Integrand size = 12, antiderivative size = 59 \[ \int \frac {1}{a+b \text {sech}(c+d x)} \, dx=\frac {x}{a}-\frac {2 b \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a \sqrt {a-b} \sqrt {a+b} d} \]
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Time = 0.05 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3868, 2738, 214} \[ \int \frac {1}{a+b \text {sech}(c+d x)} \, dx=\frac {x}{a}-\frac {2 b \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}} \]
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Rule 214
Rule 2738
Rule 3868
Rubi steps \begin{align*} \text {integral}& = \frac {x}{a}-\frac {\int \frac {1}{1+\frac {a \cosh (c+d x)}{b}} \, dx}{a} \\ & = \frac {x}{a}+\frac {(2 i) \text {Subst}\left (\int \frac {1}{1+\frac {a}{b}+\left (1-\frac {a}{b}\right ) x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{a d} \\ & = \frac {x}{a}-\frac {2 b \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a \sqrt {a-b} \sqrt {a+b} d} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.02 \[ \int \frac {1}{a+b \text {sech}(c+d x)} \, dx=\frac {\frac {c}{d}+x+\frac {2 b \arctan \left (\frac {(-a+b) \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} d}}{a} \]
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Time = 0.21 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.41
method | result | size |
derivativedivides | \(\frac {\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a}-\frac {2 b \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a \sqrt {\left (a +b \right ) \left (a -b \right )}}}{d}\) | \(83\) |
default | \(\frac {\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a}-\frac {2 b \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a \sqrt {\left (a +b \right ) \left (a -b \right )}}}{d}\) | \(83\) |
risch | \(\frac {x}{a}-\frac {b \ln \left ({\mathrm e}^{d x +c}+\frac {b \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{\sqrt {-a^{2}+b^{2}}\, a}\right )}{\sqrt {-a^{2}+b^{2}}\, d a}+\frac {b \ln \left ({\mathrm e}^{d x +c}+\frac {b \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{\sqrt {-a^{2}+b^{2}}\, a}\right )}{\sqrt {-a^{2}+b^{2}}\, d a}\) | \(136\) |
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Time = 0.29 (sec) , antiderivative size = 270, normalized size of antiderivative = 4.58 \[ \int \frac {1}{a+b \text {sech}(c+d x)} \, dx=\left [\frac {{\left (a^{2} - b^{2}\right )} d x - \sqrt {-a^{2} + b^{2}} b \log \left (\frac {a^{2} \cosh \left (d x + c\right )^{2} + a^{2} \sinh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) - a^{2} + 2 \, b^{2} + 2 \, {\left (a^{2} \cosh \left (d x + c\right ) + a b\right )} \sinh \left (d x + c\right ) + 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) + b\right )}}{a \cosh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) + 2 \, {\left (a \cosh \left (d x + c\right ) + b\right )} \sinh \left (d x + c\right ) + a}\right )}{{\left (a^{3} - a b^{2}\right )} d}, \frac {{\left (a^{2} - b^{2}\right )} d x + 2 \, \sqrt {a^{2} - b^{2}} b \arctan \left (-\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )}{{\left (a^{3} - a b^{2}\right )} d}\right ] \]
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\[ \int \frac {1}{a+b \text {sech}(c+d x)} \, dx=\int \frac {1}{a + b \operatorname {sech}{\left (c + d x \right )}}\, dx \]
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Exception generated. \[ \int \frac {1}{a+b \text {sech}(c+d x)} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.28 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.95 \[ \int \frac {1}{a+b \text {sech}(c+d x)} \, dx=-\frac {\frac {2 \, b \arctan \left (\frac {a e^{\left (d x + c\right )} + b}{\sqrt {a^{2} - b^{2}}}\right )}{\sqrt {a^{2} - b^{2}} a} - \frac {d x + c}{a}}{d} \]
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Time = 0.37 (sec) , antiderivative size = 131, normalized size of antiderivative = 2.22 \[ \int \frac {1}{a+b \text {sech}(c+d x)} \, dx=\frac {x}{a}+\frac {b\,\ln \left (\frac {2\,b\,{\mathrm {e}}^{c+d\,x}}{a^2}-\frac {2\,b\,\left (a+b\,{\mathrm {e}}^{c+d\,x}\right )}{a^2\,\sqrt {a+b}\,\sqrt {b-a}}\right )}{a\,d\,\sqrt {a+b}\,\sqrt {b-a}}-\frac {b\,\ln \left (\frac {2\,b\,{\mathrm {e}}^{c+d\,x}}{a^2}+\frac {2\,b\,\left (a+b\,{\mathrm {e}}^{c+d\,x}\right )}{a^2\,\sqrt {a+b}\,\sqrt {b-a}}\right )}{a\,d\,\sqrt {a+b}\,\sqrt {b-a}} \]
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