Optimal. Leaf size=25 \[ \left (20-\frac {3^{\frac {1}{x}} x \left (x^4\right )^{-1/x}}{\log (x)}\right )^2 \]
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Rubi [F]
time = 6.25, 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 {3^{2/x} \left (x^4\right )^{-2/x} \left (-2 x^2+\left (-8 x+2 x^2\right ) \log (x)+2 x \log (x) \log \left (\frac {x^4}{3}\right )+3^{-1/x} \left (x^4\right )^{\frac {1}{x}} \left (40 x \log (x)+(160-40 x) \log ^2(x)-40 \log ^2(x) \log \left (\frac {x^4}{3}\right )\right )\right )}{x \log ^3(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 {2\ 3^{\frac {1}{x}} \left (x^4\right )^{-2/x} \left (3^{\frac {1}{x}} x-20 \left (x^4\right )^{\frac {1}{x}} \log (x)\right ) \left (-x+\log (x) \left (-4+x+\log \left (\frac {x^4}{3}\right )\right )\right )}{x \log ^3(x)} \, dx\\ &=2 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-2/x} \left (3^{\frac {1}{x}} x-20 \left (x^4\right )^{\frac {1}{x}} \log (x)\right ) \left (-x+\log (x) \left (-4+x+\log \left (\frac {x^4}{3}\right )\right )\right )}{x \log ^3(x)} \, dx\\ &=2 \int \left (\frac {3^{2/x} \left (x^4\right )^{-2/x} \left (-x-4 \log (x)+x \log (x)+\log (x) \log \left (\frac {x^4}{3}\right )\right )}{\log ^3(x)}-\frac {20\ 3^{\frac {1}{x}} \left (x^4\right )^{-1/x} \left (-x-4 \log (x)+x \log (x)+\log (x) \log \left (\frac {x^4}{3}\right )\right )}{x \log ^2(x)}\right ) \, dx\\ &=2 \int \frac {3^{2/x} \left (x^4\right )^{-2/x} \left (-x-4 \log (x)+x \log (x)+\log (x) \log \left (\frac {x^4}{3}\right )\right )}{\log ^3(x)} \, dx-40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x} \left (-x-4 \log (x)+x \log (x)+\log (x) \log \left (\frac {x^4}{3}\right )\right )}{x \log ^2(x)} \, dx\\ &=2 \int \left (\frac {3^{2/x} \left (x^4\right )^{-2/x} (-x-4 \log (x)+x \log (x))}{\log ^3(x)}+\frac {3^{2/x} \left (x^4\right )^{-2/x} \log \left (\frac {x^4}{3}\right )}{\log ^2(x)}\right ) \, dx-40 \int \left (\frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x} (-x-4 \log (x)+x \log (x))}{x \log ^2(x)}+\frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x} \log \left (\frac {x^4}{3}\right )}{x \log (x)}\right ) \, dx\\ &=2 \int \frac {3^{2/x} \left (x^4\right )^{-2/x} (-x-4 \log (x)+x \log (x))}{\log ^3(x)} \, dx+2 \int \frac {3^{2/x} \left (x^4\right )^{-2/x} \log \left (\frac {x^4}{3}\right )}{\log ^2(x)} \, dx-40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x} (-x-4 \log (x)+x \log (x))}{x \log ^2(x)} \, dx-40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x} \log \left (\frac {x^4}{3}\right )}{x \log (x)} \, dx\\ &=2 \int \left (-\frac {3^{2/x} x \left (x^4\right )^{-2/x}}{\log ^3(x)}+\frac {3^{2/x} (-4+x) \left (x^4\right )^{-2/x}}{\log ^2(x)}\right ) \, dx+2 \int \frac {9^{\frac {1}{x}} \left (x^4\right )^{-2/x} \log \left (\frac {x^4}{3}\right )}{\log ^2(x)} \, dx-40 \int \left (-\frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x}}{\log ^2(x)}+\frac {3^{\frac {1}{x}} (-4+x) \left (x^4\right )^{-1/x}}{x \log (x)}\right ) \, dx-40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x} \log \left (\frac {x^4}{3}\right )}{x \log (x)} \, dx\\ &=-\left (2 \int \frac {3^{2/x} x \left (x^4\right )^{-2/x}}{\log ^3(x)} \, dx\right )+2 \int \frac {3^{2/x} (-4+x) \left (x^4\right )^{-2/x}}{\log ^2(x)} \, dx+2 \int \frac {9^{\frac {1}{x}} \left (x^4\right )^{-2/x} \log \left (\frac {x^4}{3}\right )}{\log ^2(x)} \, dx+40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x}}{\log ^2(x)} \, dx-40 \int \frac {3^{\frac {1}{x}} (-4+x) \left (x^4\right )^{-1/x}}{x \log (x)} \, dx-40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x} \log \left (\frac {x^4}{3}\right )}{x \log (x)} \, dx\\ &=-\left (2 \int \frac {9^{\frac {1}{x}} x \left (x^4\right )^{-2/x}}{\log ^3(x)} \, dx\right )+2 \int \frac {9^{\frac {1}{x}} (-4+x) \left (x^4\right )^{-2/x}}{\log ^2(x)} \, dx+2 \int \frac {9^{\frac {1}{x}} \left (x^4\right )^{-2/x} \log \left (\frac {x^4}{3}\right )}{\log ^2(x)} \, dx-40 \int \left (\frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x}}{\log (x)}-\frac {4\ 3^{\frac {1}{x}} \left (x^4\right )^{-1/x}}{x \log (x)}\right ) \, dx+40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x}}{\log ^2(x)} \, dx-40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x} \log \left (\frac {x^4}{3}\right )}{x \log (x)} \, dx\\ &=2 \int \left (-\frac {4\ 9^{\frac {1}{x}} \left (x^4\right )^{-2/x}}{\log ^2(x)}+\frac {9^{\frac {1}{x}} x \left (x^4\right )^{-2/x}}{\log ^2(x)}\right ) \, dx-2 \int \frac {9^{\frac {1}{x}} x \left (x^4\right )^{-2/x}}{\log ^3(x)} \, dx+2 \int \frac {9^{\frac {1}{x}} \left (x^4\right )^{-2/x} \log \left (\frac {x^4}{3}\right )}{\log ^2(x)} \, dx+40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x}}{\log ^2(x)} \, dx-40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x}}{\log (x)} \, dx-40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x} \log \left (\frac {x^4}{3}\right )}{x \log (x)} \, dx+160 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x}}{x \log (x)} \, dx\\ &=-\left (2 \int \frac {9^{\frac {1}{x}} x \left (x^4\right )^{-2/x}}{\log ^3(x)} \, dx\right )+2 \int \frac {9^{\frac {1}{x}} x \left (x^4\right )^{-2/x}}{\log ^2(x)} \, dx+2 \int \frac {9^{\frac {1}{x}} \left (x^4\right )^{-2/x} \log \left (\frac {x^4}{3}\right )}{\log ^2(x)} \, dx-8 \int \frac {9^{\frac {1}{x}} \left (x^4\right )^{-2/x}}{\log ^2(x)} \, dx+40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x}}{\log ^2(x)} \, dx-40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x}}{\log (x)} \, dx-40 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x} \log \left (\frac {x^4}{3}\right )}{x \log (x)} \, dx+160 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x}}{x \log (x)} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [F]
time = 0.45, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3^{2/x} \left (x^4\right )^{-2/x} \left (-2 x^2+\left (-8 x+2 x^2\right ) \log (x)+2 x \log (x) \log \left (\frac {x^4}{3}\right )+3^{-1/x} \left (x^4\right )^{\frac {1}{x}} \left (40 x \log (x)+(160-40 x) \log ^2(x)-40 \log ^2(x) \log \left (\frac {x^4}{3}\right )\right )\right )}{x \log ^3(x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.40, size = 42, normalized size = 1.68
method | result | size |
default | \(-\frac {40 x \,{\mathrm e}^{-\frac {\ln \left (\frac {x^{4}}{3}\right )}{x}}}{\ln \left (x \right )}+\frac {x^{2} {\mathrm e}^{-\frac {2 \ln \left (\frac {x^{4}}{3}\right )}{x}}}{\ln \left (x \right )^{2}}\) | \(42\) |
risch | \(-\frac {40 x 3^{\frac {1}{x}} x^{-\frac {4}{x}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i x^{2}\right )^{3}-2 \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )-\mathrm {csgn}\left (i x^{3}\right )^{2} \mathrm {csgn}\left (i x^{2}\right )-\mathrm {csgn}\left (i x^{3}\right )^{2} \mathrm {csgn}\left (i x \right )+\mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right ) \mathrm {csgn}\left (i x \right )-\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{4}\right )^{2}+\mathrm {csgn}\left (i x^{3}\right )^{3}-\mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right )^{2}+\mathrm {csgn}\left (i x^{4}\right )^{3}\right )}{2 x}}}{\ln \left (x \right )}+\frac {x^{2} 3^{\frac {2}{x}} x^{-\frac {8}{x}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i x^{2}\right )^{3}-2 \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )-\mathrm {csgn}\left (i x^{3}\right )^{2} \mathrm {csgn}\left (i x^{2}\right )-\mathrm {csgn}\left (i x^{3}\right )^{2} \mathrm {csgn}\left (i x \right )+\mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right ) \mathrm {csgn}\left (i x \right )-\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{4}\right )^{2}+\mathrm {csgn}\left (i x^{3}\right )^{3}-\mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right )^{2}+\mathrm {csgn}\left (i x^{4}\right )^{3}\right )}{x}}}{\ln \left (x \right )^{2}}\) | \(399\) |
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. 46 vs.
\(2 (21) = 42\).
time = 0.50, size = 46, normalized size = 1.84 \begin {gather*} \frac {x^{2} e^{\left (\frac {2 \, \log \left (3\right )}{x} - \frac {8 \, \log \left (x\right )}{x}\right )} - 40 \, x e^{\left (\frac {\log \left (3\right )}{x} - \frac {4 \, \log \left (x\right )}{x}\right )} \log \left (x\right )}{\log \left (x\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. 43 vs.
\(2 (21) = 42\).
time = 0.37, size = 43, normalized size = 1.72 \begin {gather*} -\frac {{\left (40 \, x e^{\left (-\frac {\log \left (3\right ) - 4 \, \log \left (x\right )}{x}\right )} \log \left (x\right ) - x^{2}\right )} e^{\left (\frac {2 \, {\left (\log \left (3\right ) - 4 \, \log \left (x\right )\right )}}{x}\right )}}{\log \left (x\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 44 vs.
\(2 (17) = 34\).
time = 0.15, size = 44, normalized size = 1.76 \begin {gather*} \frac {x^{2} e^{- \frac {2 \cdot \left (4 \log {\left (x \right )} - \log {\left (3 \right )}\right )}{x}} \log {\left (x \right )} - 40 x e^{- \frac {4 \log {\left (x \right )} - \log {\left (3 \right )}}{x}} \log {\left (x \right )}^{2}}{\log {\left (x \right )}^{3}} \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 = 2.53, size = 41, normalized size = 1.64 \begin {gather*} \frac {3^{1/x}\,x\,\left (3^{1/x}\,x-40\,\ln \left (x\right )\,{\left (x^4\right )}^{1/x}\right )}{{\ln \left (x\right )}^2\,{\left (x^4\right )}^{2/x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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