Integrand size = 12, antiderivative size = 51 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^2} \, dx=-3 a \sqrt {1-\frac {1}{a^2 x^2}}-\frac {2 \left (a+\frac {1}{x}\right )^2}{a \sqrt {1-\frac {1}{a^2 x^2}}}+3 a \csc ^{-1}(a x) \]
[Out]
Time = 0.06 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6304, 867, 683, 655, 222} \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^2} \, dx=-\frac {2 \left (a+\frac {1}{x}\right )^2}{a \sqrt {1-\frac {1}{a^2 x^2}}}-3 a \sqrt {1-\frac {1}{a^2 x^2}}+3 a \csc ^{-1}(a x) \]
[In]
[Out]
Rule 222
Rule 655
Rule 683
Rule 867
Rule 6304
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^2}{\left (1-\frac {x}{a}\right ) \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = -\text {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^3}{\left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {2 \left (a+\frac {1}{x}\right )^2}{a \sqrt {1-\frac {1}{a^2 x^2}}}+3 \text {Subst}\left (\int \frac {1+\frac {x}{a}}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = -3 a \sqrt {1-\frac {1}{a^2 x^2}}-\frac {2 \left (a+\frac {1}{x}\right )^2}{a \sqrt {1-\frac {1}{a^2 x^2}}}+3 \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = -3 a \sqrt {1-\frac {1}{a^2 x^2}}-\frac {2 \left (a+\frac {1}{x}\right )^2}{a \sqrt {1-\frac {1}{a^2 x^2}}}+3 a \csc ^{-1}(a x) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.80 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^2} \, dx=\frac {a \sqrt {1-\frac {1}{a^2 x^2}} (1-5 a x)}{-1+a x}+3 a \arcsin \left (\frac {1}{a x}\right ) \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(115\) vs. \(2(47)=94\).
Time = 0.14 (sec) , antiderivative size = 116, normalized size of antiderivative = 2.27
method | result | size |
risch | \(-\frac {a x -1}{x \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (3 a \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )-\frac {4 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{x -\frac {1}{a}}\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\) | \(116\) |
default | \(\frac {-\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{4} x^{4}+\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}+5 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{3} x^{3}+\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+3 a^{3} x^{3} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )-\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}-\ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-2 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a x -7 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{2} x^{2}-2 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-6 a^{2} x^{2} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )-2 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x +2 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x^{2}+2 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}+3 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a x +\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x +3 a x \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )-\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x -\ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x}{\sqrt {a^{2}}\, x \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(593\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.45 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^2} \, dx=-\frac {6 \, {\left (a^{2} x^{2} - a x\right )} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + {\left (5 \, a^{2} x^{2} + 4 \, a x - 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a x^{2} - x} \]
[In]
[Out]
\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^2} \, dx=\int \frac {1}{x^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\, dx \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.41 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^2} \, dx=-2 \, a {\left (\frac {\frac {3 \, {\left (a x - 1\right )}}{a x + 1} + 2}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + \sqrt {\frac {a x - 1}{a x + 1}}} + 3 \, \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )\right )} \]
[In]
[Out]
\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^2} \, dx=\int { \frac {1}{x^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Time = 4.51 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.12 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{x^2} \, dx=\frac {1}{x\,\sqrt {\frac {a\,x-1}{a\,x+1}}}-6\,a\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )-\frac {5\,a}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \]
[In]
[Out]