Optimal. Leaf size=74 \[ -\frac{9}{a c^3 (1-a x)}+\frac{7}{2 a c^3 (1-a x)^2}-\frac{2}{3 a c^3 (1-a x)^3}-\frac{5 \log (1-a x)}{a c^3}-\frac{x}{c^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.133356, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {6131, 6129, 77} \[ -\frac{9}{a c^3 (1-a x)}+\frac{7}{2 a c^3 (1-a x)^2}-\frac{2}{3 a c^3 (1-a x)^3}-\frac{5 \log (1-a x)}{a c^3}-\frac{x}{c^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6131
Rule 6129
Rule 77
Rubi steps
\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^3} \, dx &=-\frac{a^3 \int \frac{e^{2 \tanh ^{-1}(a x)} x^3}{(1-a x)^3} \, dx}{c^3}\\ &=-\frac{a^3 \int \frac{x^3 (1+a x)}{(1-a x)^4} \, dx}{c^3}\\ &=-\frac{a^3 \int \left (\frac{1}{a^3}+\frac{2}{a^3 (-1+a x)^4}+\frac{7}{a^3 (-1+a x)^3}+\frac{9}{a^3 (-1+a x)^2}+\frac{5}{a^3 (-1+a x)}\right ) \, dx}{c^3}\\ &=-\frac{x}{c^3}-\frac{2}{3 a c^3 (1-a x)^3}+\frac{7}{2 a c^3 (1-a x)^2}-\frac{9}{a c^3 (1-a x)}-\frac{5 \log (1-a x)}{a c^3}\\ \end{align*}
Mathematica [A] time = 0.122096, size = 63, normalized size = 0.85 \[ \frac{-6 a^4 x^4+18 a^3 x^3+36 a^2 x^2-81 a x-30 (a x-1)^3 \log (1-a x)+37}{6 a c^3 (a x-1)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.035, size = 67, normalized size = 0.9 \begin{align*} -{\frac{x}{{c}^{3}}}+{\frac{2}{3\,a{c}^{3} \left ( ax-1 \right ) ^{3}}}+{\frac{7}{2\,a{c}^{3} \left ( ax-1 \right ) ^{2}}}+9\,{\frac{1}{a{c}^{3} \left ( ax-1 \right ) }}-5\,{\frac{\ln \left ( ax-1 \right ) }{a{c}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.972135, size = 103, normalized size = 1.39 \begin{align*} \frac{54 \, a^{2} x^{2} - 87 \, a x + 37}{6 \,{\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}} - \frac{x}{c^{3}} - \frac{5 \, \log \left (a x - 1\right )}{a c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.17975, size = 219, normalized size = 2.96 \begin{align*} -\frac{6 \, a^{4} x^{4} - 18 \, a^{3} x^{3} - 36 \, a^{2} x^{2} + 81 \, a x + 30 \,{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (a x - 1\right ) - 37}{6 \,{\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.551466, size = 73, normalized size = 0.99 \begin{align*} \frac{54 a^{2} x^{2} - 87 a x + 37}{6 a^{4} c^{3} x^{3} - 18 a^{3} c^{3} x^{2} + 18 a^{2} c^{3} x - 6 a c^{3}} - \frac{x}{c^{3}} - \frac{5 \log{\left (a x - 1 \right )}}{a c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.16687, size = 69, normalized size = 0.93 \begin{align*} -\frac{x}{c^{3}} - \frac{5 \, \log \left ({\left | a x - 1 \right |}\right )}{a c^{3}} + \frac{54 \, a^{2} x^{2} - 87 \, a x + 37}{6 \,{\left (a x - 1\right )}^{3} a c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]