Integrand size = 11, antiderivative size = 42 \[ \int \frac {\text {sech}(x)}{a+b \text {sech}(x)} \, dx=\frac {2 \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b}} \]
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Time = 0.04 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3916, 2738, 211} \[ \int \frac {\text {sech}(x)}{a+b \text {sech}(x)} \, dx=\frac {2 \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b}} \]
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Rule 211
Rule 2738
Rule 3916
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {1}{1+\frac {a \cosh (x)}{b}} \, dx}{b} \\ & = \frac {2 \text {Subst}\left (\int \frac {1}{1+\frac {a}{b}-\left (1-\frac {a}{b}\right ) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{b} \\ & = \frac {2 \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.98 \[ \int \frac {\text {sech}(x)}{a+b \text {sech}(x)} \, dx=-\frac {2 \arctan \left (\frac {(-a+b) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}} \]
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Time = 0.09 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.86
method | result | size |
default | \(\frac {2 \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\) | \(36\) |
risch | \(-\frac {\ln \left ({\mathrm e}^{x}+\frac {b \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{\sqrt {-a^{2}+b^{2}}\, a}\right )}{\sqrt {-a^{2}+b^{2}}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {b \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{\sqrt {-a^{2}+b^{2}}\, a}\right )}{\sqrt {-a^{2}+b^{2}}}\) | \(109\) |
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Time = 0.25 (sec) , antiderivative size = 165, normalized size of antiderivative = 3.93 \[ \int \frac {\text {sech}(x)}{a+b \text {sech}(x)} \, dx=\left [-\frac {\sqrt {-a^{2} + b^{2}} \log \left (\frac {a^{2} \cosh \left (x\right )^{2} + a^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) - a^{2} + 2 \, b^{2} + 2 \, {\left (a^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) - 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cosh \left (x\right ) + a \sinh \left (x\right ) + b\right )}}{a \cosh \left (x\right )^{2} + a \sinh \left (x\right )^{2} + 2 \, b \cosh \left (x\right ) + 2 \, {\left (a \cosh \left (x\right ) + b\right )} \sinh \left (x\right ) + a}\right )}{a^{2} - b^{2}}, -\frac {2 \, \arctan \left (-\frac {a \cosh \left (x\right ) + a \sinh \left (x\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )}{\sqrt {a^{2} - b^{2}}}\right ] \]
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\[ \int \frac {\text {sech}(x)}{a+b \text {sech}(x)} \, dx=\int \frac {\operatorname {sech}{\left (x \right )}}{a + b \operatorname {sech}{\left (x \right )}}\, dx \]
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Exception generated. \[ \int \frac {\text {sech}(x)}{a+b \text {sech}(x)} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.26 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.76 \[ \int \frac {\text {sech}(x)}{a+b \text {sech}(x)} \, dx=\frac {2 \, \arctan \left (\frac {a e^{x} + b}{\sqrt {a^{2} - b^{2}}}\right )}{\sqrt {a^{2} - b^{2}}} \]
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Time = 1.98 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.02 \[ \int \frac {\text {sech}(x)}{a+b \text {sech}(x)} \, dx=\frac {2\,\mathrm {atan}\left (\frac {b}{\sqrt {a^2-b^2}}+\frac {a\,{\mathrm {e}}^x}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}} \]
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