Optimal. Leaf size=24 \[ 30+x^2+\frac {5}{x+\log (4)+2 \left (x-\frac {\log (x)}{e^4}\right )} \]
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Rubi [A]
time = 0.49, antiderivative size = 28, normalized size of antiderivative = 1.17, number of steps
used = 4, number of rules used = 3, integrand size = 130, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {6820, 6874,
6818} \begin {gather*} x^2+\frac {5 e^4}{3 e^4 x-2 \log (x)+e^4 \log (4)} \end {gather*}
Antiderivative was successfully verified.
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Rule 6818
Rule 6820
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {10 e^4+e^8 x \left (-15+18 x^3+12 x^2 \log (4)+2 x \log ^2(4)\right )-8 e^4 x^2 (3 x+\log (4)) \log (x)+8 x^2 \log ^2(x)}{x \left (e^4 (3 x+\log (4))-2 \log (x)\right )^2} \, dx\\ &=\int \left (2 x-\frac {5 e^4 \left (-2+3 e^4 x\right )}{x \left (3 e^4 x+e^4 \log (4)-2 \log (x)\right )^2}\right ) \, dx\\ &=x^2-\left (5 e^4\right ) \int \frac {-2+3 e^4 x}{x \left (3 e^4 x+e^4 \log (4)-2 \log (x)\right )^2} \, dx\\ &=x^2+\frac {5 e^4}{3 e^4 x+e^4 \log (4)-2 \log (x)}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.12, size = 26, normalized size = 1.08 \begin {gather*} x^2+\frac {5 e^4}{e^4 (3 x+\log (4))-2 \log (x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.23, size = 48, normalized size = 2.00
method | result | size |
risch | \(x^{2}+\frac {5 \,{\mathrm e}^{4}}{2 \,{\mathrm e}^{4} \ln \left (2\right )+3 x \,{\mathrm e}^{4}-2 \ln \left (x \right )}\) | \(27\) |
default | \(\frac {3 x^{3} {\mathrm e}^{4}-2 x^{2} \ln \left (x \right )+2 x^{2} {\mathrm e}^{4} \ln \left (2\right )+5 \,{\mathrm e}^{4}}{2 \,{\mathrm e}^{4} \ln \left (2\right )+3 x \,{\mathrm e}^{4}-2 \ln \left (x \right )}\) | \(48\) |
norman | \(\frac {3 x^{3} {\mathrm e}^{4}-2 x^{2} \ln \left (x \right )+2 x^{2} {\mathrm e}^{4} \ln \left (2\right )+5 \,{\mathrm e}^{4}}{2 \,{\mathrm e}^{4} \ln \left (2\right )+3 x \,{\mathrm e}^{4}-2 \ln \left (x \right )}\) | \(48\) |
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. 47 vs.
\(2 (23) = 46\).
time = 0.51, size = 47, normalized size = 1.96 \begin {gather*} \frac {3 \, x^{3} e^{4} + 2 \, x^{2} e^{4} \log \left (2\right ) - 2 \, x^{2} \log \left (x\right ) + 5 \, e^{4}}{3 \, x e^{4} + 2 \, e^{4} \log \left (2\right ) - 2 \, \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 46, normalized size = 1.92 \begin {gather*} \frac {2 \, x^{2} e^{4} \log \left (2\right ) - 2 \, x^{2} \log \left (x\right ) + {\left (3 \, x^{3} + 5\right )} e^{4}}{3 \, x e^{4} + 2 \, e^{4} \log \left (2\right ) - 2 \, \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 27, normalized size = 1.12 \begin {gather*} x^{2} - \frac {5 e^{4}}{- 3 x e^{4} + 2 \log {\left (x \right )} - 2 e^{4} \log {\left (2 \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 47 vs.
\(2 (23) = 46\).
time = 0.39, size = 47, normalized size = 1.96 \begin {gather*} \frac {3 \, x^{3} e^{4} + 2 \, x^{2} e^{4} \log \left (2\right ) - 2 \, x^{2} \log \left (x\right ) + 5 \, e^{4}}{3 \, x e^{4} + 2 \, e^{4} \log \left (2\right ) - 2 \, \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {10\,{\mathrm {e}}^4-{\mathrm {e}}^8\,\left (15\,x-18\,x^4\right )-\ln \left (x\right )\,\left (24\,{\mathrm {e}}^4\,x^3+16\,{\mathrm {e}}^4\,\ln \left (2\right )\,x^2\right )+8\,x^2\,{\ln \left (x\right )}^2+24\,x^3\,{\mathrm {e}}^8\,\ln \left (2\right )+8\,x^2\,{\mathrm {e}}^8\,{\ln \left (2\right )}^2}{4\,x\,{\ln \left (x\right )}^2+9\,x^3\,{\mathrm {e}}^8-\ln \left (x\right )\,\left (12\,{\mathrm {e}}^4\,x^2+8\,{\mathrm {e}}^4\,\ln \left (2\right )\,x\right )+4\,x\,{\mathrm {e}}^8\,{\ln \left (2\right )}^2+12\,x^2\,{\mathrm {e}}^8\,\ln \left (2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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