Optimal. Leaf size=15 \[ -\frac{c (a x+1)^3}{3 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.033691, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {6167, 6140, 32} \[ -\frac{c (a x+1)^3}{3 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6167
Rule 6140
Rule 32
Rubi steps
\begin{align*} \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx\\ &=-\left (c \int (1+a x)^2 \, dx\right )\\ &=-\frac{c (1+a x)^3}{3 a}\\ \end{align*}
Mathematica [A] time = 0.0121771, size = 20, normalized size = 1.33 \[ -c \left (\frac{a^2 x^3}{3}+a x^2+x\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.037, 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 = 1.05957, size = 28, normalized size = 1.87 \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 = 1.47445, size = 43, normalized size = 2.87 \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 = 0.06979, size = 20, normalized size = 1.33 \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 [A] time = 1.09652, size = 28, normalized size = 1.87 \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]