Optimal. Leaf size=109 \[ \frac {23 a}{16 c^3 (1-a x)}-\frac {a}{16 c^3 (a x+1)}+\frac {3 a}{8 c^3 (1-a x)^2}+\frac {a}{12 c^3 (1-a x)^3}+\frac {2 a \log (x)}{c^3}-\frac {9 a \log (1-a x)}{4 c^3}+\frac {a \log (a x+1)}{4 c^3}-\frac {1}{c^3 x} \]
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Rubi [A] time = 0.14, antiderivative size = 109, 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} \[ \frac {23 a}{16 c^3 (1-a x)}-\frac {a}{16 c^3 (a x+1)}+\frac {3 a}{8 c^3 (1-a x)^2}+\frac {a}{12 c^3 (1-a x)^3}+\frac {2 a \log (x)}{c^3}-\frac {9 a \log (1-a x)}{4 c^3}+\frac {a \log (a x+1)}{4 c^3}-\frac {1}{c^3 x} \]
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^2 \left (c-a^2 c x^2\right )^3} \, dx &=\frac {\int \frac {1}{x^2 (1-a x)^4 (1+a x)^2} \, dx}{c^3}\\ &=\frac {\int \left (\frac {1}{x^2}+\frac {2 a}{x}+\frac {a^2}{4 (-1+a x)^4}-\frac {3 a^2}{4 (-1+a x)^3}+\frac {23 a^2}{16 (-1+a x)^2}-\frac {9 a^2}{4 (-1+a x)}+\frac {a^2}{16 (1+a x)^2}+\frac {a^2}{4 (1+a x)}\right ) \, dx}{c^3}\\ &=-\frac {1}{c^3 x}+\frac {a}{12 c^3 (1-a x)^3}+\frac {3 a}{8 c^3 (1-a x)^2}+\frac {23 a}{16 c^3 (1-a x)}-\frac {a}{16 c^3 (1+a x)}+\frac {2 a \log (x)}{c^3}-\frac {9 a \log (1-a x)}{4 c^3}+\frac {a \log (1+a x)}{4 c^3}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 78, normalized size = 0.72 \[ \frac {\frac {69 a}{1-a x}-\frac {3 a}{a x+1}+\frac {18 a}{(a x-1)^2}-\frac {4 a}{(a x-1)^3}+96 a \log (x)-108 a \log (1-a x)+12 a \log (a x+1)-\frac {48}{x}}{48 c^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.09, size = 175, normalized size = 1.61 \[ -\frac {30 \, a^{4} x^{4} - 48 \, a^{3} x^{3} - 14 \, a^{2} x^{2} + 46 \, a x - 3 \, {\left (a^{5} x^{5} - 2 \, a^{4} x^{4} + 2 \, a^{2} x^{2} - a x\right )} \log \left (a x + 1\right ) + 27 \, {\left (a^{5} x^{5} - 2 \, a^{4} x^{4} + 2 \, a^{2} x^{2} - a x\right )} \log \left (a x - 1\right ) - 24 \, {\left (a^{5} x^{5} - 2 \, a^{4} x^{4} + 2 \, a^{2} x^{2} - a x\right )} \log \relax (x) - 12}{12 \, {\left (a^{4} c^{3} x^{5} - 2 \, a^{3} c^{3} x^{4} + 2 \, a c^{3} x^{2} - c^{3} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 88, normalized size = 0.81 \[ \frac {a \log \left ({\left | a x + 1 \right |}\right )}{4 \, c^{3}} - \frac {9 \, a \log \left ({\left | a x - 1 \right |}\right )}{4 \, c^{3}} + \frac {2 \, a \log \left ({\left | x \right |}\right )}{c^{3}} - \frac {15 \, a^{4} x^{4} - 24 \, a^{3} x^{3} - 7 \, a^{2} x^{2} + 23 \, a x - 6}{6 \, {\left (a x + 1\right )} {\left (a x - 1\right )}^{3} c^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 94, normalized size = 0.86 \[ -\frac {1}{c^{3} x}+\frac {2 a \ln \relax (x )}{c^{3}}-\frac {a}{12 c^{3} \left (a x -1\right )^{3}}+\frac {3 a}{8 c^{3} \left (a x -1\right )^{2}}-\frac {23 a}{16 c^{3} \left (a x -1\right )}-\frac {9 a \ln \left (a x -1\right )}{4 c^{3}}-\frac {a}{16 c^{3} \left (a x +1\right )}+\frac {a \ln \left (a x +1\right )}{4 c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 104, normalized size = 0.95 \[ -\frac {15 \, a^{4} x^{4} - 24 \, a^{3} x^{3} - 7 \, a^{2} x^{2} + 23 \, a x - 6}{6 \, {\left (a^{4} c^{3} x^{5} - 2 \, a^{3} c^{3} x^{4} + 2 \, a c^{3} x^{2} - c^{3} x\right )}} + \frac {a \log \left (a x + 1\right )}{4 \, c^{3}} - \frac {9 \, a \log \left (a x - 1\right )}{4 \, c^{3}} + \frac {2 \, a \log \relax (x)}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.99, size = 104, normalized size = 0.95 \[ \frac {2\,a\,\ln \relax (x)}{c^3}-\frac {-\frac {5\,a^4\,x^4}{2}+4\,a^3\,x^3+\frac {7\,a^2\,x^2}{6}-\frac {23\,a\,x}{6}+1}{-a^4\,c^3\,x^5+2\,a^3\,c^3\,x^4-2\,a\,c^3\,x^2+c^3\,x}-\frac {9\,a\,\ln \left (a\,x-1\right )}{4\,c^3}+\frac {a\,\ln \left (a\,x+1\right )}{4\,c^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.74, size = 104, normalized size = 0.95 \[ \frac {- 15 a^{4} x^{4} + 24 a^{3} x^{3} + 7 a^{2} x^{2} - 23 a x + 6}{6 a^{4} c^{3} x^{5} - 12 a^{3} c^{3} x^{4} + 12 a c^{3} x^{2} - 6 c^{3} x} + \frac {2 a \log {\relax (x )} - \frac {9 a \log {\left (x - \frac {1}{a} \right )}}{4} + \frac {a \log {\left (x + \frac {1}{a} \right )}}{4}}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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