Integrand size = 27, antiderivative size = 82 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^2} \, dx=\frac {\sqrt {c-a^2 c x^2}}{x}-a \sqrt {c} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )+2 a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \]
[Out]
Time = 0.25 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6302, 6286, 1821, 858, 223, 209, 272, 65, 214} \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^2} \, dx=-a \sqrt {c} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )+2 a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )+\frac {\sqrt {c-a^2 c x^2}}{x} \]
[In]
[Out]
Rule 65
Rule 209
Rule 214
Rule 223
Rule 272
Rule 858
Rule 1821
Rule 6286
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int \frac {e^{2 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2}}{x^2} \, dx \\ & = -\left (c \int \frac {(1+a x)^2}{x^2 \sqrt {c-a^2 c x^2}} \, dx\right ) \\ & = \frac {\sqrt {c-a^2 c x^2}}{x}+\int \frac {-2 a c-a^2 c x}{x \sqrt {c-a^2 c x^2}} \, dx \\ & = \frac {\sqrt {c-a^2 c x^2}}{x}-(2 a c) \int \frac {1}{x \sqrt {c-a^2 c x^2}} \, dx-\left (a^2 c\right ) \int \frac {1}{\sqrt {c-a^2 c x^2}} \, dx \\ & = \frac {\sqrt {c-a^2 c x^2}}{x}-(a c) \text {Subst}\left (\int \frac {1}{x \sqrt {c-a^2 c x}} \, dx,x,x^2\right )-\left (a^2 c\right ) \text {Subst}\left (\int \frac {1}{1+a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c-a^2 c x^2}}\right ) \\ & = \frac {\sqrt {c-a^2 c x^2}}{x}-a \sqrt {c} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )+\frac {2 \text {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2 c}} \, dx,x,\sqrt {c-a^2 c x^2}\right )}{a} \\ & = \frac {\sqrt {c-a^2 c x^2}}{x}-a \sqrt {c} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )+2 a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.27 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^2} \, dx=\frac {\sqrt {c-a^2 c x^2}}{x}+a \sqrt {c} \arctan \left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (-1+a^2 x^2\right )}\right )-2 a \sqrt {c} \log (x)+2 a \sqrt {c} \log \left (c+\sqrt {c} \sqrt {c-a^2 c x^2}\right ) \]
[In]
[Out]
Time = 0.66 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.24
method | result | size |
risch | \(-\frac {\left (a^{2} x^{2}-1\right ) c}{x \sqrt {-c \left (a^{2} x^{2}-1\right )}}-\left (\frac {a^{2} \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{\sqrt {a^{2} c}}-\frac {2 a \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )}{\sqrt {c}}\right ) c\) | \(102\) |
default | \(\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{c x}+2 a^{2} \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}\right )-2 a \left (\sqrt {-a^{2} c \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )\right )+2 a \left (\sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c}-\frac {a c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c}}\right )}{\sqrt {a^{2} c}}\right )\) | \(209\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.55 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^2} \, dx=\left [\frac {a \sqrt {c} x \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) + a \sqrt {c} x \log \left (-\frac {a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) + \sqrt {-a^{2} c x^{2} + c}}{x}, \frac {4 \, a \sqrt {-c} x \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + a \sqrt {-c} x \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) + 2 \, \sqrt {-a^{2} c x^{2} + c}}{2 \, x}\right ] \]
[In]
[Out]
\[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^2} \, dx=\int \frac {\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )}{x^{2} \left (a x - 1\right )}\, dx \]
[In]
[Out]
\[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^2} \, dx=\int { \frac {\sqrt {-a^{2} c x^{2} + c} {\left (a x + 1\right )}}{{\left (a x - 1\right )} x^{2}} \,d x } \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.63 \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^2} \, dx=-\frac {4 \, a c \arctan \left (-\frac {\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} - \frac {a^{2} \sqrt {-c} \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{{\left | a \right |}} - \frac {2 \, a^{2} \sqrt {-c} c}{{\left ({\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} - c\right )} {\left | a \right |}} \]
[In]
[Out]
Timed out. \[ \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^2} \, dx=\int \frac {\sqrt {c-a^2\,c\,x^2}\,\left (a\,x+1\right )}{x^2\,\left (a\,x-1\right )} \,d x \]
[In]
[Out]