Optimal. Leaf size=78 \[ -\frac {1}{c^2 x}+\frac {a}{4 c^2 (1-a x)^2}+\frac {5 a}{4 c^2 (1-a x)}+\frac {2 a \log (x)}{c^2}-\frac {17 a \log (1-a x)}{8 c^2}+\frac {a \log (1+a x)}{8 c^2} \]
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Rubi [A]
time = 0.08, antiderivative size = 78, 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 = {6285, 90}
\begin {gather*} \frac {5 a}{4 c^2 (1-a x)}+\frac {a}{4 c^2 (1-a x)^2}+\frac {2 a \log (x)}{c^2}-\frac {17 a \log (1-a x)}{8 c^2}+\frac {a \log (a x+1)}{8 c^2}-\frac {1}{c^2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 6285
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)}}{x^2 \left (c-a^2 c x^2\right )^2} \, dx &=\frac {\int \frac {1}{x^2 (1-a x)^3 (1+a x)} \, dx}{c^2}\\ &=\frac {\int \left (\frac {1}{x^2}+\frac {2 a}{x}-\frac {a^2}{2 (-1+a x)^3}+\frac {5 a^2}{4 (-1+a x)^2}-\frac {17 a^2}{8 (-1+a x)}+\frac {a^2}{8 (1+a x)}\right ) \, dx}{c^2}\\ &=-\frac {1}{c^2 x}+\frac {a}{4 c^2 (1-a x)^2}+\frac {5 a}{4 c^2 (1-a x)}+\frac {2 a \log (x)}{c^2}-\frac {17 a \log (1-a x)}{8 c^2}+\frac {a \log (1+a x)}{8 c^2}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 57, normalized size = 0.73 \begin {gather*} \frac {-\frac {8}{x}+\frac {10 a}{1-a x}+\frac {2 a}{(-1+a x)^2}+16 a \log (x)-17 a \log (1-a x)+a \log (1+a x)}{8 c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 54, normalized size = 0.69
method | result | size |
default | \(\frac {\frac {a \ln \left (a x +1\right )}{8}-\frac {1}{x}+2 a \ln \left (x \right )+\frac {a}{4 \left (a x -1\right )^{2}}-\frac {5 a}{4 \left (a x -1\right )}-\frac {17 a \ln \left (a x -1\right )}{8}}{c^{2}}\) | \(54\) |
risch | \(\frac {-\frac {9}{4} a^{2} x^{2}+\frac {7}{2} a x -1}{c^{2} x \left (a x -1\right )^{2}}+\frac {a \ln \left (a x +1\right )}{8 c^{2}}+\frac {2 a \ln \left (-x \right )}{c^{2}}-\frac {17 a \ln \left (-a x +1\right )}{8 c^{2}}\) | \(65\) |
norman | \(\frac {-\frac {1}{c}+\frac {15 a^{2} x^{2}}{4 c}-\frac {9 a^{4} x^{4}}{4 c}+\frac {2 a^{3} x^{3}}{c}-\frac {3 a^{5} x^{5}}{2 c}}{\left (a^{2} x^{2}-1\right )^{2} x c}+\frac {2 a \ln \left (x \right )}{c^{2}}-\frac {17 a \ln \left (a x -1\right )}{8 c^{2}}+\frac {a \ln \left (a x +1\right )}{8 c^{2}}\) | \(102\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 76, normalized size = 0.97 \begin {gather*} -\frac {9 \, a^{2} x^{2} - 14 \, a x + 4}{4 \, {\left (a^{2} c^{2} x^{3} - 2 \, a c^{2} x^{2} + c^{2} x\right )}} + \frac {a \log \left (a x + 1\right )}{8 \, c^{2}} - \frac {17 \, a \log \left (a x - 1\right )}{8 \, c^{2}} + \frac {2 \, a \log \left (x\right )}{c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 120, normalized size = 1.54 \begin {gather*} -\frac {18 \, a^{2} x^{2} - 28 \, a x - {\left (a^{3} x^{3} - 2 \, a^{2} x^{2} + a x\right )} \log \left (a x + 1\right ) + 17 \, {\left (a^{3} x^{3} - 2 \, a^{2} x^{2} + a x\right )} \log \left (a x - 1\right ) - 16 \, {\left (a^{3} x^{3} - 2 \, a^{2} x^{2} + a x\right )} \log \left (x\right ) + 8}{8 \, {\left (a^{2} c^{2} x^{3} - 2 \, a c^{2} x^{2} + c^{2} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.32, size = 76, normalized size = 0.97 \begin {gather*} - \frac {9 a^{2} x^{2} - 14 a x + 4}{4 a^{2} c^{2} x^{3} - 8 a c^{2} x^{2} + 4 c^{2} x} - \frac {- 2 a \log {\left (x \right )} + \frac {17 a \log {\left (x - \frac {1}{a} \right )}}{8} - \frac {a \log {\left (x + \frac {1}{a} \right )}}{8}}{c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 65, normalized size = 0.83 \begin {gather*} \frac {a \log \left ({\left | a x + 1 \right |}\right )}{8 \, c^{2}} - \frac {17 \, a \log \left ({\left | a x - 1 \right |}\right )}{8 \, c^{2}} + \frac {2 \, a \log \left ({\left | x \right |}\right )}{c^{2}} - \frac {9 \, a^{2} x^{2} - 14 \, a x + 4}{4 \, {\left (a x - 1\right )}^{2} c^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 76, normalized size = 0.97 \begin {gather*} \frac {2\,a\,\ln \left (x\right )}{c^2}-\frac {\frac {9\,a^2\,x^2}{4}-\frac {7\,a\,x}{2}+1}{a^2\,c^2\,x^3-2\,a\,c^2\,x^2+c^2\,x}-\frac {17\,a\,\ln \left (a\,x-1\right )}{8\,c^2}+\frac {a\,\ln \left (a\,x+1\right )}{8\,c^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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