Optimal. Leaf size=35 \[ \frac{c^2 (a x+1)^5}{5 a}-\frac{c^2 (a x+1)^4}{2 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0668577, antiderivative size = 35, 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 = {6167, 6140, 43} \[ \frac{c^2 (a x+1)^5}{5 a}-\frac{c^2 (a x+1)^4}{2 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6167
Rule 6140
Rule 43
Rubi steps
\begin{align*} \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx\\ &=-\left (c^2 \int (1-a x) (1+a x)^3 \, dx\right )\\ &=-\left (c^2 \int \left (2 (1+a x)^3-(1+a x)^4\right ) \, dx\right )\\ &=-\frac{c^2 (1+a x)^4}{2 a}+\frac{c^2 (1+a x)^5}{5 a}\\ \end{align*}
Mathematica [A] time = 0.0162947, size = 23, normalized size = 0.66 \[ \frac{c^2 (a x+1)^4 (2 a x-3)}{10 a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.039, size = 31, normalized size = 0.9 \begin{align*}{c}^{2} \left ({\frac{{x}^{5}{a}^{4}}{5}}+{\frac{{x}^{4}{a}^{3}}{2}}-a{x}^{2}-x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.04745, size = 51, normalized size = 1.46 \begin{align*} \frac{1}{5} \, a^{4} c^{2} x^{5} + \frac{1}{2} \, a^{3} c^{2} x^{4} - a c^{2} x^{2} - c^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.44836, size = 74, normalized size = 2.11 \begin{align*} \frac{1}{5} \, a^{4} c^{2} x^{5} + \frac{1}{2} \, a^{3} c^{2} x^{4} - a c^{2} x^{2} - c^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.081442, size = 36, normalized size = 1.03 \begin{align*} \frac{a^{4} c^{2} x^{5}}{5} + \frac{a^{3} c^{2} x^{4}}{2} - a c^{2} x^{2} - c^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.12924, size = 51, normalized size = 1.46 \begin{align*} \frac{1}{5} \, a^{4} c^{2} x^{5} + \frac{1}{2} \, a^{3} c^{2} x^{4} - a c^{2} x^{2} - c^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]