Optimal. Leaf size=99 \[ \frac {7 a^2}{4 c^2 (1-a x)}+\frac {a^2}{4 c^2 (1-a x)^2}+\frac {4 a^2 \log (x)}{c^2}-\frac {31 a^2 \log (1-a x)}{8 c^2}-\frac {a^2 \log (a x+1)}{8 c^2}-\frac {2 a}{c^2 x}-\frac {1}{2 c^2 x^2} \]
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Rubi [A] time = 0.12, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6150, 88} \[ \frac {7 a^2}{4 c^2 (1-a x)}+\frac {a^2}{4 c^2 (1-a x)^2}+\frac {4 a^2 \log (x)}{c^2}-\frac {31 a^2 \log (1-a x)}{8 c^2}-\frac {a^2 \log (a x+1)}{8 c^2}-\frac {2 a}{c^2 x}-\frac {1}{2 c^2 x^2} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6150
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)}}{x^3 \left (c-a^2 c x^2\right )^2} \, dx &=\frac {\int \frac {1}{x^3 (1-a x)^3 (1+a x)} \, dx}{c^2}\\ &=\frac {\int \left (\frac {1}{x^3}+\frac {2 a}{x^2}+\frac {4 a^2}{x}-\frac {a^3}{2 (-1+a x)^3}+\frac {7 a^3}{4 (-1+a x)^2}-\frac {31 a^3}{8 (-1+a x)}-\frac {a^3}{8 (1+a x)}\right ) \, dx}{c^2}\\ &=-\frac {1}{2 c^2 x^2}-\frac {2 a}{c^2 x}+\frac {a^2}{4 c^2 (1-a x)^2}+\frac {7 a^2}{4 c^2 (1-a x)}+\frac {4 a^2 \log (x)}{c^2}-\frac {31 a^2 \log (1-a x)}{8 c^2}-\frac {a^2 \log (1+a x)}{8 c^2}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 72, normalized size = 0.73 \[ -\frac {\frac {14 a^2}{a x-1}-\frac {2 a^2}{(a x-1)^2}-32 a^2 \log (x)+31 a^2 \log (1-a x)+a^2 \log (a x+1)+\frac {16 a}{x}+\frac {4}{x^2}}{8 c^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 141, normalized size = 1.42 \[ -\frac {30 \, a^{3} x^{3} - 44 \, a^{2} x^{2} + 8 \, a x + {\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + a^{2} x^{2}\right )} \log \left (a x + 1\right ) + 31 \, {\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + a^{2} x^{2}\right )} \log \left (a x - 1\right ) - 32 \, {\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + a^{2} x^{2}\right )} \log \relax (x) + 4}{8 \, {\left (a^{2} c^{2} x^{4} - 2 \, a c^{2} x^{3} + c^{2} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.09, size = 79, normalized size = 0.80 \[ -\frac {a^{2} \log \left ({\left | a x + 1 \right |}\right )}{8 \, c^{2}} - \frac {31 \, a^{2} \log \left ({\left | a x - 1 \right |}\right )}{8 \, c^{2}} + \frac {4 \, a^{2} \log \left ({\left | x \right |}\right )}{c^{2}} - \frac {15 \, a^{3} x^{3} - 22 \, a^{2} x^{2} + 4 \, a x + 2}{4 \, {\left (a x - 1\right )}^{2} c^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 87, normalized size = 0.88 \[ -\frac {1}{2 c^{2} x^{2}}-\frac {2 a}{c^{2} x}+\frac {4 a^{2} \ln \relax (x )}{c^{2}}+\frac {a^{2}}{4 c^{2} \left (a x -1\right )^{2}}-\frac {7 a^{2}}{4 c^{2} \left (a x -1\right )}-\frac {31 a^{2} \ln \left (a x -1\right )}{8 c^{2}}-\frac {a^{2} \ln \left (a x +1\right )}{8 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 92, normalized size = 0.93 \[ -\frac {a^{2} \log \left (a x + 1\right )}{8 \, c^{2}} - \frac {31 \, a^{2} \log \left (a x - 1\right )}{8 \, c^{2}} + \frac {4 \, a^{2} \log \relax (x)}{c^{2}} - \frac {15 \, a^{3} x^{3} - 22 \, a^{2} x^{2} + 4 \, a x + 2}{4 \, {\left (a^{2} c^{2} x^{4} - 2 \, a c^{2} x^{3} + c^{2} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 91, normalized size = 0.92 \[ \frac {4\,a^2\,\ln \relax (x)}{c^2}-\frac {\frac {15\,a^3\,x^3}{4}-\frac {11\,a^2\,x^2}{2}+a\,x+\frac {1}{2}}{a^2\,c^2\,x^4-2\,a\,c^2\,x^3+c^2\,x^2}-\frac {31\,a^2\,\ln \left (a\,x-1\right )}{8\,c^2}-\frac {a^2\,\ln \left (a\,x+1\right )}{8\,c^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.61, size = 92, normalized size = 0.93 \[ - \frac {15 a^{3} x^{3} - 22 a^{2} x^{2} + 4 a x + 2}{4 a^{2} c^{2} x^{4} - 8 a c^{2} x^{3} + 4 c^{2} x^{2}} - \frac {- 4 a^{2} \log {\relax (x )} + \frac {31 a^{2} \log {\left (x - \frac {1}{a} \right )}}{8} + \frac {a^{2} \log {\left (x + \frac {1}{a} \right )}}{8}}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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