Optimal. Leaf size=54 \[ \frac{4 a^3}{1-a x}-\frac{8 a^2}{x}+12 a^3 \log (x)-12 a^3 \log (1-a x)-\frac{2 a}{x^2}-\frac{1}{3 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0612882, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6167, 6126, 88} \[ \frac{4 a^3}{1-a x}-\frac{8 a^2}{x}+12 a^3 \log (x)-12 a^3 \log (1-a x)-\frac{2 a}{x^2}-\frac{1}{3 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6167
Rule 6126
Rule 88
Rubi steps
\begin{align*} \int \frac{e^{4 \coth ^{-1}(a x)}}{x^4} \, dx &=\int \frac{e^{4 \tanh ^{-1}(a x)}}{x^4} \, dx\\ &=\int \frac{(1+a x)^2}{x^4 (1-a x)^2} \, dx\\ &=\int \left (\frac{1}{x^4}+\frac{4 a}{x^3}+\frac{8 a^2}{x^2}+\frac{12 a^3}{x}+\frac{4 a^4}{(-1+a x)^2}-\frac{12 a^4}{-1+a x}\right ) \, dx\\ &=-\frac{1}{3 x^3}-\frac{2 a}{x^2}-\frac{8 a^2}{x}+\frac{4 a^3}{1-a x}+12 a^3 \log (x)-12 a^3 \log (1-a x)\\ \end{align*}
Mathematica [A] time = 0.0424887, size = 54, normalized size = 1. \[ \frac{4 a^3}{1-a x}-\frac{8 a^2}{x}+12 a^3 \log (x)-12 a^3 \log (1-a x)-\frac{2 a}{x^2}-\frac{1}{3 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.049, size = 51, normalized size = 0.9 \begin{align*} -{\frac{1}{3\,{x}^{3}}}-2\,{\frac{a}{{x}^{2}}}-8\,{\frac{{a}^{2}}{x}}+12\,{a}^{3}\ln \left ( x \right ) -4\,{\frac{{a}^{3}}{ax-1}}-12\,{a}^{3}\ln \left ( ax-1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.0022, size = 76, normalized size = 1.41 \begin{align*} -12 \, a^{3} \log \left (a x - 1\right ) + 12 \, a^{3} \log \left (x\right ) - \frac{36 \, a^{3} x^{3} - 18 \, a^{2} x^{2} - 5 \, a x - 1}{3 \,{\left (a x^{4} - x^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.89311, size = 173, normalized size = 3.2 \begin{align*} -\frac{36 \, a^{3} x^{3} - 18 \, a^{2} x^{2} - 5 \, a x + 36 \,{\left (a^{4} x^{4} - a^{3} x^{3}\right )} \log \left (a x - 1\right ) - 36 \,{\left (a^{4} x^{4} - a^{3} x^{3}\right )} \log \left (x\right ) - 1}{3 \,{\left (a x^{4} - x^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.462664, size = 49, normalized size = 0.91 \begin{align*} 12 a^{3} \left (\log{\left (x \right )} - \log{\left (x - \frac{1}{a} \right )}\right ) - \frac{36 a^{3} x^{3} - 18 a^{2} x^{2} - 5 a x - 1}{3 a x^{4} - 3 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.1818, size = 100, normalized size = 1.85 \begin{align*} 12 \, a^{3} \log \left ({\left | -\frac{1}{a x - 1} - 1 \right |}\right ) - \frac{4 \, a^{3}}{a x - 1} + \frac{31 \, a^{3} + \frac{69 \, a^{3}}{a x - 1} + \frac{39 \, a^{3}}{{\left (a x - 1\right )}^{2}}}{3 \,{\left (\frac{1}{a x - 1} + 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]