Optimal. Leaf size=14 \[ \frac {\tanh ^{-1}\left (\frac {e x}{d}\right )}{d e} \]
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Rubi [A] time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {208} \[ \frac {\tanh ^{-1}\left (\frac {e x}{d}\right )}{d e} \]
Antiderivative was successfully verified.
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Rule 208
Rubi steps
\begin {align*} \int \frac {1}{d^2-e^2 x^2} \, dx &=\frac {\tanh ^{-1}\left (\frac {e x}{d}\right )}{d e}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 14, normalized size = 1.00 \[ \frac {\tanh ^{-1}\left (\frac {e x}{d}\right )}{d e} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.38, size = 25, normalized size = 1.79 \[ \frac {\log \left (e x + d\right ) - \log \left (e x - d\right )}{2 \, d e} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.46, size = 38, normalized size = 2.71 \[ -\frac {e^{\left (-1\right )} \log \left (\frac {{\left | 2 \, x e^{2} - 2 \, {\left | d \right |} e \right |}}{{\left | 2 \, x e^{2} + 2 \, {\left | d \right |} e \right |}}\right )}{2 \, {\left | d \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 32, normalized size = 2.29 \[ -\frac {\ln \left (e x -d \right )}{2 d e}+\frac {\ln \left (e x +d \right )}{2 d e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.96, size = 31, normalized size = 2.21 \[ \frac {\log \left (e x + d\right )}{2 \, d e} - \frac {\log \left (e x - d\right )}{2 \, d e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.43, size = 14, normalized size = 1.00 \[ \frac {\mathrm {atanh}\left (\frac {e\,x}{d}\right )}{d\,e} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.14, size = 20, normalized size = 1.43 \[ - \frac {\frac {\log {\left (- \frac {d}{e} + x \right )}}{2} - \frac {\log {\left (\frac {d}{e} + x \right )}}{2}}{d e} \]
Verification of antiderivative is not currently implemented for this CAS.
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