Optimal. Leaf size=28 \[ e^{\frac {3}{-2+e^{\log ^2(\log (x))}-\frac {3}{x \log (1+2 x)}}} \]
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Rubi [F]
time = 8.11, 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 {(1+2 x)^{\frac {3 x}{-3-2 x \log (1+2 x)+e^{\log ^2(\log (x))} x \log (1+2 x)}} \left (-18 x \log (x)+(-9-18 x) \log (x) \log (1+2 x)+e^{\log ^2(\log (x))} \left (-6 x-12 x^2\right ) \log ^2(1+2 x) \log (\log (x))\right )}{(9+18 x) \log (x)+\left (12 x+24 x^2\right ) \log (x) \log (1+2 x)+e^{2 \log ^2(\log (x))} \left (x^2+2 x^3\right ) \log (x) \log ^2(1+2 x)+\left (4 x^2+8 x^3\right ) \log (x) \log ^2(1+2 x)+e^{\log ^2(\log (x))} \left (\left (-6 x-12 x^2\right ) \log (x) \log (1+2 x)+\left (-4 x^2-8 x^3\right ) \log (x) \log ^2(1+2 x)\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {(1+2 x)^{-1+\frac {3 x}{-3+\left (-2+e^{\log ^2(\log (x))}\right ) x \log (1+2 x)}} \left (-9 \log (x) (2 x+(1+2 x) \log (1+2 x))-6 e^{\log ^2(\log (x))} x (1+2 x) \log ^2(1+2 x) \log (\log (x))\right )}{\log (x) \left (3-\left (-2+e^{\log ^2(\log (x))}\right ) x \log (1+2 x)\right )^2} \, dx\\ &=\int \left (-\frac {6 (1+2 x)^{\frac {3 x}{-3+\left (-2+e^{\log ^2(\log (x))}\right ) x \log (1+2 x)}} \log (1+2 x) \log (\log (x))}{\log (x) \left (-3-2 x \log (1+2 x)+e^{\log ^2(\log (x))} x \log (1+2 x)\right )}-\frac {3 (1+2 x)^{-1+\frac {3 x}{-3+\left (-2+e^{\log ^2(\log (x))}\right ) x \log (1+2 x)}} \left (6 x \log (x)+3 \log (x) \log (1+2 x)+6 x \log (x) \log (1+2 x)+6 \log (1+2 x) \log (\log (x))+12 x \log (1+2 x) \log (\log (x))+4 x \log ^2(1+2 x) \log (\log (x))+8 x^2 \log ^2(1+2 x) \log (\log (x))\right )}{\log (x) \left (-3-2 x \log (1+2 x)+e^{\log ^2(\log (x))} x \log (1+2 x)\right )^2}\right ) \, dx\\ &=-\left (3 \int \frac {(1+2 x)^{-1+\frac {3 x}{-3+\left (-2+e^{\log ^2(\log (x))}\right ) x \log (1+2 x)}} \left (6 x \log (x)+3 \log (x) \log (1+2 x)+6 x \log (x) \log (1+2 x)+6 \log (1+2 x) \log (\log (x))+12 x \log (1+2 x) \log (\log (x))+4 x \log ^2(1+2 x) \log (\log (x))+8 x^2 \log ^2(1+2 x) \log (\log (x))\right )}{\log (x) \left (-3-2 x \log (1+2 x)+e^{\log ^2(\log (x))} x \log (1+2 x)\right )^2} \, dx\right )-6 \int \frac {(1+2 x)^{\frac {3 x}{-3+\left (-2+e^{\log ^2(\log (x))}\right ) x \log (1+2 x)}} \log (1+2 x) \log (\log (x))}{\log (x) \left (-3-2 x \log (1+2 x)+e^{\log ^2(\log (x))} x \log (1+2 x)\right )} \, dx\\ &=-\left (3 \int \frac {(1+2 x)^{-1+\frac {3 x}{-3+\left (-2+e^{\log ^2(\log (x))}\right ) x \log (1+2 x)}} (3 \log (x) (2 x+(1+2 x) \log (1+2 x))+2 (1+2 x) \log (1+2 x) (3+2 x \log (1+2 x)) \log (\log (x)))}{\log (x) \left (3-\left (-2+e^{\log ^2(\log (x))}\right ) x \log (1+2 x)\right )^2} \, dx\right )-6 \int \frac {(1+2 x)^{\frac {3 x}{-3+\left (-2+e^{\log ^2(\log (x))}\right ) x \log (1+2 x)}} \log (1+2 x) \log (\log (x))}{\log (x) \left (-3-2 x \log (1+2 x)+e^{\log ^2(\log (x))} x \log (1+2 x)\right )} \, dx\\ &=-\left (3 \int \left (\frac {6 x (1+2 x)^{-1+\frac {3 x}{-3+\left (-2+e^{\log ^2(\log (x))}\right ) x \log (1+2 x)}}}{\left (-3-2 x \log (1+2 x)+e^{\log ^2(\log (x))} x \log (1+2 x)\right )^2}+\frac {3 (1+2 x)^{-1+\frac {3 x}{-3+\left (-2+e^{\log ^2(\log (x))}\right ) x \log (1+2 x)}} \log (1+2 x)}{\left (-3-2 x \log (1+2 x)+e^{\log ^2(\log (x))} x \log (1+2 x)\right )^2}+\frac {6 x (1+2 x)^{-1+\frac {3 x}{-3+\left (-2+e^{\log ^2(\log (x))}\right ) x \log (1+2 x)}} \log (1+2 x)}{\left (-3-2 x \log (1+2 x)+e^{\log ^2(\log (x))} x \log (1+2 x)\right )^2}+\frac {6 (1+2 x)^{-1+\frac {3 x}{-3+\left (-2+e^{\log ^2(\log (x))}\right ) x \log (1+2 x)}} \log (1+2 x) \log (\log (x))}{\log (x) \left (-3-2 x \log (1+2 x)+e^{\log ^2(\log (x))} x \log (1+2 x)\right )^2}+\frac {12 x (1+2 x)^{-1+\frac {3 x}{-3+\left (-2+e^{\log ^2(\log (x))}\right ) x \log (1+2 x)}} \log (1+2 x) \log (\log (x))}{\log (x) \left (-3-2 x \log (1+2 x)+e^{\log ^2(\log (x))} x \log (1+2 x)\right )^2}+\frac {4 x (1+2 x)^{-1+\frac {3 x}{-3+\left (-2+e^{\log ^2(\log (x))}\right ) x \log (1+2 x)}} \log ^2(1+2 x) \log (\log (x))}{\log (x) \left (-3-2 x \log (1+2 x)+e^{\log ^2(\log (x))} x \log (1+2 x)\right )^2}+\frac {8 x^2 (1+2 x)^{-1+\frac {3 x}{-3+\left (-2+e^{\log ^2(\log (x))}\right ) x \log (1+2 x)}} \log ^2(1+2 x) \log (\log (x))}{\log (x) \left (-3-2 x \log (1+2 x)+e^{\log ^2(\log (x))} x \log (1+2 x)\right )^2}\right ) \, dx\right )-6 \int \frac {(1+2 x)^{\frac {3 x}{-3+\left (-2+e^{\log ^2(\log (x))}\right ) x \log (1+2 x)}} \log (1+2 x) \log (\log (x))}{\log (x) \left (-3-2 x \log (1+2 x)+e^{\log ^2(\log (x))} x \log (1+2 x)\right )} \, dx\\ &=-\left (6 \int \frac {(1+2 x)^{\frac {3 x}{-3+\left (-2+e^{\log ^2(\log (x))}\right ) x \log (1+2 x)}} \log (1+2 x) \log (\log (x))}{\log (x) \left (-3-2 x \log (1+2 x)+e^{\log ^2(\log (x))} x \log (1+2 x)\right )} \, dx\right )-9 \int \frac {(1+2 x)^{-1+\frac {3 x}{-3+\left (-2+e^{\log ^2(\log (x))}\right ) x \log (1+2 x)}} \log (1+2 x)}{\left (-3-2 x \log (1+2 x)+e^{\log ^2(\log (x))} x \log (1+2 x)\right )^2} \, dx-12 \int \frac {x (1+2 x)^{-1+\frac {3 x}{-3+\left (-2+e^{\log ^2(\log (x))}\right ) x \log (1+2 x)}} \log ^2(1+2 x) \log (\log (x))}{\log (x) \left (-3-2 x \log (1+2 x)+e^{\log ^2(\log (x))} x \log (1+2 x)\right )^2} \, dx-18 \int \frac {x (1+2 x)^{-1+\frac {3 x}{-3+\left (-2+e^{\log ^2(\log (x))}\right ) x \log (1+2 x)}}}{\left (-3-2 x \log (1+2 x)+e^{\log ^2(\log (x))} x \log (1+2 x)\right )^2} \, dx-18 \int \frac {x (1+2 x)^{-1+\frac {3 x}{-3+\left (-2+e^{\log ^2(\log (x))}\right ) x \log (1+2 x)}} \log (1+2 x)}{\left (-3-2 x \log (1+2 x)+e^{\log ^2(\log (x))} x \log (1+2 x)\right )^2} \, dx-18 \int \frac {(1+2 x)^{-1+\frac {3 x}{-3+\left (-2+e^{\log ^2(\log (x))}\right ) x \log (1+2 x)}} \log (1+2 x) \log (\log (x))}{\log (x) \left (-3-2 x \log (1+2 x)+e^{\log ^2(\log (x))} x \log (1+2 x)\right )^2} \, dx-24 \int \frac {x^2 (1+2 x)^{-1+\frac {3 x}{-3+\left (-2+e^{\log ^2(\log (x))}\right ) x \log (1+2 x)}} \log ^2(1+2 x) \log (\log (x))}{\log (x) \left (-3-2 x \log (1+2 x)+e^{\log ^2(\log (x))} x \log (1+2 x)\right )^2} \, dx-36 \int \frac {x (1+2 x)^{-1+\frac {3 x}{-3+\left (-2+e^{\log ^2(\log (x))}\right ) x \log (1+2 x)}} \log (1+2 x) \log (\log (x))}{\log (x) \left (-3-2 x \log (1+2 x)+e^{\log ^2(\log (x))} x \log (1+2 x)\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.18, size = 30, normalized size = 1.07 \begin {gather*} (1+2 x)^{\frac {3 x}{-3+\left (-2+e^{\log ^2(\log (x))}\right ) x \log (1+2 x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 37, normalized size = 1.32
method | result | size |
risch | \(\left (2 x +1\right )^{\frac {3 x}{x \ln \left (2 x +1\right ) {\mathrm e}^{\ln \left (\ln \left (x \right )\right )^{2}}-2 x \ln \left (2 x +1\right )-3}}\) | \(37\) |
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. 64 vs.
\(2 (28) = 56\).
time = 0.76, size = 64, normalized size = 2.29 \begin {gather*} e^{\left (\frac {9}{x e^{\left (2 \, \log \left (\log \left (x\right )\right )^{2}\right )} \log \left (2 \, x + 1\right ) - {\left (4 \, x \log \left (2 \, x + 1\right ) + 3\right )} e^{\left (\log \left (\log \left (x\right )\right )^{2}\right )} + 4 \, x \log \left (2 \, x + 1\right ) + 6} + \frac {3}{e^{\left (\log \left (\log \left (x\right )\right )^{2}\right )} - 2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 36, normalized size = 1.29 \begin {gather*} {\left (2 \, x + 1\right )}^{\frac {3 \, x}{x e^{\left (\log \left (\log \left (x\right )\right )^{2}\right )} \log \left (2 \, x + 1\right ) - 2 \, x \log \left (2 \, x + 1\right ) - 3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.04, size = 38, normalized size = 1.36 \begin {gather*} {\mathrm {e}}^{-\frac {3\,x\,\ln \left (2\,x+1\right )}{2\,x\,\ln \left (2\,x+1\right )-x\,{\mathrm {e}}^{{\ln \left (\ln \left (x\right )\right )}^2}\,\ln \left (2\,x+1\right )+3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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