Optimal. Leaf size=27 \[ \log \left (\frac {(9+x)^2+\left (x+\frac {x (3+x)}{2+e^4}\right )^2}{x}\right ) \]
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Rubi [B] time = 0.24, antiderivative size = 55, normalized size of antiderivative = 2.04, number of steps used = 3, number of rules used = 2, integrand size = 110, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {2074, 1587} \begin {gather*} \log \left (x^4+2 \left (5+e^4\right ) x^3+\left (29+14 e^4+2 e^8\right ) x^2+18 \left (2+e^4\right )^2 x+81 \left (2+e^4\right )^2\right )-\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 1587
Rule 2074
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {1}{x}+\frac {2 \left (9 \left (2+e^4\right )^2+\left (29+14 e^4+2 e^8\right ) x+3 \left (5+e^4\right ) x^2+2 x^3\right )}{81 \left (2+e^4\right )^2+18 \left (2+e^4\right )^2 x+\left (29+14 e^4+2 e^8\right ) x^2+2 \left (5+e^4\right ) x^3+x^4}\right ) \, dx\\ &=-\log (x)+2 \int \frac {9 \left (2+e^4\right )^2+\left (29+14 e^4+2 e^8\right ) x+3 \left (5+e^4\right ) x^2+2 x^3}{81 \left (2+e^4\right )^2+18 \left (2+e^4\right )^2 x+\left (29+14 e^4+2 e^8\right ) x^2+2 \left (5+e^4\right ) x^3+x^4} \, dx\\ &=-\log (x)+\log \left (81 \left (2+e^4\right )^2+18 \left (2+e^4\right )^2 x+\left (29+14 e^4+2 e^8\right ) x^2+2 \left (5+e^4\right ) x^3+x^4\right )\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.05, size = 70, normalized size = 2.59 \begin {gather*} -\log (x)+\log \left (324+324 e^4+81 e^8+72 x+72 e^4 x+18 e^8 x+29 x^2+14 e^4 x^2+2 e^8 x^2+10 x^3+2 e^4 x^3+x^4\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 54, normalized size = 2.00 \begin {gather*} \log \left (x^{4} + 10 \, x^{3} + 29 \, x^{2} + {\left (2 \, x^{2} + 18 \, x + 81\right )} e^{8} + 2 \, {\left (x^{3} + 7 \, x^{2} + 36 \, x + 162\right )} e^{4} + 72 \, x + 324\right ) - \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] 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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maple [B] time = 0.12, size = 58, normalized size = 2.15
method | result | size |
risch | \(-\ln \left (-x \right )+\ln \left (x^{4}+\left (2 \,{\mathrm e}^{4}+10\right ) x^{3}+\left (2 \,{\mathrm e}^{8}+14 \,{\mathrm e}^{4}+29\right ) x^{2}+\left (72 \,{\mathrm e}^{4}+18 \,{\mathrm e}^{8}+72\right ) x +81 \,{\mathrm e}^{8}+324 \,{\mathrm e}^{4}+324\right )\) | \(58\) |
default | \(-\ln \relax (x )+\ln \left (2 x^{2} {\mathrm e}^{8}+2 x^{3} {\mathrm e}^{4}+x^{4}+18 x \,{\mathrm e}^{8}+14 x^{2} {\mathrm e}^{4}+10 x^{3}+81 \,{\mathrm e}^{8}+72 x \,{\mathrm e}^{4}+29 x^{2}+324 \,{\mathrm e}^{4}+72 x +324\right )\) | \(64\) |
norman | \(-\ln \relax (x )+\ln \left (2 x^{2} {\mathrm e}^{8}+2 x^{3} {\mathrm e}^{4}+x^{4}+18 x \,{\mathrm e}^{8}+14 x^{2} {\mathrm e}^{4}+10 x^{3}+81 \,{\mathrm e}^{8}+72 x \,{\mathrm e}^{4}+29 x^{2}+324 \,{\mathrm e}^{4}+72 x +324\right )\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.36, size = 53, normalized size = 1.96 \begin {gather*} \log \left (x^{4} + 2 \, x^{3} {\left (e^{4} + 5\right )} + x^{2} {\left (2 \, e^{8} + 14 \, e^{4} + 29\right )} + 18 \, x {\left (e^{8} + 4 \, e^{4} + 4\right )} + 81 \, e^{8} + 324 \, e^{4} + 324\right ) - \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.81, size = 63, normalized size = 2.33 \begin {gather*} \ln \left (72\,x+324\,{\mathrm {e}}^4+81\,{\mathrm {e}}^8+72\,x\,{\mathrm {e}}^4+18\,x\,{\mathrm {e}}^8+14\,x^2\,{\mathrm {e}}^4+2\,x^3\,{\mathrm {e}}^4+2\,x^2\,{\mathrm {e}}^8+29\,x^2+10\,x^3+x^4+324\right )-\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 5.59, size = 58, normalized size = 2.15 \begin {gather*} - \log {\relax (x )} + \log {\left (x^{4} + x^{3} \left (10 + 2 e^{4}\right ) + x^{2} \left (29 + 14 e^{4} + 2 e^{8}\right ) + x \left (72 + 72 e^{4} + 18 e^{8}\right ) + 324 + 324 e^{4} + 81 e^{8} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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