Integrand size = 22, antiderivative size = 57 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {x}{c^3}+\frac {1}{2 a c^3 (1-a x)}+\frac {5 \log (1-a x)}{4 a c^3}-\frac {\log (1+a x)}{4 a c^3} \]
[Out]
Time = 0.11 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6302, 6266, 6264, 90} \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {1}{2 a c^3 (1-a x)}+\frac {5 \log (1-a x)}{4 a c^3}-\frac {\log (a x+1)}{4 a c^3}+\frac {x}{c^3} \]
[In]
[Out]
Rule 90
Rule 6264
Rule 6266
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx \\ & = \frac {a^3 \int \frac {e^{-2 \text {arctanh}(a x)} x^3}{(1-a x)^3} \, dx}{c^3} \\ & = \frac {a^3 \int \frac {x^3}{(1-a x)^2 (1+a x)} \, dx}{c^3} \\ & = \frac {a^3 \int \left (\frac {1}{a^3}+\frac {1}{2 a^3 (-1+a x)^2}+\frac {5}{4 a^3 (-1+a x)}-\frac {1}{4 a^3 (1+a x)}\right ) \, dx}{c^3} \\ & = \frac {x}{c^3}+\frac {1}{2 a c^3 (1-a x)}+\frac {5 \log (1-a x)}{4 a c^3}-\frac {\log (1+a x)}{4 a c^3} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {x}{c^3}+\frac {1}{2 a c^3 (1-a x)}+\frac {5 \log (1-a x)}{4 a c^3}-\frac {\log (1+a x)}{4 a c^3} \]
[In]
[Out]
Time = 0.51 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.84
method | result | size |
default | \(\frac {a^{3} \left (-\frac {\ln \left (a x +1\right )}{4 a^{4}}+\frac {x}{a^{3}}-\frac {1}{2 a^{4} \left (a x -1\right )}+\frac {5 \ln \left (a x -1\right )}{4 a^{4}}\right )}{c^{3}}\) | \(48\) |
risch | \(\frac {x}{c^{3}}-\frac {1}{2 a \left (a x -1\right ) c^{3}}+\frac {5 \ln \left (-a x +1\right )}{4 a \,c^{3}}-\frac {\ln \left (a x +1\right )}{4 a \,c^{3}}\) | \(51\) |
parallelrisch | \(\frac {4 a^{2} x^{2}+5 a \ln \left (a x -1\right ) x -a \ln \left (a x +1\right ) x -6 a x -5 \ln \left (a x -1\right )+\ln \left (a x +1\right )}{4 c^{3} \left (a x -1\right ) a}\) | \(63\) |
norman | \(\frac {\frac {a^{2} x^{3}}{c}+\frac {3 x}{2 c}-\frac {5 a \,x^{2}}{2 c}}{c^{2} \left (a x -1\right )^{2}}+\frac {5 \ln \left (a x -1\right )}{4 a \,c^{3}}-\frac {\ln \left (a x +1\right )}{4 a \,c^{3}}\) | \(67\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.04 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {4 \, a^{2} x^{2} - 4 \, a x - {\left (a x - 1\right )} \log \left (a x + 1\right ) + 5 \, {\left (a x - 1\right )} \log \left (a x - 1\right ) - 2}{4 \, {\left (a^{2} c^{3} x - a c^{3}\right )}} \]
[In]
[Out]
Time = 0.24 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.98 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=a^{3} \left (- \frac {1}{2 a^{5} c^{3} x - 2 a^{4} c^{3}} + \frac {x}{a^{3} c^{3}} + \frac {\frac {5 \log {\left (x - \frac {1}{a} \right )}}{4} - \frac {\log {\left (x + \frac {1}{a} \right )}}{4}}{a^{4} c^{3}}\right ) \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.93 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=-\frac {1}{2 \, {\left (a^{2} c^{3} x - a c^{3}\right )}} + \frac {x}{c^{3}} - \frac {\log \left (a x + 1\right )}{4 \, a c^{3}} + \frac {5 \, \log \left (a x - 1\right )}{4 \, a c^{3}} \]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.89 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {x}{c^{3}} - \frac {\log \left ({\left | a x + 1 \right |}\right )}{4 \, a c^{3}} + \frac {5 \, \log \left ({\left | a x - 1 \right |}\right )}{4 \, a c^{3}} - \frac {1}{2 \, {\left (a x - 1\right )} a c^{3}} \]
[In]
[Out]
Time = 0.10 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.91 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {x}{c^3}+\frac {1}{2\,a\,\left (c^3-a\,c^3\,x\right )}+\frac {5\,\ln \left (a\,x-1\right )}{4\,a\,c^3}-\frac {\ln \left (a\,x+1\right )}{4\,a\,c^3} \]
[In]
[Out]