Optimal. Leaf size=12 \[ \frac {1}{4} \tanh ^{-1}\left (\frac {e^x}{4}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2281, 212}
\begin {gather*} \frac {1}{4} \tanh ^{-1}\left (\frac {e^x}{4}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2281
Rubi steps
\begin {align*} \int \frac {e^x}{16-e^{2 x}} \, dx &=\text {Subst}\left (\int \frac {1}{16-x^2} \, dx,x,e^x\right )\\ &=\frac {1}{4} \tanh ^{-1}\left (\frac {e^x}{4}\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 12, normalized size = 1.00 \begin {gather*} \frac {1}{4} \tanh ^{-1}\left (\frac {e^x}{4}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(15\) vs.
\(2(7)=14\).
time = 0.02, size = 16, normalized size = 1.33
method | result | size |
default | \(\frac {\ln \left ({\mathrm e}^{x}+4\right )}{8}-\frac {\ln \left ({\mathrm e}^{x}-4\right )}{8}\) | \(16\) |
norman | \(\frac {\ln \left ({\mathrm e}^{x}+4\right )}{8}-\frac {\ln \left ({\mathrm e}^{x}-4\right )}{8}\) | \(16\) |
risch | \(\frac {\ln \left ({\mathrm e}^{x}+4\right )}{8}-\frac {\ln \left ({\mathrm e}^{x}-4\right )}{8}\) | \(16\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 15 vs.
\(2 (7) = 14\).
time = 0.28, size = 15, normalized size = 1.25 \begin {gather*} \frac {1}{8} \, \log \left (e^{x} + 4\right ) - \frac {1}{8} \, \log \left (e^{x} - 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 15 vs.
\(2 (7) = 14\).
time = 0.37, size = 15, normalized size = 1.25 \begin {gather*} \frac {1}{8} \, \log \left (e^{x} + 4\right ) - \frac {1}{8} \, \log \left (e^{x} - 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 15 vs.
\(2 (7) = 14\).
time = 0.03, size = 15, normalized size = 1.25 \begin {gather*} - \frac {\log {\left (e^{x} - 4 \right )}}{8} + \frac {\log {\left (e^{x} + 4 \right )}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 16 vs.
\(2 (7) = 14\).
time = 2.64, size = 16, normalized size = 1.33 \begin {gather*} \frac {1}{8} \, \log \left (e^{x} + 4\right ) - \frac {1}{8} \, \log \left ({\left | e^{x} - 4 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.65, size = 15, normalized size = 1.25 \begin {gather*} \frac {\ln \left ({\mathrm {e}}^x+4\right )}{8}-\frac {\ln \left ({\mathrm {e}}^x-4\right )}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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