Integrand size = 11, antiderivative size = 37 \[ \int \frac {\text {sech}(x)}{a+b \tanh (x)} \, dx=\frac {\arctan \left (\frac {\cosh (x) (b+a \tanh (x))}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}} \]
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Time = 0.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3590, 212} \[ \int \frac {\text {sech}(x)}{a+b \tanh (x)} \, dx=\frac {\arctan \left (\frac {\cosh (x) (a \tanh (x)+b)}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}} \]
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Rule 212
Rule 3590
Rubi steps \begin{align*} \text {integral}& = i \text {Subst}\left (\int \frac {1}{a^2-b^2-x^2} \, dx,x,\cosh (x) (-i b-i a \tanh (x))\right ) \\ & = \frac {\arctan \left (\frac {\cosh (x) (b+a \tanh (x))}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.24 \[ \int \frac {\text {sech}(x)}{a+b \tanh (x)} \, dx=\frac {2 \arctan \left (\frac {b+a \tanh \left (\frac {x}{2}\right )}{\sqrt {a-b} \sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b}} \]
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Time = 0.40 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.05
method | result | size |
default | \(\frac {2 \arctan \left (\frac {2 a \tanh \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}\) | \(39\) |
risch | \(-\frac {\ln \left ({\mathrm e}^{x}-\frac {a -b}{\sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {a -b}{\sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}}\) | \(70\) |
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none
Time = 0.25 (sec) , antiderivative size = 148, normalized size of antiderivative = 4.00 \[ \int \frac {\text {sech}(x)}{a+b \tanh (x)} \, dx=\left [-\frac {\sqrt {-a^{2} + b^{2}} \log \left (\frac {{\left (a + b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )^{2} - 2 \, \sqrt {-a^{2} + b^{2}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} - a + b}{{\left (a + b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )^{2} + a - b}\right )}{a^{2} - b^{2}}, -\frac {2 \, \arctan \left (\frac {\sqrt {a^{2} - b^{2}}}{{\left (a + b\right )} \cosh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )}\right )}{\sqrt {a^{2} - b^{2}}}\right ] \]
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\[ \int \frac {\text {sech}(x)}{a+b \tanh (x)} \, dx=\int \frac {\operatorname {sech}{\left (x \right )}}{a + b \tanh {\left (x \right )}}\, dx \]
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Exception generated. \[ \int \frac {\text {sech}(x)}{a+b \tanh (x)} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.95 \[ \int \frac {\text {sech}(x)}{a+b \tanh (x)} \, dx=\frac {2 \, \arctan \left (\frac {a e^{x} + b e^{x}}{\sqrt {a^{2} - b^{2}}}\right )}{\sqrt {a^{2} - b^{2}}} \]
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Time = 0.12 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.95 \[ \int \frac {\text {sech}(x)}{a+b \tanh (x)} \, dx=\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {a^2-b^2}}{a-b}\right )}{\sqrt {a^2-b^2}} \]
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