Optimal. Leaf size=18 \[ 2 a \log (x)-2 a \log (a x+1)+\frac {1}{x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6167, 6126, 77} \[ 2 a \log (x)-2 a \log (a x+1)+\frac {1}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 77
Rule 6126
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{-2 \coth ^{-1}(a x)}}{x^2} \, dx &=-\int \frac {e^{-2 \tanh ^{-1}(a x)}}{x^2} \, dx\\ &=-\int \frac {1-a x}{x^2 (1+a x)} \, dx\\ &=-\int \left (\frac {1}{x^2}-\frac {2 a}{x}+\frac {2 a^2}{1+a x}\right ) \, dx\\ &=\frac {1}{x}+2 a \log (x)-2 a \log (1+a x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 18, normalized size = 1.00 \[ 2 a \log (x)-2 a \log (a x+1)+\frac {1}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.52, size = 23, normalized size = 1.28 \[ -\frac {2 \, a x \log \left (a x + 1\right ) - 2 \, a x \log \relax (x) - 1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.11, size = 20, normalized size = 1.11 \[ -2 \, a \log \left ({\left | a x + 1 \right |}\right ) + 2 \, a \log \left ({\left | x \right |}\right ) + \frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 19, normalized size = 1.06 \[ \frac {1}{x}+2 a \ln \relax (x )-2 a \ln \left (a x +1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 18, normalized size = 1.00 \[ -2 \, a \log \left (a x + 1\right ) + 2 \, a \log \relax (x) + \frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.20, size = 14, normalized size = 0.78 \[ \frac {1}{x}-4\,a\,\mathrm {atanh}\left (2\,a\,x+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.13, size = 15, normalized size = 0.83 \[ 2 a \left (\log {\relax (x )} - \log {\left (x + \frac {1}{a} \right )}\right ) + \frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________