Optimal. Leaf size=23 \[ \frac {\left (1+e^{1+x}\right )^2}{x}+\log \left (\frac {1}{x}+x \log (4)\right ) \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(48\) vs. \(2(23)=46\).
time = 0.63, antiderivative size = 48, normalized size of antiderivative = 2.09, number of steps
used = 8, number of rules used = 5, integrand size = 82, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {1607, 6857,
2228, 1816, 266} \begin {gather*} \frac {\log (16) \log \left (x^2 \log (4)+1\right )}{2 \log (4)}+\frac {2 e^{x+1}}{x}+\frac {e^{2 x+2}}{x}+\frac {1}{x}-\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 266
Rule 1607
Rule 1816
Rule 2228
Rule 6857
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-1-x+\left (-x^2+x^3\right ) \log (4)+e^{1+x} \left (-2+2 x+\left (-2 x^2+2 x^3\right ) \log (4)\right )+e^{2+2 x} \left (-1+2 x+\left (-x^2+2 x^3\right ) \log (4)\right )}{x^2 \left (1+x^2 \log (4)\right )} \, dx\\ &=\int \left (\frac {2 e^{1+x} (-1+x)}{x^2}+\frac {e^{2+2 x} (-1+2 x)}{x^2}+\frac {-1-x-x^2 \log (4)+x^3 \log (4)}{x^2 \left (1+x^2 \log (4)\right )}\right ) \, dx\\ &=2 \int \frac {e^{1+x} (-1+x)}{x^2} \, dx+\int \frac {e^{2+2 x} (-1+2 x)}{x^2} \, dx+\int \frac {-1-x-x^2 \log (4)+x^3 \log (4)}{x^2 \left (1+x^2 \log (4)\right )} \, dx\\ &=\frac {2 e^{1+x}}{x}+\frac {e^{2+2 x}}{x}+\int \left (-\frac {1}{x^2}-\frac {1}{x}+\frac {x \log (16)}{1+x^2 \log (4)}\right ) \, dx\\ &=\frac {1}{x}+\frac {2 e^{1+x}}{x}+\frac {e^{2+2 x}}{x}-\log (x)+\log (16) \int \frac {x}{1+x^2 \log (4)} \, dx\\ &=\frac {1}{x}+\frac {2 e^{1+x}}{x}+\frac {e^{2+2 x}}{x}-\log (x)+\frac {\log (16) \log \left (1+x^2 \log (4)\right )}{2 \log (4)}\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(50\) vs. \(2(23)=46\).
time = 0.19, size = 50, normalized size = 2.17 \begin {gather*} \frac {4 e^{1+x} \log (4)+\log (16)+e^{2+2 x} \log (16)-2 x \log (4) \log (x)+x \log (16) \log \left (1+x^2 \log (4)\right )}{x \log (16)} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(50\) vs.
\(2(23)=46\).
time = 1.71, size = 51, normalized size = 2.22
method | result | size |
norman | \(\frac {1+{\mathrm e}^{2 x +2}+2 \,{\mathrm e}^{x +1}}{x}-\ln \left (x \right )+\ln \left (2 x^{2} \ln \left (2\right )+1\right )\) | \(34\) |
risch | \(\frac {1}{x}-\ln \left (x \right )+\ln \left (-2 x^{2} \ln \left (2\right )-1\right )+\frac {{\mathrm e}^{2 x +2}}{x}+\frac {2 \,{\mathrm e}^{x +1}}{x}\) | \(38\) |
derivativedivides | \(\ln \left (2 \ln \left (2\right ) \left (x +1\right )^{2}-4 \ln \left (2\right ) \left (x +1\right )+2 \ln \left (2\right )+1\right )+\frac {1}{x}+\frac {{\mathrm e}^{2 x +2}}{x}-\ln \left (x \right )+\frac {2 \,{\mathrm e}^{x +1}}{x}\) | \(51\) |
default | \(\ln \left (2 \ln \left (2\right ) \left (x +1\right )^{2}-4 \ln \left (2\right ) \left (x +1\right )+2 \ln \left (2\right )+1\right )+\frac {1}{x}+\frac {{\mathrm e}^{2 x +2}}{x}-\ln \left (x \right )+\frac {2 \,{\mathrm e}^{x +1}}{x}\) | \(51\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 37, normalized size = 1.61 \begin {gather*} \frac {e^{\left (2 \, x + 2\right )} + 2 \, e^{\left (x + 1\right )}}{x} + \frac {1}{x} + \log \left (2 \, x^{2} \log \left (2\right ) + 1\right ) - \frac {1}{2} \, \log \left (x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 35, normalized size = 1.52 \begin {gather*} \frac {x \log \left (2 \, x^{2} \log \left (2\right ) + 1\right ) - x \log \left (x\right ) + e^{\left (2 \, x + 2\right )} + 2 \, e^{\left (x + 1\right )} + 1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.20, size = 37, normalized size = 1.61 \begin {gather*} - \log {\left (x \right )} + \log {\left (x^{2} + \frac {1}{2 \log {\left (2 \right )}} \right )} + \frac {1}{x} + \frac {2 x e^{x + 1} + x e^{2 x + 2}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 35, normalized size = 1.52 \begin {gather*} \frac {x \log \left (2 \, x^{2} \log \left (2\right ) + 1\right ) - x \log \left (x\right ) + e^{\left (2 \, x + 2\right )} + 2 \, e^{\left (x + 1\right )} + 1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.25, size = 33, normalized size = 1.43 \begin {gather*} \ln \left (2\,\ln \left (2\right )\,x^2+1\right )-\ln \left (x\right )+\frac {2\,{\mathrm {e}}^{x+1}+{\mathrm {e}}^{2\,x+2}+1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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