Integrand size = 13, antiderivative size = 56 \[ \int \frac {\text {sech}^3(x)}{a+b \tanh (x)} \, dx=\frac {a \arctan (\sinh (x))}{b^2}-\frac {\sqrt {a^2-b^2} \arctan \left (\frac {\cosh (x) (b+a \tanh (x))}{\sqrt {a^2-b^2}}\right )}{b^2}+\frac {\text {sech}(x)}{b} \]
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Time = 0.07 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3591, 3567, 3855, 3590, 212} \[ \int \frac {\text {sech}^3(x)}{a+b \tanh (x)} \, dx=-\frac {\sqrt {a^2-b^2} \arctan \left (\frac {\cosh (x) (a \tanh (x)+b)}{\sqrt {a^2-b^2}}\right )}{b^2}+\frac {a \arctan (\sinh (x))}{b^2}+\frac {\text {sech}(x)}{b} \]
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Rule 212
Rule 3567
Rule 3590
Rule 3591
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\int \text {sech}(x) (a-b \tanh (x)) \, dx}{b^2}-\frac {\left (a^2-b^2\right ) \int \frac {\text {sech}(x)}{a+b \tanh (x)} \, dx}{b^2} \\ & = \frac {\text {sech}(x)}{b}+\frac {a \int \text {sech}(x) \, dx}{b^2}-\frac {\left (i \left (a^2-b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a^2-b^2-x^2} \, dx,x,\cosh (x) (-i b-i a \tanh (x))\right )}{b^2} \\ & = \frac {a \arctan (\sinh (x))}{b^2}-\frac {\sqrt {a^2-b^2} \arctan \left (\frac {\cosh (x) (b+a \tanh (x))}{\sqrt {a^2-b^2}}\right )}{b^2}+\frac {\text {sech}(x)}{b} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.16 \[ \int \frac {\text {sech}^3(x)}{a+b \tanh (x)} \, dx=\frac {2 a \arctan \left (\tanh \left (\frac {x}{2}\right )\right )-2 \sqrt {a-b} \sqrt {a+b} \arctan \left (\frac {b+a \tanh \left (\frac {x}{2}\right )}{\sqrt {a-b} \sqrt {a+b}}\right )+b \text {sech}(x)}{b^2} \]
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Time = 4.49 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.38
method | result | size |
default | \(\frac {\frac {2 b}{1+\tanh \left (\frac {x}{2}\right )^{2}}+2 a \arctan \left (\tanh \left (\frac {x}{2}\right )\right )}{b^{2}}+\frac {2 \left (-a^{2}+b^{2}\right ) \arctan \left (\frac {2 a \tanh \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{2} \sqrt {a^{2}-b^{2}}}\) | \(77\) |
risch | \(\frac {2 \,{\mathrm e}^{x}}{b \left (1+{\mathrm e}^{2 x}\right )}+\frac {i a \ln \left ({\mathrm e}^{x}+i\right )}{b^{2}}-\frac {i a \ln \left ({\mathrm e}^{x}-i\right )}{b^{2}}+\frac {\sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{x}-\frac {\sqrt {-a^{2}+b^{2}}}{a +b}\right )}{b^{2}}-\frac {\sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{x}+\frac {\sqrt {-a^{2}+b^{2}}}{a +b}\right )}{b^{2}}\) | \(117\) |
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Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (52) = 104\).
Time = 0.28 (sec) , antiderivative size = 309, normalized size of antiderivative = 5.52 \[ \int \frac {\text {sech}^3(x)}{a+b \tanh (x)} \, dx=\left [\frac {\sqrt {-a^{2} + b^{2}} {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )} \log \left (\frac {{\left (a + b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )^{2} - 2 \, \sqrt {-a^{2} + b^{2}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} - a + b}{{\left (a + b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )^{2} + a - b}\right ) + 2 \, {\left (a \cosh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) \sinh \left (x\right ) + a \sinh \left (x\right )^{2} + a\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + 2 \, b \cosh \left (x\right ) + 2 \, b \sinh \left (x\right )}{b^{2} \cosh \left (x\right )^{2} + 2 \, b^{2} \cosh \left (x\right ) \sinh \left (x\right ) + b^{2} \sinh \left (x\right )^{2} + b^{2}}, \frac {2 \, {\left (\sqrt {a^{2} - b^{2}} {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )} \arctan \left (\frac {\sqrt {a^{2} - b^{2}}}{{\left (a + b\right )} \cosh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )}\right ) + {\left (a \cosh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) \sinh \left (x\right ) + a \sinh \left (x\right )^{2} + a\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + b \cosh \left (x\right ) + b \sinh \left (x\right )\right )}}{b^{2} \cosh \left (x\right )^{2} + 2 \, b^{2} \cosh \left (x\right ) \sinh \left (x\right ) + b^{2} \sinh \left (x\right )^{2} + b^{2}}\right ] \]
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\[ \int \frac {\text {sech}^3(x)}{a+b \tanh (x)} \, dx=\int \frac {\operatorname {sech}^{3}{\left (x \right )}}{a + b \tanh {\left (x \right )}}\, dx \]
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Exception generated. \[ \int \frac {\text {sech}^3(x)}{a+b \tanh (x)} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.26 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.12 \[ \int \frac {\text {sech}^3(x)}{a+b \tanh (x)} \, dx=\frac {2 \, a \arctan \left (e^{x}\right )}{b^{2}} - \frac {2 \, \sqrt {a^{2} - b^{2}} \arctan \left (\frac {a e^{x} + b e^{x}}{\sqrt {a^{2} - b^{2}}}\right )}{b^{2}} + \frac {2 \, e^{x}}{b {\left (e^{\left (2 \, x\right )} + 1\right )}} \]
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Time = 4.29 (sec) , antiderivative size = 119, normalized size of antiderivative = 2.12 \[ \int \frac {\text {sech}^3(x)}{a+b \tanh (x)} \, dx=\frac {\ln \left (a\,{\mathrm {e}}^x+b\,{\mathrm {e}}^x-\sqrt {b^2-a^2}\right )\,\sqrt {-\left (a+b\right )\,\left (a-b\right )}}{b^2}-\frac {\ln \left (a\,{\mathrm {e}}^x+b\,{\mathrm {e}}^x+\sqrt {b^2-a^2}\right )\,\sqrt {-\left (a+b\right )\,\left (a-b\right )}}{b^2}+\frac {2\,{\mathrm {e}}^x}{b\,\left ({\mathrm {e}}^{2\,x}+1\right )}-\frac {a\,\ln \left ({\mathrm {e}}^x-\mathrm {i}\right )\,1{}\mathrm {i}}{b^2}+\frac {a\,\ln \left ({\mathrm {e}}^x+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{b^2} \]
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