Integrand size = 10, antiderivative size = 68 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x \, dx=\frac {1}{2} e^{\text {sech}^{-1}\left (a x^2\right )} x^2-\frac {\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \text {arctanh}\left (\sqrt {1-a^2 x^4}\right )}{2 a}+\frac {\log (x)}{a} \]
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Time = 0.03 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6470, 29, 265, 272, 65, 214} \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x \, dx=-\frac {\sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \text {arctanh}\left (\sqrt {1-a^2 x^4}\right )}{2 a}+\frac {1}{2} x^2 e^{\text {sech}^{-1}\left (a x^2\right )}+\frac {\log (x)}{a} \]
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Rule 29
Rule 65
Rule 214
Rule 265
Rule 272
Rule 6470
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} e^{\text {sech}^{-1}\left (a x^2\right )} x^2+\frac {\int \frac {1}{x} \, dx}{a}+\frac {\left (\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {1}{x \sqrt {1-a x^2} \sqrt {1+a x^2}} \, dx}{a} \\ & = \frac {1}{2} e^{\text {sech}^{-1}\left (a x^2\right )} x^2+\frac {\log (x)}{a}+\frac {\left (\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {1}{x \sqrt {1-a^2 x^4}} \, dx}{a} \\ & = \frac {1}{2} e^{\text {sech}^{-1}\left (a x^2\right )} x^2+\frac {\log (x)}{a}+\frac {\left (\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^4\right )}{4 a} \\ & = \frac {1}{2} e^{\text {sech}^{-1}\left (a x^2\right )} x^2+\frac {\log (x)}{a}-\frac {\left (\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^4}\right )}{2 a^3} \\ & = \frac {1}{2} e^{\text {sech}^{-1}\left (a x^2\right )} x^2-\frac {\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \text {arctanh}\left (\sqrt {1-a^2 x^4}\right )}{2 a}+\frac {\log (x)}{a} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.47 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x \, dx=\frac {\sqrt {\frac {1-a x^2}{1+a x^2}} \left (1+a x^2\right )+2 \log \left (a x^2\right )-\log \left (1+\sqrt {\frac {1-a x^2}{1+a x^2}}+a x^2 \sqrt {\frac {1-a x^2}{1+a x^2}}\right )}{2 a} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.11 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.87
method | result | size |
default | \(\frac {\sqrt {-\frac {a \,x^{2}-1}{a \,x^{2}}}\, x^{2} \sqrt {\frac {a \,x^{2}+1}{a \,x^{2}}}\, \left (\operatorname {csgn}\left (\frac {1}{a}\right ) a \sqrt {-\frac {x^{4} a^{2}-1}{a^{2}}}-\ln \left (\frac {2 \,\operatorname {csgn}\left (\frac {1}{a}\right ) a \sqrt {-\frac {x^{4} a^{2}-1}{a^{2}}}+2}{a^{2} x^{2}}\right )\right ) \operatorname {csgn}\left (\frac {1}{a}\right )}{2 a \sqrt {-\frac {x^{4} a^{2}-1}{a^{2}}}}+\frac {\ln \left (x \right )}{a}\) | \(127\) |
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Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (61) = 122\).
Time = 0.25 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.96 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x \, dx=\frac {2 \, a x^{2} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} - \log \left (a x^{2} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} + 1\right ) + \log \left (a x^{2} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} - 1\right ) + 4 \, \log \left (x\right )}{4 \, a} \]
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\[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x \, dx=\frac {\int \frac {1}{x}\, dx + \int a x \sqrt {-1 + \frac {1}{a x^{2}}} \sqrt {1 + \frac {1}{a x^{2}}}\, dx}{a} \]
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\[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x \, dx=\int { x {\left (\sqrt {\frac {1}{a x^{2}} + 1} \sqrt {\frac {1}{a x^{2}} - 1} + \frac {1}{a x^{2}}\right )} \,d x } \]
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Exception generated. \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x \, dx=\text {Exception raised: TypeError} \]
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Time = 7.09 (sec) , antiderivative size = 182, normalized size of antiderivative = 2.68 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x \, dx=\frac {\ln \left (x\right )}{a}-\frac {2\,\mathrm {atanh}\left (\frac {\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}}{\sqrt {\frac {1}{a\,x^2}+1}-1}\right )}{a}+\frac {\frac {5\,{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^2}+1}{\frac {8\,a\,\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}{\sqrt {\frac {1}{a\,x^2}+1}-1}+\frac {8\,a\,{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^3}{{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^3}}+\frac {\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}}{8\,a\,\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )} \]
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