Optimal. Leaf size=17 \[ \left (x-\log \left (\log ^2\left (-e^{2+x}\right )\right )\right )^2 \]
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Rubi [A]
time = 0.08, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.060, Rules used = {6820, 12, 6818}
\begin {gather*} \left (x-\log \left (\log ^2\left (-e^{x+2}\right )\right )\right )^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 6818
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (2-\log \left (-e^{2+x}\right )\right ) \left (-x+\log \left (\log ^2\left (-e^{2+x}\right )\right )\right )}{\log \left (-e^{2+x}\right )} \, dx\\ &=2 \int \frac {\left (2-\log \left (-e^{2+x}\right )\right ) \left (-x+\log \left (\log ^2\left (-e^{2+x}\right )\right )\right )}{\log \left (-e^{2+x}\right )} \, dx\\ &=\left (x-\log \left (\log ^2\left (-e^{2+x}\right )\right )\right )^2\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(74\) vs. \(2(17)=34\).
time = 0.05, size = 74, normalized size = 4.35 \begin {gather*} -4 x+x^2+4 \log \left (-e^{2+x}\right )+4 \left (-x+\log \left (-e^{2+x}\right )\right ) \log \left (\log \left (-e^{2+x}\right )\right )-2 \log \left (-e^{2+x}\right ) \log \left (\log ^2\left (-e^{2+x}\right )\right )+\log ^2\left (\log ^2\left (-e^{2+x}\right )\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. \(99\) vs.
\(2(16)=32\).
time = 7.46, size = 100, normalized size = 5.88
method | result | size |
default | \(-2 \ln \left (\ln \left (-{\mathrm e}^{2+x}\right )^{2}\right ) \ln \left (-{\mathrm e}^{2+x}\right )+4 \ln \left (-{\mathrm e}^{2+x}\right )+4 \ln \left (\ln \left (-{\mathrm e}^{2+x}\right )\right ) \ln \left (\ln \left (-{\mathrm e}^{2+x}\right )^{2}\right )-4 \ln \left (\ln \left (-{\mathrm e}^{2+x}\right )\right )^{2}+x^{2}-4 x +4 \ln \left (\ln \left (-{\mathrm e}^{2+x}\right )\right ) \left (\ln \left (-{\mathrm e}^{2+x}\right )-x -2\right )+8 \ln \left (\ln \left (-{\mathrm e}^{2+x}\right )\right )\) | \(100\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 27, normalized size = 1.59 \begin {gather*} x^{2} - 4 \, x \log \left (\log \left (-e^{x}\right ) + 2\right ) + 4 \, \log \left (\log \left (-e^{x}\right ) + 2\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 0.40, size = 49, normalized size = 2.88 \begin {gather*} x^{2} - 2 \, x \log \left (-\pi ^{2} + 2 i \, \pi {\left (x + 2\right )} + x^{2} + 4 \, x + 4\right ) + \log \left (-\pi ^{2} + 2 i \, \pi {\left (x + 2\right )} + x^{2} + 4 \, x + 4\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 0.41, size = 51, normalized size = 3.00 \begin {gather*} x^{2} - 2 \, x \log \left (4 i \, \pi - \pi ^{2} + 2 i \, \pi x + x^{2} + 4 \, x + 4\right ) + \log \left (4 i \, \pi - \pi ^{2} + 2 i \, \pi x + x^{2} + 4 \, x + 4\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.83, size = 16, normalized size = 0.94 \begin {gather*} {\left (x-\ln \left ({\left (x+2+\pi \,1{}\mathrm {i}\right )}^2\right )\right )}^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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