Optimal. Leaf size=57 \[ \frac {4}{a^4 (1-a x)}+\frac {16 \log (1-a x)}{a^4}+\frac {12 x}{a^3}+\frac {4 x^2}{a^2}+\frac {4 x^3}{3 a}+\frac {x^4}{4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6167, 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 88
Rule 6126
Rule 6167
Rubi steps
\begin {align*} \int e^{4 \coth ^{-1}(a x)} x^3 \, dx &=\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 = 0.05, size = 57, normalized size = 1.00 \[ \frac {4}{a^4 (1-a x)}+\frac {16 \log (1-a x)}{a^4}+\frac {12 x}{a^3}+\frac {4 x^2}{a^2}+\frac {4 x^3}{3 a}+\frac {x^4}{4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.64, size = 66, normalized size = 1.16 \[ \frac {3 \, a^{5} x^{5} + 13 \, a^{4} x^{4} + 32 \, a^{3} x^{3} + 96 \, a^{2} x^{2} - 144 \, a x + 192 \, {\left (a x - 1\right )} \log \left (a x - 1\right ) - 48}{12 \, {\left (a^{5} x - a^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.14, size = 78, normalized size = 1.37 \[ \frac {{\left (a x - 1\right )}^{4} {\left (\frac {28}{a x - 1} + \frac {114}{{\left (a x - 1\right )}^{2}} + \frac {300}{{\left (a x - 1\right )}^{3}} + 3\right )}}{12 \, a^{4}} - \frac {16 \, \log \left (\frac {{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2} {\left | a \right |}}\right )}{a^{4}} - \frac {4}{{\left (a x - 1\right )} a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 52, normalized size = 0.91 \[ \frac {x^{4}}{4}+\frac {4 x^{3}}{3 a}+\frac {4 x^{2}}{a^{2}}+\frac {12 x}{a^{3}}+\frac {16 \ln \left (a x -1\right )}{a^{4}}-\frac {4}{a^{4} \left (a x -1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 58, normalized size = 1.02 \[ -\frac {4}{a^{5} x - a^{4}} + \frac {3 \, a^{3} x^{4} + 16 \, a^{2} x^{3} + 48 \, a x^{2} + 144 \, x}{12 \, a^{3}} + \frac {16 \, \log \left (a x - 1\right )}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.04, size = 57, normalized size = 1.00 \[ \frac {16\,\ln \left (a\,x-1\right )}{a^4}-\frac {4}{a\,\left (a^4\,x-a^3\right )}+\frac {12\,x}{a^3}+\frac {x^4}{4}+\frac {4\,x^3}{3\,a}+\frac {4\,x^2}{a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.15, size = 49, normalized size = 0.86 \[ \frac {x^{4}}{4} - \frac {4}{a^{5} x - a^{4}} + \frac {4 x^{3}}{3 a} + \frac {4 x^{2}}{a^{2}} + \frac {12 x}{a^{3}} + \frac {16 \log {\left (a x - 1 \right )}}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________