Optimal. Leaf size=75 \[ \frac {7}{4 a c^2 (1-a x)}-\frac {1}{4 a c^2 (1-a x)^2}+\frac {17 \log (1-a x)}{8 a c^2}-\frac {\log (a x+1)}{8 a c^2}+\frac {x}{c^2} \]
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Rubi [A] time = 0.18, antiderivative size = 75, 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} \[ \frac {7}{4 a c^2 (1-a x)}-\frac {1}{4 a c^2 (1-a x)^2}+\frac {17 \log (1-a x)}{8 a c^2}-\frac {\log (a x+1)}{8 a c^2}+\frac {x}{c^2} \]
Antiderivative was successfully verified.
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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)^3 (1+a x)} \, dx}{c^2}\\ &=-\frac {a^4 \int \left (-\frac {1}{a^4}-\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)}+\frac {1}{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 {17 \log (1-a x)}{8 a c^2}-\frac {\log (1+a x)}{8 a c^2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 75, normalized size = 1.00 \[ \frac {7}{4 a c^2 (1-a x)}-\frac {1}{4 a c^2 (1-a x)^2}+\frac {17 \log (1-a x)}{8 a c^2}-\frac {\log (a x+1)}{8 a c^2}+\frac {x}{c^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 93, normalized size = 1.24 \[ \frac {8 \, a^{3} x^{3} - 16 \, a^{2} x^{2} - 6 \, a x - {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x + 1\right ) + 17 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x - 1\right ) + 12}{8 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 57, normalized size = 0.76 \[ \frac {x}{c^{2}} - \frac {\log \left ({\left | a x + 1 \right |}\right )}{8 \, a c^{2}} + \frac {17 \, \log \left ({\left | a x - 1 \right |}\right )}{8 \, a c^{2}} - \frac {7 \, a x - 6}{4 \, {\left (a x - 1\right )}^{2} a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 65, normalized size = 0.87 \[ \frac {x}{c^{2}}-\frac {1}{4 a \,c^{2} \left (a x -1\right )^{2}}-\frac {7}{4 a \,c^{2} \left (a x -1\right )}+\frac {17 \ln \left (a x -1\right )}{8 a \,c^{2}}-\frac {\ln \left (a x +1\right )}{8 a \,c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 69, normalized size = 0.92 \[ -\frac {7 \, a x - 6}{4 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}} + \frac {x}{c^{2}} - \frac {\log \left (a x + 1\right )}{8 \, a c^{2}} + \frac {17 \, \log \left (a x - 1\right )}{8 \, a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 68, normalized size = 0.91 \[ \frac {x}{c^2}-\frac {\frac {7\,x}{4}-\frac {3}{2\,a}}{a^2\,c^2\,x^2-2\,a\,c^2\,x+c^2}+\frac {17\,\ln \left (a\,x-1\right )}{8\,a\,c^2}-\frac {\ln \left (a\,x+1\right )}{8\,a\,c^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.37, size = 73, normalized size = 0.97 \[ a^{4} \left (\frac {- 7 a x + 6}{4 a^{7} c^{2} x^{2} - 8 a^{6} c^{2} x + 4 a^{5} c^{2}} + \frac {x}{a^{4} c^{2}} + \frac {\frac {17 \log {\left (x - \frac {1}{a} \right )}}{8} - \frac {\log {\left (x + \frac {1}{a} \right )}}{8}}{a^{5} c^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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