Optimal. Leaf size=49 \[ \frac {1}{4 a c^2 (a x+1)}+\frac {1}{4 a c^2 (a x+1)^2}-\frac {\tanh ^{-1}(a x)}{4 a c^2} \]
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Rubi [A] time = 0.08, antiderivative size = 49, 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, 6140, 44, 207} \[ \frac {1}{4 a c^2 (a x+1)}+\frac {1}{4 a c^2 (a x+1)^2}-\frac {\tanh ^{-1}(a x)}{4 a c^2} \]
Antiderivative was successfully verified.
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Rule 44
Rule 207
Rule 6140
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx &=-\int \frac {e^{-2 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx\\ &=-\frac {\int \frac {1}{(1-a x) (1+a x)^3} \, dx}{c^2}\\ &=-\frac {\int \left (\frac {1}{2 (1+a x)^3}+\frac {1}{4 (1+a x)^2}-\frac {1}{4 \left (-1+a^2 x^2\right )}\right ) \, dx}{c^2}\\ &=\frac {1}{4 a c^2 (1+a x)^2}+\frac {1}{4 a c^2 (1+a x)}+\frac {\int \frac {1}{-1+a^2 x^2} \, dx}{4 c^2}\\ &=\frac {1}{4 a c^2 (1+a x)^2}+\frac {1}{4 a c^2 (1+a x)}-\frac {\tanh ^{-1}(a x)}{4 a c^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 33, normalized size = 0.67 \[ \frac {a x+(a x+1)^2 \left (-\tanh ^{-1}(a x)\right )+2}{4 a (a c x+c)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 76, normalized size = 1.55 \[ \frac {2 \, a x - {\left (a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (a x + 1\right ) + {\left (a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (a x - 1\right ) + 4}{8 \, {\left (a^{3} c^{2} x^{2} + 2 \, a^{2} c^{2} x + a c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 51, normalized size = 1.04 \[ -\frac {\log \left ({\left | a x + 1 \right |}\right )}{8 \, a c^{2}} + \frac {\log \left ({\left | a x - 1 \right |}\right )}{8 \, a c^{2}} + \frac {a x + 2}{4 \, {\left (a x + 1\right )}^{2} a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 60, normalized size = 1.22 \[ \frac {\ln \left (a x -1\right )}{8 c^{2} a}+\frac {1}{4 a \,c^{2} \left (a x +1\right )^{2}}+\frac {1}{4 a \,c^{2} \left (a x +1\right )}-\frac {\ln \left (a x +1\right )}{8 a \,c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 63, normalized size = 1.29 \[ \frac {a x + 2}{4 \, {\left (a^{3} c^{2} x^{2} + 2 \, a^{2} c^{2} x + a c^{2}\right )}} - \frac {\log \left (a x + 1\right )}{8 \, a c^{2}} + \frac {\log \left (a x - 1\right )}{8 \, a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 46, normalized size = 0.94 \[ \frac {\frac {x}{4}+\frac {1}{2\,a}}{a^2\,c^2\,x^2+2\,a\,c^2\,x+c^2}-\frac {\mathrm {atanh}\left (a\,x\right )}{4\,a\,c^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.28, size = 54, normalized size = 1.10 \[ \frac {a x + 2}{4 a^{3} c^{2} x^{2} + 8 a^{2} c^{2} x + 4 a c^{2}} + \frac {\frac {\log {\left (x - \frac {1}{a} \right )}}{8} - \frac {\log {\left (x + \frac {1}{a} \right )}}{8}}{a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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