Integrand size = 10, antiderivative size = 35 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^2} \, dx=-\frac {e^{\text {sech}^{-1}(a x)}}{2 x}+a \text {arctanh}\left (\sqrt {\frac {1-a x}{1+a x}}\right ) \]
[Out]
Leaf count is larger than twice the leaf count of optimal. \(99\) vs. \(2(35)=70\).
Time = 0.03 (sec) , antiderivative size = 99, normalized size of antiderivative = 2.83, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6470, 30, 105, 12, 94, 214} \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^2} \, dx=\frac {1}{2} a \sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \text {arctanh}\left (\sqrt {1-a x} \sqrt {a x+1}\right )+\frac {\sqrt {1-a x}}{2 a x^2 \sqrt {\frac {1}{a x+1}}}+\frac {1}{2 a x^2}-\frac {e^{\text {sech}^{-1}(a x)}}{x} \]
[In]
[Out]
Rule 12
Rule 30
Rule 94
Rule 105
Rule 214
Rule 6470
Rubi steps \begin{align*} \text {integral}& = -\frac {e^{\text {sech}^{-1}(a x)}}{x}-\frac {\int \frac {1}{x^3} \, dx}{a}-\frac {\left (\sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {1}{x^3 \sqrt {1-a x} \sqrt {1+a x}} \, dx}{a} \\ & = \frac {1}{2 a x^2}-\frac {e^{\text {sech}^{-1}(a x)}}{x}+\frac {\sqrt {1-a x}}{2 a x^2 \sqrt {\frac {1}{1+a x}}}-\frac {\left (\sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {a^2}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{2 a} \\ & = \frac {1}{2 a x^2}-\frac {e^{\text {sech}^{-1}(a x)}}{x}+\frac {\sqrt {1-a x}}{2 a x^2 \sqrt {\frac {1}{1+a x}}}-\frac {1}{2} \left (a \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {1}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx \\ & = \frac {1}{2 a x^2}-\frac {e^{\text {sech}^{-1}(a x)}}{x}+\frac {\sqrt {1-a x}}{2 a x^2 \sqrt {\frac {1}{1+a x}}}+\frac {1}{2} \left (a^2 \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \text {Subst}\left (\int \frac {1}{a-a x^2} \, dx,x,\sqrt {1-a x} \sqrt {1+a x}\right ) \\ & = \frac {1}{2 a x^2}-\frac {e^{\text {sech}^{-1}(a x)}}{x}+\frac {\sqrt {1-a x}}{2 a x^2 \sqrt {\frac {1}{1+a x}}}+\frac {1}{2} a \sqrt {\frac {1}{1+a x}} \sqrt {1+a x} \text {arctanh}\left (\sqrt {1-a x} \sqrt {1+a x}\right ) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(93\) vs. \(2(35)=70\).
Time = 0.07 (sec) , antiderivative size = 93, normalized size of antiderivative = 2.66 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^2} \, dx=\frac {1}{2} \left (-\frac {1}{a x^2}-\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)}{a x^2}-a \log (x)+a \log \left (1+\sqrt {\frac {1-a x}{1+a x}}+a x \sqrt {\frac {1-a x}{1+a x}}\right )\right ) \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 91, normalized size of antiderivative = 2.60
method | result | size |
default | \(\frac {\sqrt {\frac {a x +1}{a x}}\, \sqrt {-\frac {a x -1}{a x}}\, \left (a^{2} x^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )-\sqrt {-a^{2} x^{2}+1}\right )}{2 x \sqrt {-a^{2} x^{2}+1}}-\frac {1}{2 a \,x^{2}}\) | \(91\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (56) = 112\).
Time = 0.25 (sec) , antiderivative size = 128, normalized size of antiderivative = 3.66 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^2} \, dx=\frac {a^{2} x^{2} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 1\right ) - a^{2} x^{2} \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 1\right ) - 2 \, a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 2}{4 \, a x^{2}} \]
[In]
[Out]
Time = 2.66 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.03 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^2} \, dx=- a \left (2 \sqrt {-1 + \frac {1}{a x}} \left (\frac {\left (1 + \frac {1}{a x}\right )^{\frac {3}{2}}}{4} - \frac {\sqrt {1 + \frac {1}{a x}}}{4}\right ) - \log {\left (2 \sqrt {-1 + \frac {1}{a x}} + 2 \sqrt {1 + \frac {1}{a x}} \right )}\right ) - \frac {1}{2 a x^{2}} \]
[In]
[Out]
\[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^2} \, dx=\int { \frac {\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}}{x^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^2} \, dx=\int { \frac {\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}}{x^{2}} \,d x } \]
[In]
[Out]
Time = 4.88 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.03 \[ \int \frac {e^{\text {sech}^{-1}(a x)}}{x^2} \, dx=\frac {a\,\ln \left (\sqrt {\frac {1}{a\,x}-1}\,\sqrt {\frac {1}{a\,x}+1}+\frac {1}{a\,x}\right )}{2}-\frac {1}{2\,a\,x^2}-\frac {\sqrt {\frac {1}{a\,x}-1}\,\sqrt {\frac {1}{a\,x}+1}}{2\,x} \]
[In]
[Out]