Integrand size = 16, antiderivative size = 66 \[ \int \frac {x}{a+b \text {sech}\left (c+d x^2\right )} \, dx=\frac {x^2}{2 a}-\frac {b \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {a+b}}\right )}{a \sqrt {a-b} \sqrt {a+b} d} \]
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Time = 0.08 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5544, 3868, 2738, 214} \[ \int \frac {x}{a+b \text {sech}\left (c+d x^2\right )} \, dx=\frac {x^2}{2 a}-\frac {b \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}} \]
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Rule 214
Rule 2738
Rule 3868
Rule 5544
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{a+b \text {sech}(c+d x)} \, dx,x,x^2\right ) \\ & = \frac {x^2}{2 a}-\frac {\text {Subst}\left (\int \frac {1}{1+\frac {a \cosh (c+d x)}{b}} \, dx,x,x^2\right )}{2 a} \\ & = \frac {x^2}{2 a}+\frac {i \text {Subst}\left (\int \frac {1}{1+\frac {a}{b}+\left (1-\frac {a}{b}\right ) x^2} \, dx,x,i \tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )\right )}{a d} \\ & = \frac {x^2}{2 a}-\frac {b \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {a+b}}\right )}{a \sqrt {a-b} \sqrt {a+b} d} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.02 \[ \int \frac {x}{a+b \text {sech}\left (c+d x^2\right )} \, dx=\frac {\frac {c}{d}+x^2+\frac {2 b \arctan \left (\frac {(-a+b) \tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} d}}{2 a} \]
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Time = 0.20 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.36
method | result | size |
derivativedivides | \(\frac {-\frac {2 b \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a \sqrt {\left (a +b \right ) \left (a -b \right )}}+\frac {\ln \left (1+\tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )\right )}{a}-\frac {\ln \left (\tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )-1\right )}{a}}{2 d}\) | \(90\) |
default | \(\frac {-\frac {2 b \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a \sqrt {\left (a +b \right ) \left (a -b \right )}}+\frac {\ln \left (1+\tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )\right )}{a}-\frac {\ln \left (\tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )-1\right )}{a}}{2 d}\) | \(90\) |
risch | \(\frac {x^{2}}{2 a}-\frac {b \ln \left ({\mathrm e}^{d \,x^{2}+c}+\frac {b \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{\sqrt {-a^{2}+b^{2}}\, a}\right )}{2 \sqrt {-a^{2}+b^{2}}\, d a}+\frac {b \ln \left ({\mathrm e}^{d \,x^{2}+c}+\frac {b \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{\sqrt {-a^{2}+b^{2}}\, a}\right )}{2 \sqrt {-a^{2}+b^{2}}\, d a}\) | \(144\) |
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Time = 0.28 (sec) , antiderivative size = 304, normalized size of antiderivative = 4.61 \[ \int \frac {x}{a+b \text {sech}\left (c+d x^2\right )} \, dx=\left [\frac {{\left (a^{2} - b^{2}\right )} d x^{2} - \sqrt {-a^{2} + b^{2}} b \log \left (\frac {a^{2} \cosh \left (d x^{2} + c\right )^{2} + a^{2} \sinh \left (d x^{2} + c\right )^{2} + 2 \, a b \cosh \left (d x^{2} + c\right ) - a^{2} + 2 \, b^{2} + 2 \, {\left (a^{2} \cosh \left (d x^{2} + c\right ) + a b\right )} \sinh \left (d x^{2} + c\right ) + 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cosh \left (d x^{2} + c\right ) + a \sinh \left (d x^{2} + c\right ) + b\right )}}{a \cosh \left (d x^{2} + c\right )^{2} + a \sinh \left (d x^{2} + c\right )^{2} + 2 \, b \cosh \left (d x^{2} + c\right ) + 2 \, {\left (a \cosh \left (d x^{2} + c\right ) + b\right )} \sinh \left (d x^{2} + c\right ) + a}\right )}{2 \, {\left (a^{3} - a b^{2}\right )} d}, \frac {{\left (a^{2} - b^{2}\right )} d x^{2} + 2 \, \sqrt {a^{2} - b^{2}} b \arctan \left (-\frac {a \cosh \left (d x^{2} + c\right ) + a \sinh \left (d x^{2} + c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )}{2 \, {\left (a^{3} - a b^{2}\right )} d}\right ] \]
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\[ \int \frac {x}{a+b \text {sech}\left (c+d x^2\right )} \, dx=\int \frac {x}{a + b \operatorname {sech}{\left (c + d x^{2} \right )}}\, dx \]
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Exception generated. \[ \int \frac {x}{a+b \text {sech}\left (c+d x^2\right )} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.27 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.92 \[ \int \frac {x}{a+b \text {sech}\left (c+d x^2\right )} \, dx=-\frac {b \arctan \left (\frac {a e^{\left (d x^{2} + c\right )} + b}{\sqrt {a^{2} - b^{2}}}\right )}{\sqrt {a^{2} - b^{2}} a d} + \frac {d x^{2} + c}{2 \, a d} \]
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Time = 0.20 (sec) , antiderivative size = 171, normalized size of antiderivative = 2.59 \[ \int \frac {x}{a+b \text {sech}\left (c+d x^2\right )} \, dx=\frac {x^2}{2\,a}-\frac {\mathrm {atan}\left (\frac {a\,d\,\sqrt {b^2}}{\sqrt {a^4\,d^2-a^2\,b^2\,d^2}}+\frac {b\,{\mathrm {e}}^{d\,x^2}\,{\mathrm {e}}^c\,\sqrt {a^4\,d^2-a^2\,b^2\,d^2}}{a^2\,d\,\sqrt {b^2}}+\frac {a^2\,b\,d\,{\mathrm {e}}^{d\,x^2}\,{\mathrm {e}}^c\,\sqrt {b^2}\,\sqrt {a^4\,d^2-a^2\,b^2\,d^2}}{a^6\,d^2-a^4\,b^2\,d^2}\right )\,\sqrt {b^2}}{\sqrt {a^4\,d^2-a^2\,b^2\,d^2}} \]
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