Optimal. Leaf size=17 \[ \frac{c^2 (a x+1)^5}{5 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0571792, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {6167, 6140, 32} \[ \frac{c^2 (a x+1)^5}{5 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6167
Rule 6140
Rule 32
Rubi steps
\begin{align*} \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx &=\int e^{4 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx\\ &=c^2 \int (1+a x)^4 \, dx\\ &=\frac{c^2 (1+a x)^5}{5 a}\\ \end{align*}
Mathematica [B] time = 0.0283111, size = 37, normalized size = 2.18 \[ c^2 \left (\frac{a^4 x^5}{5}+a^3 x^4+2 a^2 x^3+2 a x^2+x\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.04, size = 16, normalized size = 0.9 \begin{align*}{\frac{{c}^{2} \left ( ax+1 \right ) ^{5}}{5\,a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.08093, size = 63, normalized size = 3.71 \begin{align*} \frac{1}{5} \, a^{4} c^{2} x^{5} + a^{3} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{3} + 2 \, a c^{2} x^{2} + c^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.53714, size = 93, normalized size = 5.47 \begin{align*} \frac{1}{5} \, a^{4} c^{2} x^{5} + a^{3} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{3} + 2 \, a c^{2} x^{2} + c^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 0.091581, size = 48, normalized size = 2.82 \begin{align*} \frac{a^{4} c^{2} x^{5}}{5} + a^{3} c^{2} x^{4} + 2 a^{2} c^{2} x^{3} + 2 a c^{2} x^{2} + c^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.11551, size = 86, normalized size = 5.06 \begin{align*} \frac{{\left (c^{2} + \frac{10 \, c^{2}}{a x - 1} + \frac{40 \, c^{2}}{{\left (a x - 1\right )}^{2}} + \frac{80 \, c^{2}}{{\left (a x - 1\right )}^{3}} + \frac{80 \, c^{2}}{{\left (a x - 1\right )}^{4}}\right )}{\left (a x - 1\right )}^{5}}{5 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]