (a + b\right )} \cosh \relax (x)^{2} + 2 \, {\left (a + b\right )} \cosh \relax (x) \sinh \relax (x) + {\left (a + b\right )} \sinh \relax (x)^{2} + 2 \, \sqrt {-a^{2} + b^{2}} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} - a + b}{{\left (a + b\right )} \cosh \relax (x)^{2} + 2 \, {\left (a + b\right )} \cosh \relax (x) \sinh \relax (x) + {\left (a + b\right )} \sinh \relax (x)^{2} + a - b}\right ) + {\left (a^{2} - b^{2}\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) - {\left (a^{2} - b^{2}\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right )}{a^{3} - a b^{2}}, \frac {2 \, \sqrt {a^{2} - b^{2}} b \arctan \left (\frac {\sqrt {a^{2} - b^{2}}}{{\left (a + b\right )} \cosh \relax (x) + {\left (a + b\right )} \sinh \relax (x)}\right ) - {\left (a^{2} - b^{2}\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) + {\left (a^{2} - b^{2}\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right )}{a^{3} - a b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 60, normalized size = 1.15 \[ -\frac {2 \, b \arctan \left (\frac {a e^{x} + b e^{x}}{\sqrt {a^{2} - b^{2}}}\right )}{\sqrt {a^{2} - b^{2}} a} - \frac {\log \left (e^{x} + 1\right )}{a} + \frac {\log \left ({\left | e^{x} - 1 \right |}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 53, normalized size = 1.02 \[ -\frac {2 b \arctan \left (\frac {2 a \tanh \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a \sqrt {a^{2}-b^{2}}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right )}{a} \]
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 = 1.40, size = 177, normalized size = 3.40 \[ \frac {\ln \left (32\,a\,b-32\,a^2+32\,a^2\,{\mathrm {e}}^x-32\,a\,b\,{\mathrm {e}}^x\right )}{a}-\frac {\ln \left (32\,a\,b-32\,a^2-32\,a^2\,{\mathrm {e}}^x+32\,a\,b\,{\mathrm {e}}^x\right )}{a}-\frac {b\,\ln \left (32\,a\,b^2\,{\mathrm {e}}^x+32\,a^2\,b\,{\mathrm {e}}^x-32\,a\,b\,\sqrt {b^2-a^2}\right )\,\sqrt {b^2-a^2}}{a\,b^2-a^3}+\frac {b\,\ln \left (32\,a\,b^2\,{\mathrm {e}}^x+32\,a^2\,b\,{\mathrm {e}}^x+32\,a\,b\,\sqrt {b^2-a^2}\right )\,\sqrt {b^2-a^2}}{a\,b^2-a^3} \]
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 \[ \int \frac {\operatorname {csch}{\relax (x )}}{a + b \tanh {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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