Optimal. Leaf size=35 \[ 9 \left (-4+\frac {x}{-4+\frac {i \pi +\log \left (-4+4 e^5\right )}{x \log (\log (x))}}\right )^2 \]
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Rubi [F]
time = 5.34, 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 {72 x \left (i \pi +\log \left (-4+4 e^5\right )\right )^2+\left (\left (-288 x^2-18 x^3\right ) \left (i \pi +\log \left (-4+4 e^5\right )\right )+144 x \left (i \pi +\log \left (-4+4 e^5\right )\right )^2 \log (x)\right ) \log (\log (x))+\left (-864 x^2-36 x^3\right ) \left (i \pi +\log \left (-4+4 e^5\right )\right ) \log (x) \log ^2(\log (x))+\left (1152 x^3+72 x^4\right ) \log (x) \log ^3(\log (x))}{-\left (i \pi +\log \left (-4+4 e^5\right )\right )^3 \log (x)+12 x \left (i \pi +\log \left (-4+4 e^5\right )\right )^2 \log (x) \log (\log (x))-48 x^2 \left (i \pi +\log \left (-4+4 e^5\right )\right ) \log (x) \log ^2(\log (x))+64 x^3 \log (x) \log ^3(\log (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 {18 x \left (4 i \pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )-x (16+x) \log (\log (x))\right ) \left (-i \pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )+2 \log (x) \log (\log (x)) \left (-i \pi -\log \left (4 \left (-1+e^5\right )\right )+2 x \log (\log (x))\right )\right )}{\log (x) \left (i \pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )-4 x \log (\log (x))\right )^3} \, dx\\ &=18 \int \frac {x \left (4 i \pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )-x (16+x) \log (\log (x))\right ) \left (-i \pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )+2 \log (x) \log (\log (x)) \left (-i \pi -\log \left (4 \left (-1+e^5\right )\right )+2 x \log (\log (x))\right )\right )}{\log (x) \left (i \pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )-4 x \log (\log (x))\right )^3} \, dx\\ &=18 \int \left (\frac {16+x}{16}+\frac {x \left (\pi -i \log \left (-4 \left (1-e^5\right )\right )\right )^2 \left (-4 x-i \pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right ) \log (x)\right )}{16 \log (x) \left (i \pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )-4 x \log (\log (x))\right )^3}+\frac {x \left (-i \pi -\log \left (-4 \left (1-e^5\right )\right )\right )}{16 \left (i \pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )-4 x \log (\log (x))\right )}+\frac {(16+x) \left (\pi -i \log \left (-4 \left (1-e^5\right )\right )\right ) \left (4 i x-\pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right ) \log (x)\right )}{16 \log (x) \left (\pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )+4 i x \log (\log (x))\right )^2}\right ) \, dx\\ &=\frac {9}{16} (16+x)^2+\frac {1}{8} \left (9 \left (\pi -i \log \left (-4 \left (1-e^5\right )\right )\right )\right ) \int \frac {(16+x) \left (4 i x-\pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right ) \log (x)\right )}{\log (x) \left (\pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )+4 i x \log (\log (x))\right )^2} \, dx+\frac {1}{8} \left (9 \left (\pi -i \log \left (-4 \left (1-e^5\right )\right )\right )^2\right ) \int \frac {x \left (-4 x-i \pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right ) \log (x)\right )}{\log (x) \left (i \pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )-4 x \log (\log (x))\right )^3} \, dx-\frac {1}{8} \left (9 \left (i \pi +\log \left (-4 \left (1-e^5\right )\right )\right )\right ) \int \frac {x}{i \pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )-4 x \log (\log (x))} \, dx\\ &=\frac {9}{16} (16+x)^2+\frac {1}{8} \left (9 \left (\pi -i \log \left (-4 \left (1-e^5\right )\right )\right )\right ) \int \left (\frac {16 \left (4 i x-\pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right ) \log (x)\right )}{\log (x) \left (\pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )+4 i x \log (\log (x))\right )^2}+\frac {x \left (4 i x-\pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right ) \log (x)\right )}{\log (x) \left (\pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )+4 i x \log (\log (x))\right )^2}\right ) \, dx+\frac {1}{8} \left (9 \left (\pi -i \log \left (-4 \left (1-e^5\right )\right )\right )^2\right ) \int \left (\frac {x \left (-i \pi -\log \left (-4 \left (1-e^5\right )\right )\right )}{\left (i \pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )-4 x \log (\log (x))\right )^3}+\frac {4 x^2}{\log (x) \left (-i \pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )+4 x \log (\log (x))\right )^3}\right ) \, dx-\frac {1}{8} \left (9 \left (i \pi +\log \left (-4 \left (1-e^5\right )\right )\right )\right ) \int \frac {x}{i \pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )-4 x \log (\log (x))} \, dx\\ &=\frac {9}{16} (16+x)^2+\frac {1}{8} \left (9 \left (\pi -i \log \left (-4 \left (1-e^5\right )\right )\right )\right ) \int \frac {x \left (4 i x-\pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right ) \log (x)\right )}{\log (x) \left (\pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )+4 i x \log (\log (x))\right )^2} \, dx+\left (18 \left (\pi -i \log \left (-4 \left (1-e^5\right )\right )\right )\right ) \int \frac {4 i x-\pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right ) \log (x)}{\log (x) \left (\pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )+4 i x \log (\log (x))\right )^2} \, dx+\frac {1}{2} \left (9 \left (\pi -i \log \left (-4 \left (1-e^5\right )\right )\right )^2\right ) \int \frac {x^2}{\log (x) \left (-i \pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )+4 x \log (\log (x))\right )^3} \, dx-\frac {1}{8} \left (9 \left (i \pi +\log \left (-4 \left (1-e^5\right )\right )\right )\right ) \int \frac {x}{i \pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )-4 x \log (\log (x))} \, dx+\frac {1}{8} \left (9 \left (i \pi +\log \left (-4 \left (1-e^5\right )\right )\right )^3\right ) \int \frac {x}{\left (i \pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )-4 x \log (\log (x))\right )^3} \, dx\\ &=\frac {9}{16} (16+x)^2+\frac {1}{8} \left (9 \left (\pi -i \log \left (-4 \left (1-e^5\right )\right )\right )\right ) \int \left (\frac {x \left (-\pi +i \log \left (-4 \left (1-e^5\right )\right )\right )}{\left (\pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )+4 i x \log (\log (x))\right )^2}+\frac {4 i x^2}{\log (x) \left (\pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )+4 i x \log (\log (x))\right )^2}\right ) \, dx+\left (18 \left (\pi -i \log \left (-4 \left (1-e^5\right )\right )\right )\right ) \int \left (\frac {-\pi +i \log \left (-4 \left (1-e^5\right )\right )}{\left (\pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )+4 i x \log (\log (x))\right )^2}+\frac {4 i x}{\log (x) \left (\pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )+4 i x \log (\log (x))\right )^2}\right ) \, dx+\frac {1}{2} \left (9 \left (\pi -i \log \left (-4 \left (1-e^5\right )\right )\right )^2\right ) \int \frac {x^2}{\log (x) \left (-i \pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )+4 x \log (\log (x))\right )^3} \, dx-\frac {1}{8} \left (9 \left (i \pi +\log \left (-4 \left (1-e^5\right )\right )\right )\right ) \int \frac {x}{i \pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )-4 x \log (\log (x))} \, dx+\frac {1}{8} \left (9 \left (i \pi +\log \left (-4 \left (1-e^5\right )\right )\right )^3\right ) \int \frac {x}{\left (i \pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )-4 x \log (\log (x))\right )^3} \, dx\\ &=\frac {9}{16} (16+x)^2-\frac {1}{8} \left (9 \left (\pi -i \log \left (-4 \left (1-e^5\right )\right )\right )^2\right ) \int \frac {x}{\left (\pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )+4 i x \log (\log (x))\right )^2} \, dx+\frac {1}{2} \left (9 \left (\pi -i \log \left (-4 \left (1-e^5\right )\right )\right )^2\right ) \int \frac {x^2}{\log (x) \left (-i \pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )+4 x \log (\log (x))\right )^3} \, dx-\left (18 \left (\pi -i \log \left (-4 \left (1-e^5\right )\right )\right )^2\right ) \int \frac {1}{\left (\pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )+4 i x \log (\log (x))\right )^2} \, dx-\frac {1}{8} \left (9 \left (i \pi +\log \left (-4 \left (1-e^5\right )\right )\right )\right ) \int \frac {x}{i \pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )-4 x \log (\log (x))} \, dx+\frac {1}{2} \left (9 \left (i \pi +\log \left (-4 \left (1-e^5\right )\right )\right )\right ) \int \frac {x^2}{\log (x) \left (\pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )+4 i x \log (\log (x))\right )^2} \, dx+\left (72 \left (i \pi +\log \left (-4 \left (1-e^5\right )\right )\right )\right ) \int \frac {x}{\log (x) \left (\pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )+4 i x \log (\log (x))\right )^2} \, dx+\frac {1}{8} \left (9 \left (i \pi +\log \left (-4 \left (1-e^5\right )\right )\right )^3\right ) \int \frac {x}{\left (i \pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )-4 x \log (\log (x))\right )^3} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.08, size = 56, normalized size = 1.60 \begin {gather*} -\frac {9 x^2 \log (\log (x)) \left (-8 i \pi -8 \log \left (4 \left (-1+e^5\right )\right )+x (32+x) \log (\log (x))\right )}{\left (\pi -i \log \left (4 \left (-1+e^5\right )\right )+4 i x \log (\log (x))\right )^2} \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. \(137\) vs.
\(2(51)=102\).
time = 0.49, size = 138, normalized size = 3.94
method | result | size |
risch | \(\frac {9 x^{2}}{16}+18 x -\frac {9 x \left (-16 x^{2} \ln \left (\ln \left (x \right )\right ) \ln \left (2\right )-8 x^{2} \ln \left (\ln \left (x \right )\right ) \ln \left (1-{\mathrm e}^{5}\right )+4 x \ln \left (2\right )^{2}+4 x \ln \left (2\right ) \ln \left (1-{\mathrm e}^{5}\right )-256 x \ln \left (\ln \left (x \right )\right ) \ln \left (2\right )+x \ln \left (1-{\mathrm e}^{5}\right )^{2}-128 x \ln \left (\ln \left (x \right )\right ) \ln \left (1-{\mathrm e}^{5}\right )+128 \ln \left (2\right )^{2}+128 \ln \left (2\right ) \ln \left (1-{\mathrm e}^{5}\right )+32 \ln \left (1-{\mathrm e}^{5}\right )^{2}\right )}{16 \left (-4 x \ln \left (\ln \left (x \right )\right )+2 \ln \left (2\right )+\ln \left (1-{\mathrm e}^{5}\right )\right )^{2}}\) | \(138\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.73, size = 188, normalized size = 5.37 \begin {gather*} -\frac {9 \, {\left (8 \, {\left (i \, \pi + 2 \, \log \left (2\right ) + \log \left (e^{4} + e^{3} + e^{2} + e + 1\right ) + \log \left (e - 1\right )\right )} x^{2} \log \left (\log \left (x\right )\right ) - {\left (x^{4} + 32 \, x^{3}\right )} \log \left (\log \left (x\right )\right )^{2}\right )}}{16 \, x^{2} \log \left (\log \left (x\right )\right )^{2} - 8 \, {\left (i \, \pi + 2 \, \log \left (2\right ) + \log \left (e^{4} + e^{3} + e^{2} + e + 1\right ) + \log \left (e - 1\right )\right )} x \log \left (\log \left (x\right )\right ) - \pi ^{2} + 2 i \, \pi {\left (2 \, \log \left (2\right ) + \log \left (e^{4} + e^{3} + e^{2} + e + 1\right ) + \log \left (e - 1\right )\right )} + 4 \, {\left (\log \left (e^{4} + e^{3} + e^{2} + e + 1\right ) + \log \left (e - 1\right )\right )} \log \left (2\right ) + 4 \, \log \left (2\right )^{2} + \log \left (e^{4} + e^{3} + e^{2} + e + 1\right )^{2} + 2 \, \log \left (e^{4} + e^{3} + e^{2} + e + 1\right ) \log \left (e - 1\right ) + \log \left (e - 1\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 69 vs.
\(2 (28) = 56\).
time = 0.32, size = 69, normalized size = 1.97 \begin {gather*} -\frac {9 \, {\left (8 \, x^{2} \log \left (-4 \, e^{5} + 4\right ) \log \left (\log \left (x\right )\right ) - {\left (x^{4} + 32 \, x^{3}\right )} \log \left (\log \left (x\right )\right )^{2}\right )}}{16 \, x^{2} \log \left (\log \left (x\right )\right )^{2} - 8 \, x \log \left (-4 \, e^{5} + 4\right ) \log \left (\log \left (x\right )\right ) + \log \left (-4 \, e^{5} + 4\right )^{2}} \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 [B] Leaf count of result is larger than twice the leaf count of optimal. 72 vs.
\(2 (28) = 56\).
time = 0.45, size = 72, normalized size = 2.06 \begin {gather*} \frac {9 \, {\left (x^{4} \log \left (\log \left (x\right )\right )^{2} + 32 \, x^{3} \log \left (\log \left (x\right )\right )^{2} - 8 \, x^{2} \log \left (-4 \, e^{5} + 4\right ) \log \left (\log \left (x\right )\right )\right )}}{16 \, x^{2} \log \left (\log \left (x\right )\right )^{2} - 8 \, x \log \left (-4 \, e^{5} + 4\right ) \log \left (\log \left (x\right )\right ) + \log \left (-4 \, e^{5} + 4\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.99, size = 2500, normalized size = 71.43 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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