Optimal. Leaf size=16 \[ -\frac{c (1-a x)^3}{3 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0189674, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {6140, 32} \[ -\frac{c (1-a x)^3}{3 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6140
Rule 32
Rubi steps
\begin{align*} \int e^{-2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx &=c \int (1-a x)^2 \, dx\\ &=-\frac{c (1-a x)^3}{3 a}\\ \end{align*}
Mathematica [A] time = 0.0103685, size = 20, normalized size = 1.25 \[ c \left (\frac{a^2 x^3}{3}-a x^2+x\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.026, size = 14, normalized size = 0.9 \begin{align*}{\frac{c \left ( ax-1 \right ) ^{3}}{3\,a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.950734, size = 27, normalized size = 1.69 \begin{align*} \frac{1}{3} \, a^{2} c x^{3} - a c x^{2} + c x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.13448, size = 42, normalized size = 2.62 \begin{align*} \frac{1}{3} \, a^{2} c x^{3} - a c x^{2} + c x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.150038, size = 19, normalized size = 1.19 \begin{align*} \frac{a^{2} c x^{3}}{3} - a c x^{2} + c x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.1363, size = 46, normalized size = 2.88 \begin{align*} \frac{{\left (a x + 1\right )}^{3}{\left (c - \frac{6 \, c}{a x + 1} + \frac{12 \, c}{{\left (a x + 1\right )}^{2}}\right )}}{3 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]