Optimal. Leaf size=32 \[ -\frac {1}{2 x^2}+\frac {2 a}{x}+2 a^2 \log (x)-2 a^2 \log (1+a x) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6261, 78}
\begin {gather*} 2 a^2 \log (x)-2 a^2 \log (a x+1)+\frac {2 a}{x}-\frac {1}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 78
Rule 6261
Rubi steps
\begin {align*} \int \frac {e^{-2 \tanh ^{-1}(a x)}}{x^3} \, dx &=\int \frac {1-a x}{x^3 (1+a x)} \, dx\\ &=\int \left (\frac {1}{x^3}-\frac {2 a}{x^2}+\frac {2 a^2}{x}-\frac {2 a^3}{1+a x}\right ) \, dx\\ &=-\frac {1}{2 x^2}+\frac {2 a}{x}+2 a^2 \log (x)-2 a^2 \log (1+a x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 32, normalized size = 1.00 \begin {gather*} -\frac {1}{2 x^2}+\frac {2 a}{x}+2 a^2 \log (x)-2 a^2 \log (1+a x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.68, size = 31, normalized size = 0.97
method | result | size |
default | \(-\frac {1}{2 x^{2}}+\frac {2 a}{x}+2 a^{2} \ln \left (x \right )-2 a^{2} \ln \left (a x +1\right )\) | \(31\) |
risch | \(\frac {2 a x -\frac {1}{2}}{x^{2}}+2 a^{2} \ln \left (-x \right )-2 a^{2} \ln \left (a x +1\right )\) | \(32\) |
norman | \(\frac {-\frac {1}{2}-2 a^{3} x^{3}+\frac {3}{2} a x}{\left (a x +1\right ) x^{2}}+2 a^{2} \ln \left (x \right )-2 a^{2} \ln \left (a x +1\right )\) | \(45\) |
meijerg | \(-a^{2} \left (-\frac {2 a x}{2 a x +2}-\ln \left (a x +1\right )+1+\ln \left (x \right )+\ln \left (a \right )\right )+a^{2} \left (-\frac {4 a x}{4 a x +4}-3 \ln \left (a x +1\right )+1+3 \ln \left (x \right )+3 \ln \left (a \right )-\frac {1}{2 a^{2} x^{2}}+\frac {2}{a x}\right )\) | \(83\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.25, size = 30, normalized size = 0.94 \begin {gather*} -2 \, a^{2} \log \left (a x + 1\right ) + 2 \, a^{2} \log \left (x\right ) + \frac {4 \, a x - 1}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.33, size = 35, normalized size = 1.09 \begin {gather*} -\frac {4 \, a^{2} x^{2} \log \left (a x + 1\right ) - 4 \, a^{2} x^{2} \log \left (x\right ) - 4 \, a x + 1}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.08, size = 27, normalized size = 0.84 \begin {gather*} - 2 a^{2} \left (- \log {\left (x \right )} + \log {\left (x + \frac {1}{a} \right )}\right ) - \frac {- 4 a x + 1}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.41, size = 50, normalized size = 1.56 \begin {gather*} 2 \, a^{2} \log \left ({\left | -\frac {1}{a x + 1} + 1 \right |}\right ) + \frac {5 \, a^{2} - \frac {6 \, a^{2}}{a x + 1}}{2 \, {\left (\frac {1}{a x + 1} - 1\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.80, size = 23, normalized size = 0.72 \begin {gather*} \frac {2\,a\,x-\frac {1}{2}}{x^2}-4\,a^2\,\mathrm {atanh}\left (2\,a\,x+1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________