Optimal. Leaf size=51 \[ \frac {1}{4 a^2 c^2 (1-a x)^2}-\frac {1}{4 a^2 c^2 (1-a x)}-\frac {\tanh ^{-1}(a x)}{4 a^2 c^2} \]
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Rubi [A]
time = 0.05, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6285, 78, 213}
\begin {gather*} -\frac {1}{4 a^2 c^2 (1-a x)}+\frac {1}{4 a^2 c^2 (1-a x)^2}-\frac {\tanh ^{-1}(a x)}{4 a^2 c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 213
Rule 6285
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)} x}{\left (c-a^2 c x^2\right )^2} \, dx &=\frac {\int \frac {x}{(1-a x)^3 (1+a x)} \, dx}{c^2}\\ &=\frac {\int \left (-\frac {1}{2 a (-1+a x)^3}-\frac {1}{4 a (-1+a x)^2}+\frac {1}{4 a \left (-1+a^2 x^2\right )}\right ) \, dx}{c^2}\\ &=\frac {1}{4 a^2 c^2 (1-a x)^2}-\frac {1}{4 a^2 c^2 (1-a x)}+\frac {\int \frac {1}{-1+a^2 x^2} \, dx}{4 a c^2}\\ &=\frac {1}{4 a^2 c^2 (1-a x)^2}-\frac {1}{4 a^2 c^2 (1-a x)}-\frac {\tanh ^{-1}(a x)}{4 a^2 c^2}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 34, normalized size = 0.67 \begin {gather*} \frac {a x-(-1+a x)^2 \tanh ^{-1}(a x)}{4 a^2 c^2 (-1+a x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 52, normalized size = 1.02
method | result | size |
risch | \(\frac {x}{4 a \,c^{2} \left (a x -1\right )^{2}}+\frac {\ln \left (-a x +1\right )}{8 a^{2} c^{2}}-\frac {\ln \left (a x +1\right )}{8 a^{2} c^{2}}\) | \(47\) |
default | \(\frac {-\frac {\ln \left (a x +1\right )}{8 a^{2}}+\frac {1}{4 a^{2} \left (a x -1\right )^{2}}+\frac {1}{4 a^{2} \left (a x -1\right )}+\frac {\ln \left (a x -1\right )}{8 a^{2}}}{c^{2}}\) | \(52\) |
norman | \(\frac {\frac {x}{4 a c}+\frac {a \,x^{3}}{4 c}+\frac {x^{2}}{2 c}}{\left (a^{2} x^{2}-1\right )^{2} c}+\frac {\ln \left (a x -1\right )}{8 a^{2} c^{2}}-\frac {\ln \left (a x +1\right )}{8 a^{2} c^{2}}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 59, normalized size = 1.16 \begin {gather*} \frac {x}{4 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}} - \frac {\log \left (a x + 1\right )}{8 \, a^{2} c^{2}} + \frac {\log \left (a x - 1\right )}{8 \, a^{2} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 77, normalized size = 1.51 \begin {gather*} \frac {2 \, a x - {\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 )}{8 \, {\left (a^{4} c^{2} x^{2} - 2 \, a^{3} c^{2} x + a^{2} c^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.14, size = 53, normalized size = 1.04 \begin {gather*} \frac {x}{4 a^{3} c^{2} x^{2} - 8 a^{2} c^{2} x + 4 a c^{2}} - \frac {- \frac {\log {\left (x - \frac {1}{a} \right )}}{8} + \frac {\log {\left (x + \frac {1}{a} \right )}}{8}}{a^{2} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 47, normalized size = 0.92 \begin {gather*} -\frac {\log \left ({\left | a x + 1 \right |}\right )}{8 \, a^{2} c^{2}} + \frac {\log \left ({\left | a x - 1 \right |}\right )}{8 \, a^{2} c^{2}} + \frac {x}{4 \, {\left (a x - 1\right )}^{2} a c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.92, size = 42, normalized size = 0.82 \begin {gather*} \frac {x}{4\,a\,\left (a^2\,c^2\,x^2-2\,a\,c^2\,x+c^2\right )}-\frac {\mathrm {atanh}\left (a\,x\right )}{4\,a^2\,c^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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