Optimal. Leaf size=17 \[ -\frac{c^2 (a x+1)^4}{2 x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0692598, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {6150, 74} \[ -\frac{c^2 (a x+1)^4}{2 x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6150
Rule 74
Rubi steps
\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^2}{x^3} \, dx &=c^2 \int \frac{(1-a x) (1+a x)^3}{x^3} \, dx\\ &=-\frac{c^2 (1+a x)^4}{2 x^2}\\ \end{align*}
Mathematica [A] time = 0.0148526, size = 17, normalized size = 1. \[ -\frac{c^2 (a x+1)^4}{2 x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.031, size = 31, normalized size = 1.8 \begin{align*}{c}^{2} \left ( -{\frac{{x}^{2}{a}^{4}}{2}}-2\,x{a}^{3}-2\,{\frac{a}{x}}-{\frac{1}{2\,{x}^{2}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 0.948993, size = 50, normalized size = 2.94 \begin{align*} -\frac{1}{2} \, a^{4} c^{2} x^{2} - 2 \, a^{3} c^{2} x - \frac{4 \, a c^{2} x + c^{2}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.64385, size = 78, normalized size = 4.59 \begin{align*} -\frac{a^{4} c^{2} x^{4} + 4 \, a^{3} c^{2} x^{3} + 4 \, a c^{2} x + c^{2}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 0.315029, size = 39, normalized size = 2.29 \begin{align*} - \frac{a^{4} c^{2} x^{2}}{2} - 2 a^{3} c^{2} x - \frac{4 a c^{2} x + c^{2}}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.11806, size = 50, normalized size = 2.94 \begin{align*} -\frac{1}{2} \, a^{4} c^{2} x^{2} - 2 \, a^{3} c^{2} x - \frac{4 \, a c^{2} x + c^{2}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]