Optimal. Leaf size=25 \[ 2 x+\frac {x}{-1-e^x+\frac {375}{2} \log ^2(5) \log (x)} \]
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Rubi [F]
time = 0.70, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {4+8 e^{2 x}+e^x (12+4 x)-750 \log ^2(5)+\left (-2250 \log ^2(5)-3000 e^x \log ^2(5)\right ) \log (x)+281250 \log ^4(5) \log ^2(x)}{4+8 e^x+4 e^{2 x}+\left (-1500 \log ^2(5)-1500 e^x \log ^2(5)\right ) \log (x)+140625 \log ^4(5) \log ^2(x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8 e^{2 x}+4 e^x (3+x)+4 \left (1-\frac {375 \log ^2(5)}{2}\right )-750 \left (3+4 e^x\right ) \log ^2(5) \log (x)+281250 \log ^4(5) \log ^2(x)}{\left (2+2 e^x-375 \log ^2(5) \log (x)\right )^2} \, dx\\ &=\int \left (2+\frac {2 (-1+x)}{2+2 e^x-375 \log ^2(5) \log (x)}+\frac {2 \left (-2 x-375 \log ^2(5)+375 x \log ^2(5) \log (x)\right )}{\left (2+2 e^x-375 \log ^2(5) \log (x)\right )^2}\right ) \, dx\\ &=2 x+2 \int \frac {-1+x}{2+2 e^x-375 \log ^2(5) \log (x)} \, dx+2 \int \frac {-2 x-375 \log ^2(5)+375 x \log ^2(5) \log (x)}{\left (2+2 e^x-375 \log ^2(5) \log (x)\right )^2} \, dx\\ &=2 x+2 \int \left (-\frac {1}{2+2 e^x-375 \log ^2(5) \log (x)}+\frac {x}{2+2 e^x-375 \log ^2(5) \log (x)}\right ) \, dx+2 \int \left (-\frac {2 x}{\left (2+2 e^x-375 \log ^2(5) \log (x)\right )^2}-\frac {375 \log ^2(5)}{\left (-2-2 e^x+375 \log ^2(5) \log (x)\right )^2}+\frac {375 x \log ^2(5) \log (x)}{\left (-2-2 e^x+375 \log ^2(5) \log (x)\right )^2}\right ) \, dx\\ &=2 x-2 \int \frac {1}{2+2 e^x-375 \log ^2(5) \log (x)} \, dx+2 \int \frac {x}{2+2 e^x-375 \log ^2(5) \log (x)} \, dx-4 \int \frac {x}{\left (2+2 e^x-375 \log ^2(5) \log (x)\right )^2} \, dx-\left (750 \log ^2(5)\right ) \int \frac {1}{\left (-2-2 e^x+375 \log ^2(5) \log (x)\right )^2} \, dx+\left (750 \log ^2(5)\right ) \int \frac {x \log (x)}{\left (-2-2 e^x+375 \log ^2(5) \log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.18, size = 24, normalized size = 0.96 \begin {gather*} 2 x+\frac {2 x}{-2-2 e^x+375 \log ^2(5) \log (x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.30, size = 24, normalized size = 0.96
method | result | size |
risch | \(2 x +\frac {2 x}{375 \ln \left (x \right ) \ln \left (5\right )^{2}-2 \,{\mathrm e}^{x}-2}\) | \(24\) |
norman | \(\frac {-2 x -4 \,{\mathrm e}^{x} x +750 x \ln \left (5\right )^{2} \ln \left (x \right )}{375 \ln \left (x \right ) \ln \left (5\right )^{2}-2 \,{\mathrm e}^{x}-2}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 36, normalized size = 1.44 \begin {gather*} \frac {2 \, {\left (375 \, x \log \left (5\right )^{2} \log \left (x\right ) - 2 \, x e^{x} - x\right )}}{375 \, \log \left (5\right )^{2} \log \left (x\right ) - 2 \, e^{x} - 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 36, normalized size = 1.44 \begin {gather*} \frac {2 \, {\left (375 \, x \log \left (5\right )^{2} \log \left (x\right ) - 2 \, x e^{x} - x\right )}}{375 \, \log \left (5\right )^{2} \log \left (x\right ) - 2 \, e^{x} - 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 20, normalized size = 0.80 \begin {gather*} 2 x - \frac {x}{e^{x} - \frac {375 \log {\left (5 \right )}^{2} \log {\left (x \right )}}{2} + 1} \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. 49 vs.
\(2 (23) = 46\).
time = 0.47, size = 49, normalized size = 1.96 \begin {gather*} \frac {2 \, {\left (375 \, x \log \left (5\right )^{2} \log \left (x\right ) + 375 \, \log \left (5\right )^{2} \log \left (x\right ) - 2 \, x e^{x} - x - 2 \, e^{x} - 2\right )}}{375 \, \log \left (5\right )^{2} \log \left (x\right ) - 2 \, e^{x} - 2} \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 {281250\,{\ln \left (5\right )}^4\,{\ln \left (x\right )}^2+\left (-3000\,{\mathrm {e}}^x\,{\ln \left (5\right )}^2-2250\,{\ln \left (5\right )}^2\right )\,\ln \left (x\right )+8\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,\left (4\,x+12\right )-750\,{\ln \left (5\right )}^2+4}{140625\,{\ln \left (5\right )}^4\,{\ln \left (x\right )}^2+\left (-1500\,{\mathrm {e}}^x\,{\ln \left (5\right )}^2-1500\,{\ln \left (5\right )}^2\right )\,\ln \left (x\right )+4\,{\mathrm {e}}^{2\,x}+8\,{\mathrm {e}}^x+4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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