Optimal. Leaf size=40 \[ \frac {c^3}{2 a^3 x^2}-\frac {c^3}{a^2 x}+\frac {c^3 \log (x)}{a}+c^3 (-x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, 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 = {6131, 6129, 75} \[ \frac {c^3}{2 a^3 x^2}-\frac {c^3}{a^2 x}+\frac {c^3 \log (x)}{a}+c^3 (-x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 75
Rule 6129
Rule 6131
Rubi steps
\begin {align*} \int e^{2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx &=-\frac {c^3 \int \frac {e^{2 \tanh ^{-1}(a x)} (1-a x)^3}{x^3} \, dx}{a^3}\\ &=-\frac {c^3 \int \frac {(1-a x)^2 (1+a x)}{x^3} \, dx}{a^3}\\ &=-\frac {c^3 \int \left (a^3+\frac {1}{x^3}-\frac {a}{x^2}-\frac {a^2}{x}\right ) \, dx}{a^3}\\ &=\frac {c^3}{2 a^3 x^2}-\frac {c^3}{a^2 x}-c^3 x+\frac {c^3 \log (x)}{a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.13, size = 42, normalized size = 1.05 \[ \frac {c^3}{2 a^3 x^2}-\frac {c^3}{a^2 x}+\frac {c^3 \log (a x)}{a}+c^3 (-x) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 45, normalized size = 1.12 \[ -\frac {2 \, a^{3} c^{3} x^{3} - 2 \, a^{2} c^{3} x^{2} \log \relax (x) + 2 \, a c^{3} x - c^{3}}{2 \, a^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.20, size = 38, normalized size = 0.95 \[ -c^{3} x + \frac {c^{3} \log \left ({\left | x \right |}\right )}{a} - \frac {2 \, a c^{3} x - c^{3}}{2 \, a^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 39, normalized size = 0.98 \[ \frac {c^{3}}{2 x^{2} a^{3}}-\frac {c^{3}}{a^{2} x}-c^{3} x +\frac {c^{3} \ln \relax (x )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.05, size = 35, normalized size = 0.88 \[ -\frac {c^3\,\left (2\,a\,x+2\,a^3\,x^3-2\,a^2\,x^2\,\ln \relax (x)-1\right )}{2\,a^3\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.17, size = 37, normalized size = 0.92 \[ \frac {- a^{3} c^{3} x + a^{2} c^{3} \log {\relax (x )} - \frac {2 a c^{3} x - c^{3}}{2 x^{2}}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________