Optimal. Leaf size=26 \[ \frac{1}{2} a c x^2+\frac{4 c \log (a x+1)}{a}-3 c x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0230231, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {6129, 43} \[ \frac{1}{2} a c x^2+\frac{4 c \log (a x+1)}{a}-3 c x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6129
Rule 43
Rubi steps
\begin{align*} \int e^{-2 \tanh ^{-1}(a x)} (c-a c x) \, dx &=c \int \frac{(1-a x)^2}{1+a x} \, dx\\ &=c \int \left (-3+a x+\frac{4}{1+a x}\right ) \, dx\\ &=-3 c x+\frac{1}{2} a c x^2+\frac{4 c \log (1+a x)}{a}\\ \end{align*}
Mathematica [A] time = 0.0134928, size = 25, normalized size = 0.96 \[ c \left (\frac{a x^2}{2}+\frac{4 \log (a x+1)}{a}-3 x\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.036, size = 25, normalized size = 1. \begin{align*} -3\,cx+{\frac{ac{x}^{2}}{2}}+4\,{\frac{c\ln \left ( ax+1 \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.94396, size = 32, normalized size = 1.23 \begin{align*} \frac{1}{2} \, a c x^{2} - 3 \, c x + \frac{4 \, c \log \left (a x + 1\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.60623, size = 65, normalized size = 2.5 \begin{align*} \frac{a^{2} c x^{2} - 6 \, a c x + 8 \, c \log \left (a x + 1\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.286534, size = 24, normalized size = 0.92 \begin{align*} \frac{a c x^{2}}{2} - 3 c x + \frac{4 c \log{\left (a x + 1 \right )}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.23473, size = 68, normalized size = 2.62 \begin{align*} \frac{{\left (a x + 1\right )}^{2}{\left (c - \frac{8 \, c}{a x + 1}\right )}}{2 \, a} - \frac{4 \, c \log \left (\frac{{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2}{\left | a \right |}}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]