Optimal. Leaf size=63 \[ \frac{x^3}{3 a^2 c}+\frac{x^2}{a^3 c}+\frac{3 x}{a^4 c}+\frac{1}{a^5 c (1-a x)}+\frac{4 \log (1-a x)}{a^5 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.111975, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {6150, 43} \[ \frac{x^3}{3 a^2 c}+\frac{x^2}{a^3 c}+\frac{3 x}{a^4 c}+\frac{1}{a^5 c (1-a x)}+\frac{4 \log (1-a x)}{a^5 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6150
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a x)} x^4}{c-a^2 c x^2} \, dx &=\frac{\int \frac{x^4}{(1-a x)^2} \, dx}{c}\\ &=\frac{\int \left (\frac{3}{a^4}+\frac{2 x}{a^3}+\frac{x^2}{a^2}+\frac{1}{a^4 (-1+a x)^2}+\frac{4}{a^4 (-1+a x)}\right ) \, dx}{c}\\ &=\frac{3 x}{a^4 c}+\frac{x^2}{a^3 c}+\frac{x^3}{3 a^2 c}+\frac{1}{a^5 c (1-a x)}+\frac{4 \log (1-a x)}{a^5 c}\\ \end{align*}
Mathematica [A] time = 0.037359, size = 49, normalized size = 0.78 \[ \frac{a^3 x^3+3 a^2 x^2+9 a x+\frac{3}{1-a x}+12 \log (1-a x)}{3 a^5 c} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.034, size = 61, normalized size = 1. \begin{align*}{\frac{{x}^{3}}{3\,{a}^{2}c}}+{\frac{{x}^{2}}{{a}^{3}c}}+3\,{\frac{x}{{a}^{4}c}}-{\frac{1}{c{a}^{5} \left ( ax-1 \right ) }}+4\,{\frac{\ln \left ( ax-1 \right ) }{c{a}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.959523, size = 77, normalized size = 1.22 \begin{align*} -\frac{1}{a^{6} c x - a^{5} c} + \frac{a^{2} x^{3} + 3 \, a x^{2} + 9 \, x}{3 \, a^{4} c} + \frac{4 \, \log \left (a x - 1\right )}{a^{5} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.94782, size = 131, normalized size = 2.08 \begin{align*} \frac{a^{4} x^{4} + 2 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 9 \, a x + 12 \,{\left (a x - 1\right )} \log \left (a x - 1\right ) - 3}{3 \,{\left (a^{6} c x - a^{5} c\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.357478, size = 53, normalized size = 0.84 \begin{align*} - \frac{1}{a^{6} c x - a^{5} c} + \frac{x^{3}}{3 a^{2} c} + \frac{x^{2}}{a^{3} c} + \frac{3 x}{a^{4} c} + \frac{4 \log{\left (a x - 1 \right )}}{a^{5} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.18937, size = 95, normalized size = 1.51 \begin{align*} \frac{4 \, \log \left ({\left | a x - 1 \right |}\right )}{a^{5} c} - \frac{1}{{\left (a x - 1\right )} a^{5} c} + \frac{a^{4} c^{2} x^{3} + 3 \, a^{3} c^{2} x^{2} + 9 \, a^{2} c^{2} x}{3 \, a^{6} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]