Optimal. Leaf size=33 \[ -\frac{1}{2 a c^3 (1-a x)}-\frac{\tanh ^{-1}(a x)}{2 a c^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0626256, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6167, 6129, 44, 207} \[ -\frac{1}{2 a c^3 (1-a x)}-\frac{\tanh ^{-1}(a x)}{2 a c^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6167
Rule 6129
Rule 44
Rule 207
Rubi steps
\begin{align*} \int \frac{e^{-2 \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx &=-\int \frac{e^{-2 \tanh ^{-1}(a x)}}{(c-a c x)^3} \, dx\\ &=-\frac{\int \frac{1}{(1-a x)^2 (1+a x)} \, dx}{c^3}\\ &=-\frac{\int \left (\frac{1}{2 (-1+a x)^2}-\frac{1}{2 \left (-1+a^2 x^2\right )}\right ) \, dx}{c^3}\\ &=-\frac{1}{2 a c^3 (1-a x)}+\frac{\int \frac{1}{-1+a^2 x^2} \, dx}{2 c^3}\\ &=-\frac{1}{2 a c^3 (1-a x)}-\frac{\tanh ^{-1}(a x)}{2 a c^3}\\ \end{align*}
Mathematica [A] time = 0.0199579, size = 32, normalized size = 0.97 \[ -\frac{\frac{1}{2 a (1-a x)}+\frac{\tanh ^{-1}(a x)}{2 a}}{c^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.051, size = 45, normalized size = 1.4 \begin{align*} -{\frac{\ln \left ( ax+1 \right ) }{4\,a{c}^{3}}}+{\frac{1}{2\,a{c}^{3} \left ( ax-1 \right ) }}+{\frac{\ln \left ( ax-1 \right ) }{4\,a{c}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.01879, size = 65, normalized size = 1.97 \begin{align*} \frac{1}{2 \,{\left (a^{2} c^{3} x - a c^{3}\right )}} - \frac{\log \left (a x + 1\right )}{4 \, a c^{3}} + \frac{\log \left (a x - 1\right )}{4 \, a c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.52915, size = 108, normalized size = 3.27 \begin{align*} -\frac{{\left (a x - 1\right )} \log \left (a x + 1\right ) -{\left (a x - 1\right )} \log \left (a x - 1\right ) - 2}{4 \,{\left (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 = 0.653654, size = 39, normalized size = 1.18 \begin{align*} \frac{1}{2 a^{2} c^{3} x - 2 a c^{3}} - \frac{- \frac{\log{\left (x - \frac{1}{a} \right )}}{4} + \frac{\log{\left (x + \frac{1}{a} \right )}}{4}}{a c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.1426, size = 62, normalized size = 1.88 \begin{align*} -\frac{\log \left ({\left | a x + 1 \right |}\right )}{4 \, a c^{3}} + \frac{\log \left ({\left | a x - 1 \right |}\right )}{4 \, a c^{3}} + \frac{1}{2 \,{\left (a x - 1\right )} a c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]