Optimal. Leaf size=73 \[ -\frac {7}{4 a c^2 (a x+1)}+\frac {1}{4 a c^2 (a x+1)^2}+\frac {\log (1-a x)}{8 a c^2}-\frac {17 \log (a x+1)}{8 a c^2}+\frac {x}{c^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.17, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6167, 6157, 6150, 88} \[ -\frac {7}{4 a c^2 (a x+1)}+\frac {1}{4 a c^2 (a x+1)^2}+\frac {\log (1-a x)}{8 a c^2}-\frac {17 \log (a x+1)}{8 a c^2}+\frac {x}{c^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 88
Rule 6150
Rule 6157
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx &=-\int \frac {e^{-2 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx\\ &=-\frac {a^4 \int \frac {e^{-2 \tanh ^{-1}(a x)} x^4}{\left (1-a^2 x^2\right )^2} \, dx}{c^2}\\ &=-\frac {a^4 \int \frac {x^4}{(1-a x) (1+a x)^3} \, dx}{c^2}\\ &=-\frac {a^4 \int \left (-\frac {1}{a^4}-\frac {1}{8 a^4 (-1+a x)}+\frac {1}{2 a^4 (1+a x)^3}-\frac {7}{4 a^4 (1+a x)^2}+\frac {17}{8 a^4 (1+a x)}\right ) \, dx}{c^2}\\ &=\frac {x}{c^2}+\frac {1}{4 a c^2 (1+a x)^2}-\frac {7}{4 a c^2 (1+a x)}+\frac {\log (1-a x)}{8 a c^2}-\frac {17 \log (1+a x)}{8 a c^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 70, normalized size = 0.96 \[ \frac {2 \left (4 a^3 x^3+8 a^2 x^2-3 a x-6\right )+(a x+1)^2 \log (1-a x)-17 (a x+1)^2 \log (a x+1)}{8 a (a c x+c)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.54, size = 92, normalized size = 1.26 \[ \frac {8 \, a^{3} x^{3} + 16 \, a^{2} x^{2} - 6 \, a x - 17 \, {\left (a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (a x + 1\right ) + {\left (a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (a x - 1\right ) - 12}{8 \, {\left (a^{3} c^{2} x^{2} + 2 \, a^{2} c^{2} x + a c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.13, size = 57, normalized size = 0.78 \[ \frac {x}{c^{2}} - \frac {17 \, \log \left ({\left | a x + 1 \right |}\right )}{8 \, a c^{2}} + \frac {\log \left ({\left | a x - 1 \right |}\right )}{8 \, a c^{2}} - \frac {7 \, a x + 6}{4 \, {\left (a x + 1\right )}^{2} a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 65, normalized size = 0.89 \[ \frac {x}{c^{2}}+\frac {\ln \left (a x -1\right )}{8 a \,c^{2}}+\frac {1}{4 a \,c^{2} \left (a x +1\right )^{2}}-\frac {7}{4 a \,c^{2} \left (a x +1\right )}-\frac {17 \ln \left (a x +1\right )}{8 a \,c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 69, normalized size = 0.95 \[ -\frac {7 \, a x + 6}{4 \, {\left (a^{3} c^{2} x^{2} + 2 \, a^{2} c^{2} x + a c^{2}\right )}} + \frac {x}{c^{2}} - \frac {17 \, \log \left (a x + 1\right )}{8 \, a c^{2}} + \frac {\log \left (a x - 1\right )}{8 \, a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.31, size = 68, normalized size = 0.93 \[ \frac {x}{c^2}-\frac {\frac {7\,x}{4}+\frac {3}{2\,a}}{a^2\,c^2\,x^2+2\,a\,c^2\,x+c^2}+\frac {\ln \left (a\,x-1\right )}{8\,a\,c^2}-\frac {17\,\ln \left (a\,x+1\right )}{8\,a\,c^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.38, size = 75, normalized size = 1.03 \[ a^{4} \left (\frac {- 7 a x - 6}{4 a^{7} c^{2} x^{2} + 8 a^{6} c^{2} x + 4 a^{5} c^{2}} + \frac {x}{a^{4} c^{2}} + \frac {\frac {\log {\left (x - \frac {1}{a} \right )}}{8} - \frac {17 \log {\left (x + \frac {1}{a} \right )}}{8}}{a^{5} c^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________