Integrand size = 12, antiderivative size = 80 \[ \int \frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{x} \, dx=-\frac {1}{2 a x^2}-\frac {\sqrt {1-a x^2}}{2 a x^2 \sqrt {\frac {1}{1+a x^2}}}-\frac {1}{2} \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \arcsin \left (a x^2\right ) \]
[Out]
Time = 0.04 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.16, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6469, 265, 281, 283, 222} \[ \int \frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{x} \, dx=-\frac {\sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \sqrt {1-a^2 x^4}}{2 a x^2}-\frac {1}{2} \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \arcsin \left (a x^2\right )-\frac {1}{2 a x^2} \]
[In]
[Out]
Rule 222
Rule 265
Rule 281
Rule 283
Rule 6469
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{2 a x^2}+\frac {\left (\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {\sqrt {1-a x^2} \sqrt {1+a x^2}}{x^3} \, dx}{a} \\ & = -\frac {1}{2 a x^2}+\frac {\left (\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {\sqrt {1-a^2 x^4}}{x^3} \, dx}{a} \\ & = -\frac {1}{2 a x^2}+\frac {\left (\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {1-a^2 x^2}}{x^2} \, dx,x,x^2\right )}{2 a} \\ & = -\frac {1}{2 a x^2}-\frac {\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \sqrt {1-a^2 x^4}}{2 a x^2}-\frac {1}{2} \left (a \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx,x,x^2\right ) \\ & = -\frac {1}{2 a x^2}-\frac {\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \sqrt {1-a^2 x^4}}{2 a x^2}-\frac {1}{2} \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \arcsin \left (a x^2\right ) \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.28 \[ \int \frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{x} \, dx=-\frac {1}{2} e^{\text {sech}^{-1}\left (a x^2\right )}+\arctan \left (e^{\text {sech}^{-1}\left (a x^2\right )}\right ) \]
[In]
[Out]
Time = 0.11 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.29
method | result | size |
default | \(-\frac {\sqrt {-\frac {a \,x^{2}-1}{a \,x^{2}}}\, \sqrt {\frac {a \,x^{2}+1}{a \,x^{2}}}\, \left (\arctan \left (\frac {x^{2}}{\sqrt {-\frac {x^{4} a^{2}-1}{a^{2}}}}\right ) x^{2}+\sqrt {-\frac {x^{4} a^{2}-1}{a^{2}}}\right )}{2 \sqrt {-\frac {x^{4} a^{2}-1}{a^{2}}}}-\frac {1}{2 a \,x^{2}}\) | \(103\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (44) = 88\).
Time = 0.25 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.28 \[ \int \frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{x} \, dx=-\frac {a x^{2} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} - 2 \, a x^{2} \arctan \left (\frac {a x^{2} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} - 1}{a x^{2}}\right ) + 1}{2 \, a x^{2}} \]
[In]
[Out]
\[ \int \frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{x} \, dx=\frac {\int \frac {1}{x^{3}}\, dx + \int \frac {a \sqrt {-1 + \frac {1}{a x^{2}}} \sqrt {1 + \frac {1}{a x^{2}}}}{x}\, dx}{a} \]
[In]
[Out]
\[ \int \frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{x} \, dx=\int { \frac {\sqrt {\frac {1}{a x^{2}} + 1} \sqrt {\frac {1}{a x^{2}} - 1} + \frac {1}{a x^{2}}}{x} \,d x } \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 252 vs. \(2 (44) = 88\).
Time = 1.73 (sec) , antiderivative size = 252, normalized size of antiderivative = 3.15 \[ \int \frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{x} \, dx=-\frac {{\left (\pi + 2 \, \arctan \left (\frac {\sqrt {a^{2} x^{2} + a} {\left (\frac {{\left (\sqrt {2} \sqrt {a} - \sqrt {-a^{2} x^{2} + a}\right )}^{2}}{a^{2} x^{2} + a} - 1\right )}}{2 \, {\left (\sqrt {2} \sqrt {a} - \sqrt {-a^{2} x^{2} + a}\right )}}\right )\right )} a^{3} + \frac {4 \, a^{3} {\left (\frac {\sqrt {2} \sqrt {a} - \sqrt {-a^{2} x^{2} + a}}{\sqrt {a^{2} x^{2} + a}} - \frac {\sqrt {a^{2} x^{2} + a}}{\sqrt {2} \sqrt {a} - \sqrt {-a^{2} x^{2} + a}}\right )}}{{\left (\frac {\sqrt {2} \sqrt {a} - \sqrt {-a^{2} x^{2} + a}}{\sqrt {a^{2} x^{2} + a}} - \frac {\sqrt {a^{2} x^{2} + a}}{\sqrt {2} \sqrt {a} - \sqrt {-a^{2} x^{2} + a}}\right )}^{2} - 4} + \frac {a^{2}}{x^{2}}}{2 \, a^{3}} \]
[In]
[Out]
Time = 7.40 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.31 \[ \int \frac {e^{\text {sech}^{-1}\left (a x^2\right )}}{x} \, dx=-\frac {\ln \left (\frac {{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^2}+1\right )\,1{}\mathrm {i}}{2}+\frac {\ln \left (\frac {\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}}{\sqrt {\frac {1}{a\,x^2}+1}-1}\right )\,1{}\mathrm {i}}{2}-\frac {1}{2\,a\,x^2}+\frac {{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^2\,8{}\mathrm {i}}{{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^2\,\left (2+\frac {2\,{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^4}-\frac {4\,{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^2}\right )} \]
[In]
[Out]