Optimal. Leaf size=29 \[ 5-\log ^2\left (\frac {4}{\left (e^x+e^{1+x+x^2}\right )^2-x}\right ) \]
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Rubi [A]
time = 0.45, antiderivative size = 42, normalized size of antiderivative = 1.45, number of steps
used = 1, number of rules used = 2, integrand size = 117, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {6816, 6818}
\begin {gather*} -\log ^2\left (\frac {4}{2 e^{x^2+2 x+1}+e^{2 x^2+2 x+2}-x+e^{2 x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 6816
Rule 6818
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=-\log ^2\left (\frac {4}{e^{2 x}+2 e^{1+2 x+x^2}+e^{2+2 x+2 x^2}-x}\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.07, size = 37, normalized size = 1.28 \begin {gather*} -\log ^2\left (\frac {4}{e^{2 x}+2 e^{(1+x)^2}+e^{2 \left (1+x+x^2\right )}-x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.95, size = 391, normalized size = 13.48
method | result | size |
risch | \(-\ln \left (-{\mathrm e}^{2 x^{2}+2 x +2}-2 \,{\mathrm e}^{\left (x +1\right )^{2}}-{\mathrm e}^{2 x}+x \right )^{2}+4 \ln \left (-{\mathrm e}^{2 x^{2}+2 x +2}-2 \,{\mathrm e}^{\left (x +1\right )^{2}}-{\mathrm e}^{2 x}+x \right )+2 i \pi \ln \left ({\mathrm e}^{2 x^{2}+2 x +2}+2 \,{\mathrm e}^{\left (x +1\right )^{2}}+{\mathrm e}^{2 x}-x \right )-2 i \pi \ln \left ({\mathrm e}^{2 x^{2}+2 x +2}+2 \,{\mathrm e}^{\left (x +1\right )^{2}}+{\mathrm e}^{2 x}-x \right ) \mathrm {csgn}\left (\frac {i}{-{\mathrm e}^{2 x^{2}+2 x +2}-2 \,{\mathrm e}^{\left (x +1\right )^{2}}-{\mathrm e}^{2 x}+x}\right )^{2}-4 i \pi \mathrm {csgn}\left (\frac {i}{-{\mathrm e}^{2 x^{2}+2 x +2}-2 \,{\mathrm e}^{\left (x +1\right )^{2}}-{\mathrm e}^{2 x}+x}\right )^{3}+4 i \pi \mathrm {csgn}\left (\frac {i}{-{\mathrm e}^{2 x^{2}+2 x +2}-2 \,{\mathrm e}^{\left (x +1\right )^{2}}-{\mathrm e}^{2 x}+x}\right )^{2}-4 i \pi +2 i \pi \ln \left ({\mathrm e}^{2 x^{2}+2 x +2}+2 \,{\mathrm e}^{\left (x +1\right )^{2}}+{\mathrm e}^{2 x}-x \right ) \mathrm {csgn}\left (\frac {i}{-{\mathrm e}^{2 x^{2}+2 x +2}-2 \,{\mathrm e}^{\left (x +1\right )^{2}}-{\mathrm e}^{2 x}+x}\right )^{3}+8+4 \ln \left (2\right ) \ln \left ({\mathrm e}^{2 x^{2}+2 x +2}+2 \,{\mathrm e}^{\left (x +1\right )^{2}}+{\mathrm e}^{2 x}-x \right )-8 \ln \left (2\right )-4 \ln \left ({\mathrm e}^{2 x^{2}+2 x +2}+2 \,{\mathrm e}^{\left (x +1\right )^{2}}+{\mathrm e}^{2 x}-x \right )\) | \(391\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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. 72 vs.
\(2 (27) = 54\).
time = 0.36, size = 72, normalized size = 2.48 \begin {gather*} -\log \left (-\frac {4 \, e^{\left (2 \, x^{2} + 2 \, x + 2\right )}}{{\left (x - 2 \, e^{\left (x^{2} + 2 \, x + 1\right )}\right )} e^{\left (2 \, x^{2} + 2 \, x + 2\right )} - e^{\left (4 \, x^{2} + 4 \, x + 4\right )} - e^{\left (2 \, x^{2} + 4 \, x + 2\right )}}\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.51, size = 37, normalized size = 1.28 \begin {gather*} - \log {\left (\frac {4}{- x + e^{2 x} + 2 e^{x} e^{x^{2} + x + 1} + e^{2 x^{2} + 2 x + 2}} \right )}^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.43, size = 42, normalized size = 1.45 \begin {gather*} -{\ln \left (\frac {4}{{\mathrm {e}}^{2\,x}-x+2\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{x^2}\,\mathrm {e}+{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^2\,{\mathrm {e}}^{2\,x^2}}\right )}^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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