Optimal. Leaf size=57 \[ \frac{4 x^2}{a^2}+\frac{12 x}{a^3}+\frac{4}{a^4 (1-a x)}+\frac{16 \log (1-a x)}{a^4}+\frac{4 x^3}{3 a}+\frac{x^4}{4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0495242, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6126, 88} \[ \frac{4 x^2}{a^2}+\frac{12 x}{a^3}+\frac{4}{a^4 (1-a x)}+\frac{16 \log (1-a x)}{a^4}+\frac{4 x^3}{3 a}+\frac{x^4}{4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6126
Rule 88
Rubi steps
\begin{align*} \int e^{4 \tanh ^{-1}(a x)} x^3 \, dx &=\int \frac{x^3 (1+a x)^2}{(1-a x)^2} \, dx\\ &=\int \left (\frac{12}{a^3}+\frac{8 x}{a^2}+\frac{4 x^2}{a}+x^3+\frac{4}{a^3 (-1+a x)^2}+\frac{16}{a^3 (-1+a x)}\right ) \, dx\\ &=\frac{12 x}{a^3}+\frac{4 x^2}{a^2}+\frac{4 x^3}{3 a}+\frac{x^4}{4}+\frac{4}{a^4 (1-a x)}+\frac{16 \log (1-a x)}{a^4}\\ \end{align*}
Mathematica [A] time = 0.0450642, size = 57, normalized size = 1. \[ \frac{4 x^2}{a^2}+\frac{12 x}{a^3}+\frac{4}{a^4 (1-a x)}+\frac{16 \log (1-a x)}{a^4}+\frac{4 x^3}{3 a}+\frac{x^4}{4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.039, size = 52, normalized size = 0.9 \begin{align*}{\frac{{x}^{4}}{4}}+{\frac{4\,{x}^{3}}{3\,a}}+4\,{\frac{{x}^{2}}{{a}^{2}}}+12\,{\frac{x}{{a}^{3}}}-4\,{\frac{1}{{a}^{4} \left ( ax-1 \right ) }}+16\,{\frac{\ln \left ( ax-1 \right ) }{{a}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.94969, size = 78, normalized size = 1.37 \begin{align*} -\frac{4}{a^{5} x - a^{4}} + \frac{3 \, a^{3} x^{4} + 16 \, a^{2} x^{3} + 48 \, a x^{2} + 144 \, x}{12 \, a^{3}} + \frac{16 \, \log \left (a x - 1\right )}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.87873, size = 155, normalized size = 2.72 \begin{align*} \frac{3 \, a^{5} x^{5} + 13 \, a^{4} x^{4} + 32 \, a^{3} x^{3} + 96 \, a^{2} x^{2} - 144 \, a x + 192 \,{\left (a x - 1\right )} \log \left (a x - 1\right ) - 48}{12 \,{\left (a^{5} x - a^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.442445, size = 49, normalized size = 0.86 \begin{align*} \frac{x^{4}}{4} - \frac{4}{a^{5} x - a^{4}} + \frac{4 x^{3}}{3 a} + \frac{4 x^{2}}{a^{2}} + \frac{12 x}{a^{3}} + \frac{16 \log{\left (a x - 1 \right )}}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.17888, size = 82, normalized size = 1.44 \begin{align*} \frac{16 \, \log \left ({\left | a x - 1 \right |}\right )}{a^{4}} - \frac{4}{{\left (a x - 1\right )} a^{4}} + \frac{3 \, a^{8} x^{4} + 16 \, a^{7} x^{3} + 48 \, a^{6} x^{2} + 144 \, a^{5} x}{12 \, a^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]