Optimal. Leaf size=16 \[ -\frac {c (1-a x)^3}{3 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6140, 32} \[ -\frac {c (1-a x)^3}{3 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 32
Rule 6140
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 = 0.01, 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]
________________________________________________________________________________________
fricas [A] time = 0.67, size = 20, normalized size = 1.25 \[ \frac {1}{3} \, a^{2} c x^{3} - a c x^{2} + c x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.18, size = 34, normalized size = 2.12 \[ \frac {{\left (a x + 1\right )}^{3} {\left (c - \frac {6 \, c}{a x + 1} + \frac {12 \, c}{{\left (a x + 1\right )}^{2}}\right )}}{3 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 14, normalized size = 0.88 \[ \frac {c \left (a x -1\right )^{3}}{3 a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.38, size = 20, normalized size = 1.25 \[ \frac {1}{3} \, a^{2} c x^{3} - a c x^{2} + c x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.04, size = 17, normalized size = 1.06 \[ \frac {c\,x\,\left (a^2\,x^2-3\,a\,x+3\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.07, size = 19, normalized size = 1.19 \[ \frac {a^{2} c x^{3}}{3} - a c x^{2} + c x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________