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 = 0.0984824, antiderivative size = 40, normalized size of antiderivative = 1., 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 6131
Rule 6129
Rule 75
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 = 0.120661, 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]
________________________________________________________________________________________
Maple [A] time = 0.037, size = 39, normalized size = 1. \begin{align*}{\frac{{c}^{3}}{2\,{x}^{2}{a}^{3}}}-{\frac{{c}^{3}}{{a}^{2}x}}-{c}^{3}x+{\frac{{c}^{3}\ln \left ( x \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.948866, size = 50, normalized size = 1.25 \begin{align*} -c^{3} x + \frac{c^{3} \log \left (x\right )}{a} - \frac{2 \, a c^{3} x - c^{3}}{2 \, a^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.01761, size = 99, normalized size = 2.48 \begin{align*} -\frac{2 \, a^{3} c^{3} x^{3} - 2 \, a^{2} c^{3} x^{2} \log \left (x\right ) + 2 \, a c^{3} x - c^{3}}{2 \, a^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.336779, size = 37, normalized size = 0.92 \begin{align*} \frac{- a^{3} c^{3} x + a^{2} c^{3} \log{\left (x \right )} - \frac{2 a c^{3} x - c^{3}}{2 x^{2}}}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.138, size = 51, normalized size = 1.27 \begin{align*} -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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]