Optimal. Leaf size=23 \[ (1+x) \log \left (\left (3+x-\left (4-x^{\frac {1}{x}}\right )^2\right )^2\right ) \]
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Rubi [F] time = 4.78, 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 {-2 x^2-2 x^3+x^{2/x} (4+4 x+(-4-4 x) \log (x))+x^{\frac {1}{x}} (-16-16 x+(16+16 x) \log (x))+\left (13 x^2-x^3-8 x^{2+\frac {1}{x}}+x^{2+\frac {2}{x}}\right ) \log \left (169-26 x+x^2+(90-2 x) x^{2/x}-16 x^{3/x}+x^{4/x}+x^{\frac {1}{x}} (-208+16 x)\right )}{13 x^2-x^3-8 x^{2+\frac {1}{x}}+x^{2+\frac {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 {-2 x^2-2 x^3+16 x^{\frac {1}{x}} (1+x) (-1+\log (x))-4 x^{2/x} (1+x) (-1+\log (x))-x^2 \left (-13+x+8 x^{\frac {1}{x}}-x^{2/x}\right ) \log \left (\left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )^2\right )}{x^2 \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )} \, dx\\ &=\int \left (-\frac {2 (1+x) \left (26-2 x+x^2-8 x^{\frac {1}{x}}-26 \log (x)+2 x \log (x)+8 x^{\frac {1}{x}} \log (x)\right )}{x^2 \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )}+\frac {4+4 x-4 \log (x)-4 x \log (x)+x^2 \log \left (\left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )^2\right )}{x^2}\right ) \, dx\\ &=-\left (2 \int \frac {(1+x) \left (26-2 x+x^2-8 x^{\frac {1}{x}}-26 \log (x)+2 x \log (x)+8 x^{\frac {1}{x}} \log (x)\right )}{x^2 \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )} \, dx\right )+\int \frac {4+4 x-4 \log (x)-4 x \log (x)+x^2 \log \left (\left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )^2\right )}{x^2} \, dx\\ &=-\left (2 \int \left (\frac {26-2 x+x^2-8 x^{\frac {1}{x}}-26 \log (x)+2 x \log (x)+8 x^{\frac {1}{x}} \log (x)}{x^2 \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )}+\frac {26-2 x+x^2-8 x^{\frac {1}{x}}-26 \log (x)+2 x \log (x)+8 x^{\frac {1}{x}} \log (x)}{x \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )}\right ) \, dx\right )+\int \left (-\frac {4 (1+x) (-1+\log (x))}{x^2}+\log \left (\left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )^2\right )\right ) \, dx\\ &=-\left (2 \int \frac {26-2 x+x^2-8 x^{\frac {1}{x}}-26 \log (x)+2 x \log (x)+8 x^{\frac {1}{x}} \log (x)}{x^2 \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )} \, dx\right )-2 \int \frac {26-2 x+x^2-8 x^{\frac {1}{x}}-26 \log (x)+2 x \log (x)+8 x^{\frac {1}{x}} \log (x)}{x \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )} \, dx-4 \int \frac {(1+x) (-1+\log (x))}{x^2} \, dx+\int \log \left (\left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )^2\right ) \, dx\\ &=-4 (1-\log (x)) \left (\frac {1}{x}-\log (x)\right )+x \log \left (\left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )^2\right )-2 \int \left (\frac {1}{13-x-8 x^{\frac {1}{x}}+x^{2/x}}+\frac {26}{x^2 \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )}-\frac {2}{x \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )}-\frac {8 x^{-2+\frac {1}{x}}}{13-x-8 x^{\frac {1}{x}}+x^{2/x}}-\frac {26 \log (x)}{x^2 \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )}+\frac {2 \log (x)}{x \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )}+\frac {8 x^{-2+\frac {1}{x}} \log (x)}{13-x-8 x^{\frac {1}{x}}+x^{2/x}}\right ) \, dx-2 \int \left (-\frac {2}{13-x-8 x^{\frac {1}{x}}+x^{2/x}}+\frac {26}{x \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )}+\frac {x}{13-x-8 x^{\frac {1}{x}}+x^{2/x}}-\frac {8 x^{-1+\frac {1}{x}}}{13-x-8 x^{\frac {1}{x}}+x^{2/x}}+\frac {2 \log (x)}{13-x-8 x^{\frac {1}{x}}+x^{2/x}}-\frac {26 \log (x)}{x \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )}+\frac {8 x^{-1+\frac {1}{x}} \log (x)}{13-x-8 x^{\frac {1}{x}}+x^{2/x}}\right ) \, dx+4 \int \frac {-1+x \log (x)}{x^2} \, dx-\int \frac {2 \left (-x^2-8 x^{\frac {1}{x}}+2 x^{2/x}-2 x^{\frac {1}{x}} \left (-4+x^{\frac {1}{x}}\right ) \log (x)\right )}{x \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )} \, dx\\ &=-4 (1-\log (x)) \left (\frac {1}{x}-\log (x)\right )+x \log \left (\left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )^2\right )-2 \int \frac {1}{13-x-8 x^{\frac {1}{x}}+x^{2/x}} \, dx-2 \int \frac {x}{13-x-8 x^{\frac {1}{x}}+x^{2/x}} \, dx-2 \int \frac {-x^2-8 x^{\frac {1}{x}}+2 x^{2/x}-2 x^{\frac {1}{x}} \left (-4+x^{\frac {1}{x}}\right ) \log (x)}{x \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )} \, dx+4 \int \frac {1}{13-x-8 x^{\frac {1}{x}}+x^{2/x}} \, dx+4 \int \frac {1}{x \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )} \, dx-4 \int \frac {\log (x)}{13-x-8 x^{\frac {1}{x}}+x^{2/x}} \, dx-4 \int \frac {\log (x)}{x \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )} \, dx+4 \int \left (-\frac {1}{x^2}+\frac {\log (x)}{x}\right ) \, dx+16 \int \frac {x^{-2+\frac {1}{x}}}{13-x-8 x^{\frac {1}{x}}+x^{2/x}} \, dx+16 \int \frac {x^{-1+\frac {1}{x}}}{13-x-8 x^{\frac {1}{x}}+x^{2/x}} \, dx-16 \int \frac {x^{-2+\frac {1}{x}} \log (x)}{13-x-8 x^{\frac {1}{x}}+x^{2/x}} \, dx-16 \int \frac {x^{-1+\frac {1}{x}} \log (x)}{13-x-8 x^{\frac {1}{x}}+x^{2/x}} \, dx-52 \int \frac {1}{x^2 \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )} \, dx-52 \int \frac {1}{x \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )} \, dx+52 \int \frac {\log (x)}{x^2 \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )} \, dx+52 \int \frac {\log (x)}{x \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )} \, dx\\ &=\frac {4}{x}-4 (1-\log (x)) \left (\frac {1}{x}-\log (x)\right )+x \log \left (\left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )^2\right )-2 \int \frac {1}{13-x-8 x^{\frac {1}{x}}+x^{2/x}} \, dx-2 \int \frac {x}{13-x-8 x^{\frac {1}{x}}+x^{2/x}} \, dx-2 \int \left (-\frac {2 (-1+\log (x))}{x}-\frac {26-2 x+x^2-8 x^{\frac {1}{x}}-26 \log (x)+2 x \log (x)+8 x^{\frac {1}{x}} \log (x)}{x \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )}\right ) \, dx+4 \int \frac {1}{13-x-8 x^{\frac {1}{x}}+x^{2/x}} \, dx+4 \int \frac {1}{x \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )} \, dx+4 \int \frac {\log (x)}{x} \, dx+4 \int \frac {\int \frac {1}{13-x-8 x^{\frac {1}{x}}+x^{2/x}} \, dx}{x} \, dx+4 \int \frac {\int \frac {1}{x \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )} \, dx}{x} \, dx+16 \int \frac {x^{-2+\frac {1}{x}}}{13-x-8 x^{\frac {1}{x}}+x^{2/x}} \, dx+16 \int \frac {x^{-1+\frac {1}{x}}}{13-x-8 x^{\frac {1}{x}}+x^{2/x}} \, dx+16 \int \frac {\int \frac {x^{-2+\frac {1}{x}}}{13-x-8 x^{\frac {1}{x}}+x^{2/x}} \, dx}{x} \, dx+16 \int \frac {\int \frac {x^{-1+\frac {1}{x}}}{13-x-8 x^{\frac {1}{x}}+x^{2/x}} \, dx}{x} \, dx-52 \int \frac {1}{x^2 \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )} \, dx-52 \int \frac {1}{x \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )} \, dx-52 \int \frac {\int \frac {1}{x^2 \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )} \, dx}{x} \, dx-52 \int \frac {\int \frac {1}{x \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )} \, dx}{x} \, dx-(4 \log (x)) \int \frac {1}{13-x-8 x^{\frac {1}{x}}+x^{2/x}} \, dx-(4 \log (x)) \int \frac {1}{x \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )} \, dx-(16 \log (x)) \int \frac {x^{-2+\frac {1}{x}}}{13-x-8 x^{\frac {1}{x}}+x^{2/x}} \, dx-(16 \log (x)) \int \frac {x^{-1+\frac {1}{x}}}{13-x-8 x^{\frac {1}{x}}+x^{2/x}} \, dx+(52 \log (x)) \int \frac {1}{x^2 \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )} \, dx+(52 \log (x)) \int \frac {1}{x \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )} \, dx\\ &=\frac {4}{x}-4 (1-\log (x)) \left (\frac {1}{x}-\log (x)\right )+2 \log ^2(x)+x \log \left (\left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )^2\right )-2 \int \frac {1}{13-x-8 x^{\frac {1}{x}}+x^{2/x}} \, dx-2 \int \frac {x}{13-x-8 x^{\frac {1}{x}}+x^{2/x}} \, dx+2 \int \frac {26-2 x+x^2-8 x^{\frac {1}{x}}-26 \log (x)+2 x \log (x)+8 x^{\frac {1}{x}} \log (x)}{x \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )} \, dx+4 \int \frac {1}{13-x-8 x^{\frac {1}{x}}+x^{2/x}} \, dx+4 \int \frac {1}{x \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )} \, dx+4 \int \frac {-1+\log (x)}{x} \, dx+4 \int \frac {\int \frac {1}{13-x-8 x^{\frac {1}{x}}+x^{2/x}} \, dx}{x} \, dx+4 \int \frac {\int \frac {1}{x \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )} \, dx}{x} \, dx+16 \int \frac {x^{-2+\frac {1}{x}}}{13-x-8 x^{\frac {1}{x}}+x^{2/x}} \, dx+16 \int \frac {x^{-1+\frac {1}{x}}}{13-x-8 x^{\frac {1}{x}}+x^{2/x}} \, dx+16 \int \frac {\int \frac {x^{-2+\frac {1}{x}}}{13-x-8 x^{\frac {1}{x}}+x^{2/x}} \, dx}{x} \, dx+16 \int \frac {\int \frac {x^{-1+\frac {1}{x}}}{13-x-8 x^{\frac {1}{x}}+x^{2/x}} \, dx}{x} \, dx-52 \int \frac {1}{x^2 \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )} \, dx-52 \int \frac {1}{x \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )} \, dx-52 \int \frac {\int \frac {1}{x^2 \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )} \, dx}{x} \, dx-52 \int \frac {\int \frac {1}{x \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )} \, dx}{x} \, dx-(4 \log (x)) \int \frac {1}{13-x-8 x^{\frac {1}{x}}+x^{2/x}} \, dx-(4 \log (x)) \int \frac {1}{x \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )} \, dx-(16 \log (x)) \int \frac {x^{-2+\frac {1}{x}}}{13-x-8 x^{\frac {1}{x}}+x^{2/x}} \, dx-(16 \log (x)) \int \frac {x^{-1+\frac {1}{x}}}{13-x-8 x^{\frac {1}{x}}+x^{2/x}} \, dx+(52 \log (x)) \int \frac {1}{x^2 \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )} \, dx+(52 \log (x)) \int \frac {1}{x \left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [B] time = 0.50, size = 47, normalized size = 2.04 \begin {gather*} 2 \log \left (-13+x+8 x^{\frac {1}{x}}-x^{2/x}\right )+x \log \left (\left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )^2\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.56, size = 51, normalized size = 2.22 \begin {gather*} {\left (x + 1\right )} \log \left (-2 \, {\left (x - 45\right )} x^{\frac {2}{x}} + 16 \, {\left (x - 13\right )} x^{\left (\frac {1}{x}\right )} + x^{2} + x^{\frac {4}{x}} - 16 \, x^{\frac {3}{x}} - 26 \, x + 169\right ) \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*} \mathit {undef} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.21, size = 189, normalized size = 8.22
method | result | size |
risch | \(2 x \ln \left (-x^{\frac {2}{x}}+8 x^{\frac {1}{x}}+x -13\right )-\frac {i \pi x \mathrm {csgn}\left (i \left (-x^{\frac {2}{x}}+8 x^{\frac {1}{x}}+x -13\right )\right )^{2} \mathrm {csgn}\left (i \left (-x^{\frac {2}{x}}+8 x^{\frac {1}{x}}+x -13\right )^{2}\right )}{2}+i \pi x \,\mathrm {csgn}\left (i \left (-x^{\frac {2}{x}}+8 x^{\frac {1}{x}}+x -13\right )\right ) \mathrm {csgn}\left (i \left (-x^{\frac {2}{x}}+8 x^{\frac {1}{x}}+x -13\right )^{2}\right )^{2}-\frac {i \pi x \mathrm {csgn}\left (i \left (-x^{\frac {2}{x}}+8 x^{\frac {1}{x}}+x -13\right )^{2}\right )^{3}}{2}+2 \ln \left (x^{\frac {2}{x}}-8 x^{\frac {1}{x}}-x +13\right )\) | \(189\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 25, normalized size = 1.09 \begin {gather*} 2 \, {\left (x + 1\right )} \log \left (x^{\frac {2}{x}} - x - 8 \, x^{\left (\frac {1}{x}\right )} + 13\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.48, size = 84, normalized size = 3.65 \begin {gather*} 2\,\ln \left (x-x^{2/x}+8\,x^{1/x}-13\right )+x\,\ln \left (90\,x^{2/x}-26\,x-16\,x^{3/x}+x^{4/x}+16\,x\,x^{1/x}-208\,x^{1/x}+x^2-2\,x\,x^{2/x}+169\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.45, size = 78, normalized size = 3.39 \begin {gather*} x \log {\left (x^{2} - 26 x + \left (90 - 2 x\right ) e^{\frac {2 \log {\relax (x )}}{x}} + \left (16 x - 208\right ) e^{\frac {\log {\relax (x )}}{x}} + e^{\frac {4 \log {\relax (x )}}{x}} - 16 e^{\frac {3 \log {\relax (x )}}{x}} + 169 \right )} + 2 \log {\left (- x + e^{\frac {2 \log {\relax (x )}}{x}} - 8 e^{\frac {\log {\relax (x )}}{x}} + 13 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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