Optimal. Leaf size=15 \[ \frac{c (a x+1)^3}{3 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0190191, antiderivative size = 15, 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} \[ \frac{c (a x+1)^3}{3 a} \]
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 = 0.0119897, size = 19, normalized size = 1.27 \[ c \left (\frac{a^2 x^3}{3}+a x^2+x\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.025, 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 = 0.941747, size = 26, normalized size = 1.73 \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 = 1.83929, size = 42, normalized size = 2.8 \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 = 0.076215, size = 19, normalized size = 1.27 \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 = 1.14134, size = 26, normalized size = 1.73 \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]