Optimal. Leaf size=40 \[ -\frac {c^2}{3 a^4 x^3}+\frac {c^2}{a^3 x^2}+\frac {2 c^2 \log (x)}{a}+c^2 (-x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6157, 6150, 75} \[ \frac {c^2}{a^3 x^2}-\frac {c^2}{3 a^4 x^3}+\frac {2 c^2 \log (x)}{a}+c^2 (-x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 75
Rule 6150
Rule 6157
Rubi steps
\begin {align*} \int e^{-2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx &=\frac {c^2 \int \frac {e^{-2 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^2}{x^4} \, dx}{a^4}\\ &=\frac {c^2 \int \frac {(1-a x)^3 (1+a x)}{x^4} \, dx}{a^4}\\ &=\frac {c^2 \int \left (-a^4+\frac {1}{x^4}-\frac {2 a}{x^3}+\frac {2 a^3}{x}\right ) \, dx}{a^4}\\ &=-\frac {c^2}{3 a^4 x^3}+\frac {c^2}{a^3 x^2}-c^2 x+\frac {2 c^2 \log (x)}{a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 40, normalized size = 1.00 \[ -\frac {c^2}{3 a^4 x^3}+\frac {c^2}{a^3 x^2}+\frac {2 c^2 \log (x)}{a}+c^2 (-x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.52, size = 43, normalized size = 1.08 \[ -\frac {3 \, a^{4} c^{2} x^{4} - 6 \, a^{3} c^{2} x^{3} \log \relax (x) - 3 \, a c^{2} x + c^{2}}{3 \, a^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.22, size = 112, normalized size = 2.80 \[ -\frac {2 \, c^{2} \log \left (\frac {{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2} {\left | a \right |}}\right )}{a} + \frac {2 \, c^{2} \log \left ({\left | -\frac {1}{a x + 1} + 1 \right |}\right )}{a} + \frac {{\left (3 \, c^{2} - \frac {5 \, c^{2}}{a x + 1} - \frac {3 \, c^{2}}{{\left (a x + 1\right )}^{2}} + \frac {6 \, c^{2}}{{\left (a x + 1\right )}^{3}}\right )} {\left (a x + 1\right )}}{3 \, a {\left (\frac {1}{a x + 1} - 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 39, normalized size = 0.98 \[ -\frac {c^{2}}{3 a^{4} x^{3}}+\frac {c^{2}}{x^{2} a^{3}}-c^{2} x +\frac {2 c^{2} \ln \relax (x )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 38, normalized size = 0.95 \[ -c^{2} x + \frac {2 \, c^{2} \log \relax (x)}{a} + \frac {3 \, a c^{2} x - c^{2}}{3 \, a^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.89, size = 35, normalized size = 0.88 \[ \frac {c^2\,\left (3\,a\,x-3\,a^4\,x^4+6\,a^3\,x^3\,\ln \relax (x)-1\right )}{3\,a^4\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.19, size = 39, normalized size = 0.98 \[ \frac {- a^{4} c^{2} x + 2 a^{3} c^{2} \log {\relax (x )} - \frac {- 3 a c^{2} x + c^{2}}{3 x^{3}}}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________