Optimal. Leaf size=21 \[ 1+2 x+x \left (-1+5 \log \left (\frac {1}{\left (e^4+2 x\right )^2}\right )\right ) \]
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Rubi [A]
time = 0.08, antiderivative size = 36, normalized size of antiderivative = 1.71, number of steps
used = 7, number of rules used = 5, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6820, 6874, 45,
2436, 2332} \begin {gather*} x+\frac {5}{2} \left (2 x+e^4\right ) \log \left (\frac {1}{\left (2 x+e^4\right )^2}\right )+5 e^4 \log \left (2 x+e^4\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2332
Rule 2436
Rule 6820
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^4-18 x+5 \left (e^4+2 x\right ) \log \left (\frac {1}{\left (e^4+2 x\right )^2}\right )}{e^4+2 x} \, dx\\ &=\int \left (\frac {e^4-18 x}{e^4+2 x}+5 \log \left (\frac {1}{\left (e^4+2 x\right )^2}\right )\right ) \, dx\\ &=5 \int \log \left (\frac {1}{\left (e^4+2 x\right )^2}\right ) \, dx+\int \frac {e^4-18 x}{e^4+2 x} \, dx\\ &=\frac {5}{2} \text {Subst}\left (\int \log \left (\frac {1}{x^2}\right ) \, dx,x,e^4+2 x\right )+\int \left (-9+\frac {10 e^4}{e^4+2 x}\right ) \, dx\\ &=x+\frac {5}{2} \left (e^4+2 x\right ) \log \left (\frac {1}{\left (e^4+2 x\right )^2}\right )+5 e^4 \log \left (e^4+2 x\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 36, normalized size = 1.71 \begin {gather*} x+\frac {5}{2} \left (e^4+2 x\right ) \log \left (\frac {1}{\left (e^4+2 x\right )^2}\right )+5 e^4 \log \left (e^4+2 x\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 24, normalized size = 1.14
method | result | size |
risch | \(x +5 \ln \left (\frac {1}{{\mathrm e}^{8}+4 x \,{\mathrm e}^{4}+4 x^{2}}\right ) x\) | \(22\) |
default | \(x +5 \ln \left (\frac {1}{{\mathrm e}^{8}+4 x \,{\mathrm e}^{4}+4 x^{2}}\right ) x\) | \(24\) |
norman | \(x +5 \ln \left (\frac {1}{{\mathrm e}^{8}+4 x \,{\mathrm e}^{4}+4 x^{2}}\right ) x\) | \(24\) |
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. 108 vs.
\(2 (20) = 40\).
time = 0.26, size = 108, normalized size = 5.14 \begin {gather*} -\frac {5}{2} \, e^{4} \log \left (4 \, x^{2} + 4 \, x e^{4} + e^{8}\right ) \log \left (2 \, x + e^{4}\right ) - \frac {5}{2} \, e^{4} \log \left (2 \, x + e^{4}\right )^{2} + \frac {5}{2} \, {\left (\log \left (4 \, x^{2} + 4 \, x e^{4} + e^{8}\right ) \log \left (2 \, x + e^{4}\right ) - \log \left (2 \, x + e^{4}\right )^{2}\right )} e^{4} + \frac {5}{2} \, {\left (e^{4} \log \left (2 \, x + e^{4}\right ) - 2 \, x\right )} \log \left (4 \, x^{2} + 4 \, x e^{4} + e^{8}\right ) + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 21, normalized size = 1.00 \begin {gather*} 5 \, x \log \left (\frac {1}{4 \, x^{2} + 4 \, x e^{4} + e^{8}}\right ) + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 22, normalized size = 1.05 \begin {gather*} 5 x \log {\left (\frac {1}{4 x^{2} + 4 x e^{4} + e^{8}} \right )} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 19, normalized size = 0.90 \begin {gather*} -5 \, x \log \left (4 \, x^{2} + 4 \, x e^{4} + e^{8}\right ) + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.28, size = 21, normalized size = 1.00 \begin {gather*} x+5\,x\,\ln \left (\frac {1}{4\,x^2+4\,{\mathrm {e}}^4\,x+{\mathrm {e}}^8}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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