Optimal. Leaf size=37 \[ \frac {\tan ^{-1}\left (\frac {\cosh (x) (a \tanh (x)+b)}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}} \]
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Rubi [A] time = 0.03, antiderivative size = 37, 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} \[ \frac {\tan ^{-1}\left (\frac {\cosh (x) (a \tanh (x)+b)}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 3509
Rubi steps
\begin {align*} \int \frac {\text {sech}(x)}{a+b \tanh (x)} \, dx &=i \operatorname {Subst}\left (\int \frac {1}{a^2-b^2-x^2} \, dx,x,\cosh (x) (-i b-i a \tanh (x))\right )\\ &=\frac {\tan ^{-1}\left (\frac {\cosh (x) (b+a \tanh (x))}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 46, normalized size = 1.24 \[ \frac {2 \tan ^{-1}\left (\frac {a \tanh \left (\frac {x}{2}\right )+b}{\sqrt {a-b} \sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 148, normalized size = 4.00 \[ \left [-\frac {\sqrt {-a^{2} + b^{2}} \log \left (\frac {{\left (a + b\right )} \cosh \relax (x)^{2} + 2 \, {\left (a + b\right )} \cosh \relax (x) \sinh \relax (x) + {\left (a + b\right )} \sinh \relax (x)^{2} - 2 \, \sqrt {-a^{2} + b^{2}} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} - a + b}{{\left (a + b\right )} \cosh \relax (x)^{2} + 2 \, {\left (a + b\right )} \cosh \relax (x) \sinh \relax (x) + {\left (a + b\right )} \sinh \relax (x)^{2} + a - b}\right )}{a^{2} - b^{2}}, -\frac {2 \, \arctan \left (\frac {\sqrt {a^{2} - b^{2}}}{{\left (a + b\right )} \cosh \relax (x) + {\left (a + b\right )} \sinh \relax (x)}\right )}{\sqrt {a^{2} - b^{2}}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 35, normalized size = 0.95 \[ \frac {2 \, \arctan \left (\frac {a e^{x} + b e^{x}}{\sqrt {a^{2} - b^{2}}}\right )}{\sqrt {a^{2} - b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 39, normalized size = 1.05 \[ \frac {2 \arctan \left (\frac {2 a \tanh \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}} \]
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 = 0.13, size = 35, normalized size = 0.95 \[ \frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {a^2-b^2}}{a-b}\right )}{\sqrt {a^2-b^2}} \]
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 \[ \int \frac {\operatorname {sech}{\relax (x )}}{a + b \tanh {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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