Optimal. Leaf size=54 \[ \frac {\tan ^{-1}(\sinh (x))}{a}-\frac {2 b \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a \sqrt {a-b} \sqrt {a+b}} \]
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Rubi [A] time = 0.07, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2747, 3770, 2659, 208} \[ \frac {\tan ^{-1}(\sinh (x))}{a}-\frac {2 b \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a \sqrt {a-b} \sqrt {a+b}} \]
Antiderivative was successfully verified.
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Rule 208
Rule 2659
Rule 2747
Rule 3770
Rubi steps
\begin {align*} \int \frac {\text {sech}(x)}{a+b \cosh (x)} \, dx &=\frac {\int \text {sech}(x) \, dx}{a}-\frac {b \int \frac {1}{a+b \cosh (x)} \, dx}{a}\\ &=\frac {\tan ^{-1}(\sinh (x))}{a}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {1}{a+b-(a-b) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{a}\\ &=\frac {\tan ^{-1}(\sinh (x))}{a}-\frac {2 b \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a \sqrt {a-b} \sqrt {a+b}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 54, normalized size = 1.00 \[ \frac {2 \left (\frac {b \tan ^{-1}\left (\frac {(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {b^2-a^2}}\right )}{\sqrt {b^2-a^2}}+\tan ^{-1}\left (\tanh \left (\frac {x}{2}\right )\right )\right )}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 227, normalized size = 4.20 \[ \left [\frac {\sqrt {a^{2} - b^{2}} b \log \left (\frac {b^{2} \cosh \relax (x)^{2} + b^{2} \sinh \relax (x)^{2} + 2 \, a b \cosh \relax (x) + 2 \, a^{2} - b^{2} + 2 \, {\left (b^{2} \cosh \relax (x) + a b\right )} \sinh \relax (x) + 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cosh \relax (x) + b \sinh \relax (x) + a\right )}}{b \cosh \relax (x)^{2} + b \sinh \relax (x)^{2} + 2 \, a \cosh \relax (x) + 2 \, {\left (b \cosh \relax (x) + a\right )} \sinh \relax (x) + b}\right ) + 2 \, {\left (a^{2} - b^{2}\right )} \arctan \left (\cosh \relax (x) + \sinh \relax (x)\right )}{a^{3} - a b^{2}}, \frac {2 \, {\left (\sqrt {-a^{2} + b^{2}} b \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cosh \relax (x) + b \sinh \relax (x) + a\right )}}{a^{2} - b^{2}}\right ) + {\left (a^{2} - b^{2}\right )} \arctan \left (\cosh \relax (x) + \sinh \relax (x)\right )\right )}}{a^{3} - a b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 45, normalized size = 0.83 \[ -\frac {2 \, b \arctan \left (\frac {b e^{x} + a}{\sqrt {-a^{2} + b^{2}}}\right )}{\sqrt {-a^{2} + b^{2}} a} + \frac {2 \, \arctan \left (e^{x}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 51, normalized size = 0.94 \[ -\frac {2 b \arctanh \left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a \sqrt {\left (a +b \right ) \left (a -b \right )}}+\frac {2 \arctan \left (\tanh \left (\frac {x}{2}\right )\right )}{a} \]
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 = 3.46, size = 286, normalized size = 5.30 \[ \frac {b\,\ln \left (64\,a^4\,b-64\,a^2\,b^3+128\,a^5\,{\mathrm {e}}^x+32\,a\,b^3\,\sqrt {a^2-b^2}-64\,a^3\,b\,\sqrt {a^2-b^2}+32\,a\,b^4\,{\mathrm {e}}^x-128\,a^4\,{\mathrm {e}}^x\,\sqrt {a^2-b^2}-160\,a^3\,b^2\,{\mathrm {e}}^x+96\,a^2\,b^2\,{\mathrm {e}}^x\,\sqrt {a^2-b^2}\right )}{a\,\sqrt {a^2-b^2}}-\frac {b\,\ln \left (64\,a^4\,b-64\,a^2\,b^3+128\,a^5\,{\mathrm {e}}^x-32\,a\,b^3\,\sqrt {a^2-b^2}+64\,a^3\,b\,\sqrt {a^2-b^2}+32\,a\,b^4\,{\mathrm {e}}^x+128\,a^4\,{\mathrm {e}}^x\,\sqrt {a^2-b^2}-160\,a^3\,b^2\,{\mathrm {e}}^x-96\,a^2\,b^2\,{\mathrm {e}}^x\,\sqrt {a^2-b^2}\right )}{a\,\sqrt {a^2-b^2}}-\frac {\ln \left ({\mathrm {e}}^x-\mathrm {i}\right )\,1{}\mathrm {i}-\ln \left ({\mathrm {e}}^x+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{a} \]
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 \[ \int \frac {\operatorname {sech}{\relax (x )}}{a + b \cosh {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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