Optimal. Leaf size=51 \[ \frac {1}{4 a c^4 (1-a x)}+\frac {1}{4 a c^4 (1-a x)^2}+\frac {\tanh ^{-1}(a x)}{4 a c^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6129, 44, 207} \[ \frac {1}{4 a c^4 (1-a x)}+\frac {1}{4 a c^4 (1-a x)^2}+\frac {\tanh ^{-1}(a x)}{4 a c^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rule 207
Rule 6129
Rubi steps
\begin {align*} \int \frac {e^{-2 \tanh ^{-1}(a x)}}{(c-a c x)^4} \, dx &=\frac {\int \frac {1}{(1-a x)^3 (1+a x)} \, dx}{c^4}\\ &=\frac {\int \left (-\frac {1}{2 (-1+a x)^3}+\frac {1}{4 (-1+a x)^2}-\frac {1}{4 \left (-1+a^2 x^2\right )}\right ) \, dx}{c^4}\\ &=\frac {1}{4 a c^4 (1-a x)^2}+\frac {1}{4 a c^4 (1-a x)}-\frac {\int \frac {1}{-1+a^2 x^2} \, dx}{4 c^4}\\ &=\frac {1}{4 a c^4 (1-a x)^2}+\frac {1}{4 a c^4 (1-a x)}+\frac {\tanh ^{-1}(a x)}{4 a c^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 35, normalized size = 0.69 \[ \frac {-a x+(a x-1)^2 \tanh ^{-1}(a x)+2}{4 a c^4 (a x-1)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.52, size = 76, normalized size = 1.49 \[ -\frac {2 \, a x - {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x + 1\right ) + {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x - 1\right ) - 4}{8 \, {\left (a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.58, size = 58, normalized size = 1.14 \[ -\frac {\log \left ({\left | -\frac {2}{a x + 1} + 1 \right |}\right )}{8 \, a c^{4}} - \frac {\frac {3}{a} - \frac {8}{{\left (a x + 1\right )} a}}{16 \, c^{4} {\left (\frac {2}{a x + 1} - 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 60, normalized size = 1.18 \[ \frac {1}{4 c^{4} a \left (a x -1\right )^{2}}-\frac {1}{4 c^{4} a \left (a x -1\right )}-\frac {\ln \left (a x -1\right )}{8 c^{4} a}+\frac {\ln \left (a x +1\right )}{8 a \,c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.33, size = 63, normalized size = 1.24 \[ -\frac {a x - 2}{4 \, {\left (a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}} + \frac {\log \left (a x + 1\right )}{8 \, a c^{4}} - \frac {\log \left (a x - 1\right )}{8 \, a c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.82, size = 47, normalized size = 0.92 \[ \frac {\mathrm {atanh}\left (a\,x\right )}{4\,a\,c^4}-\frac {\frac {x}{4}-\frac {1}{2\,a}}{a^2\,c^4\,x^2-2\,a\,c^4\,x+c^4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.28, size = 56, normalized size = 1.10 \[ - \frac {a x - 2}{4 a^{3} c^{4} x^{2} - 8 a^{2} c^{4} x + 4 a c^{4}} - \frac {\frac {\log {\left (x - \frac {1}{a} \right )}}{8} - \frac {\log {\left (x + \frac {1}{a} \right )}}{8}}{a c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________