Optimal. Leaf size=22 \[ \frac {x}{\frac {13}{3}-e^2 x-\log \left (4+e^2\right )} \]
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Rubi [A]
time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.45, number of steps
used = 4, number of rules used = 4, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {12, 2006, 27,
32} \begin {gather*} \frac {13-3 \log \left (4+e^2\right )}{e^2 \left (-3 e^2 x+13-3 \log \left (4+e^2\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 32
Rule 2006
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\left (3 \left (13-3 \log \left (4+e^2\right )\right )\right ) \int \frac {1}{169-78 e^2 x+9 e^4 x^2+\left (-78+18 e^2 x\right ) \log \left (4+e^2\right )+9 \log ^2\left (4+e^2\right )} \, dx\\ &=\left (3 \left (13-3 \log \left (4+e^2\right )\right )\right ) \int \frac {1}{9 e^4 x^2-6 e^2 x \left (13-3 \log \left (4+e^2\right )\right )+\left (13-3 \log \left (4+e^2\right )\right )^2} \, dx\\ &=\left (3 \left (13-3 \log \left (4+e^2\right )\right )\right ) \int \frac {1}{\left (-13+3 e^2 x+3 \log \left (4+e^2\right )\right )^2} \, dx\\ &=\frac {13-3 \log \left (4+e^2\right )}{e^2 \left (13-3 e^2 x-3 \log \left (4+e^2\right )\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 35, normalized size = 1.59 \begin {gather*} -\frac {39-9 \log \left (4+e^2\right )}{3 e^2 \left (-13+3 e^2 x+3 \log \left (4+e^2\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.56, size = 31, normalized size = 1.41
method | result | size |
gosper | \(\frac {\left (3 \ln \left (4+{\mathrm e}^{2}\right )-13\right ) {\mathrm e}^{-2}}{3 \,{\mathrm e}^{2} x +3 \ln \left (4+{\mathrm e}^{2}\right )-13}\) | \(31\) |
norman | \(\frac {\left (3 \ln \left (4+{\mathrm e}^{2}\right )-13\right ) {\mathrm e}^{-2}}{3 \,{\mathrm e}^{2} x +3 \ln \left (4+{\mathrm e}^{2}\right )-13}\) | \(31\) |
risch | \(\frac {{\mathrm e}^{-2} \ln \left (4+{\mathrm e}^{2}\right )}{{\mathrm e}^{2} x +\ln \left (4+{\mathrm e}^{2}\right )-\frac {13}{3}}-\frac {13 \,{\mathrm e}^{-2}}{3 \left ({\mathrm e}^{2} x +\ln \left (4+{\mathrm e}^{2}\right )-\frac {13}{3}\right )}\) | \(40\) |
meijerg | \(\frac {9 \ln \left (4+{\mathrm e}^{2}\right ) \left (-3 \ln \left (4+{\mathrm e}^{2}\right )+13\right ) x}{\left (3 \ln \left (4+{\mathrm e}^{2}\right )-13\right )^{3} \left (1+\frac {3 x \,{\mathrm e}^{2}}{3 \ln \left (4+{\mathrm e}^{2}\right )-13}\right )}-\frac {39 \left (-3 \ln \left (4+{\mathrm e}^{2}\right )+13\right ) x}{\left (3 \ln \left (4+{\mathrm e}^{2}\right )-13\right )^{3} \left (1+\frac {3 x \,{\mathrm e}^{2}}{3 \ln \left (4+{\mathrm e}^{2}\right )-13}\right )}\) | \(93\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 31, normalized size = 1.41 \begin {gather*} \frac {3 \, \log \left (e^{2} + 4\right ) - 13}{3 \, x e^{4} + 3 \, e^{2} \log \left (e^{2} + 4\right ) - 13 \, e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.51, size = 31, normalized size = 1.41 \begin {gather*} \frac {3 \, \log \left (e^{2} + 4\right ) - 13}{3 \, x e^{4} + 3 \, e^{2} \log \left (e^{2} + 4\right ) - 13 \, e^{2}} \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. 34 vs.
\(2 (15) = 30\).
time = 0.13, size = 34, normalized size = 1.55 \begin {gather*} - \frac {39 - 9 \log {\left (4 + e^{2} \right )}}{9 x e^{4} - 39 e^{2} + 9 e^{2} \log {\left (4 + e^{2} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 28, normalized size = 1.27 \begin {gather*} \frac {{\left (3 \, \log \left (e^{2} + 4\right ) - 13\right )} e^{\left (-2\right )}}{3 \, x e^{2} + 3 \, \log \left (e^{2} + 4\right ) - 13} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.18, size = 28, normalized size = 1.27 \begin {gather*} \frac {{\mathrm {e}}^{-2}\,\left (3\,\ln \left ({\mathrm {e}}^2+4\right )-13\right )}{3\,\ln \left ({\mathrm {e}}^2+4\right )+3\,x\,{\mathrm {e}}^2-13} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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