Optimal. Leaf size=30 \[ (5 (-x+x (-4+2 x))+\log (x)) \left (x (1+x)^2+\log (2+\log (x))\right ) \]
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Rubi [F]
time = 1.24, 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 {-23 x-86 x^2-238 x^3-40 x^4+100 x^5+\left (1+3 x-40 x^2-113 x^3-20 x^4+50 x^5\right ) \log (x)+\left (x+4 x^2+3 x^3\right ) \log ^2(x)+\left (2-50 x+40 x^2+\left (1-25 x+20 x^2\right ) \log (x)\right ) \log (2+\log (x))}{2 x+x \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 {-23 x-86 x^2-238 x^3-40 x^4+100 x^5+\left (1+3 x-40 x^2-113 x^3-20 x^4+50 x^5\right ) \log (x)+\left (x+4 x^2+3 x^3\right ) \log ^2(x)+\left (2-50 x+40 x^2+\left (1-25 x+20 x^2\right ) \log (x)\right ) \log (2+\log (x))}{x (2+\log (x))} \, dx\\ &=\int \left (-\frac {23}{2+\log (x)}-\frac {86 x}{2+\log (x)}-\frac {238 x^2}{2+\log (x)}-\frac {40 x^3}{2+\log (x)}+\frac {100 x^4}{2+\log (x)}+\frac {\left (1+3 x-40 x^2-113 x^3-20 x^4+50 x^5\right ) \log (x)}{x (2+\log (x))}+\frac {(1+x) (1+3 x) \log ^2(x)}{2+\log (x)}+\frac {\left (1-25 x+20 x^2\right ) \log (2+\log (x))}{x}\right ) \, dx\\ &=-\left (23 \int \frac {1}{2+\log (x)} \, dx\right )-40 \int \frac {x^3}{2+\log (x)} \, dx-86 \int \frac {x}{2+\log (x)} \, dx+100 \int \frac {x^4}{2+\log (x)} \, dx-238 \int \frac {x^2}{2+\log (x)} \, dx+\int \frac {\left (1+3 x-40 x^2-113 x^3-20 x^4+50 x^5\right ) \log (x)}{x (2+\log (x))} \, dx+\int \frac {(1+x) (1+3 x) \log ^2(x)}{2+\log (x)} \, dx+\int \frac {\left (1-25 x+20 x^2\right ) \log (2+\log (x))}{x} \, dx\\ &=-\left (23 \text {Subst}\left (\int \frac {e^x}{2+x} \, dx,x,\log (x)\right )\right )-40 \text {Subst}\left (\int \frac {e^{4 x}}{2+x} \, dx,x,\log (x)\right )-86 \text {Subst}\left (\int \frac {e^{2 x}}{2+x} \, dx,x,\log (x)\right )+100 \text {Subst}\left (\int \frac {e^{5 x}}{2+x} \, dx,x,\log (x)\right )-238 \text {Subst}\left (\int \frac {e^{3 x}}{2+x} \, dx,x,\log (x)\right )+\int \left (-2 (1+x) (1+3 x)+(1+x) (1+3 x) \log (x)+\frac {4 (1+x) (1+3 x)}{2+\log (x)}\right ) \, dx+\int \left (\frac {1+3 x-40 x^2-113 x^3-20 x^4+50 x^5}{x}-\frac {2 \left (1+3 x-40 x^2-113 x^3-20 x^4+50 x^5\right )}{x (2+\log (x))}\right ) \, dx+\int \left (-25 \log (2+\log (x))+\frac {\log (2+\log (x))}{x}+20 x \log (2+\log (x))\right ) \, dx\\ &=-\frac {23 \text {Ei}(2+\log (x))}{e^2}-\frac {86 \text {Ei}(2 (2+\log (x)))}{e^4}-\frac {238 \text {Ei}(3 (2+\log (x)))}{e^6}-\frac {40 \text {Ei}(4 (2+\log (x)))}{e^8}+\frac {100 \text {Ei}(5 (2+\log (x)))}{e^{10}}-2 \int (1+x) (1+3 x) \, dx-2 \int \frac {1+3 x-40 x^2-113 x^3-20 x^4+50 x^5}{x (2+\log (x))} \, dx+4 \int \frac {(1+x) (1+3 x)}{2+\log (x)} \, dx+20 \int x \log (2+\log (x)) \, dx-25 \int \log (2+\log (x)) \, dx+\int \frac {1+3 x-40 x^2-113 x^3-20 x^4+50 x^5}{x} \, dx+\int (1+x) (1+3 x) \log (x) \, dx+\int \frac {\log (2+\log (x))}{x} \, dx\\ &=-2 x (1+x)^2-\frac {23 \text {Ei}(2+\log (x))}{e^2}-\frac {86 \text {Ei}(2 (2+\log (x)))}{e^4}-\frac {238 \text {Ei}(3 (2+\log (x)))}{e^6}-\frac {40 \text {Ei}(4 (2+\log (x)))}{e^8}+\frac {100 \text {Ei}(5 (2+\log (x)))}{e^{10}}-2 \int \frac {1+3 x-40 x^2-113 x^3-20 x^4+50 x^5}{x (2+\log (x))} \, dx+4 \int \left (\frac {1}{2+\log (x)}+\frac {4 x}{2+\log (x)}+\frac {3 x^2}{2+\log (x)}\right ) \, dx+20 \int x \log (2+\log (x)) \, dx-25 \int \log (2+\log (x)) \, dx+\int \left (3+\frac {1}{x}-40 x-113 x^2-20 x^3+50 x^4\right ) \, dx+\int \left (\log (x)+4 x \log (x)+3 x^2 \log (x)\right ) \, dx+\text {Subst}(\int \log (2+x) \, dx,x,\log (x))\\ &=3 x-20 x^2-\frac {113 x^3}{3}-5 x^4+10 x^5-2 x (1+x)^2-\frac {23 \text {Ei}(2+\log (x))}{e^2}-\frac {86 \text {Ei}(2 (2+\log (x)))}{e^4}-\frac {238 \text {Ei}(3 (2+\log (x)))}{e^6}-\frac {40 \text {Ei}(4 (2+\log (x)))}{e^8}+\frac {100 \text {Ei}(5 (2+\log (x)))}{e^{10}}+\log (x)-2 \int \frac {1+3 x-40 x^2-113 x^3-20 x^4+50 x^5}{x (2+\log (x))} \, dx+3 \int x^2 \log (x) \, dx+4 \int x \log (x) \, dx+4 \int \frac {1}{2+\log (x)} \, dx+12 \int \frac {x^2}{2+\log (x)} \, dx+16 \int \frac {x}{2+\log (x)} \, dx+20 \int x \log (2+\log (x)) \, dx-25 \int \log (2+\log (x)) \, dx+\int \log (x) \, dx+\text {Subst}(\int \log (x) \, dx,x,2+\log (x))\\ &=2 x-21 x^2-38 x^3-5 x^4+10 x^5-2 x (1+x)^2-\frac {23 \text {Ei}(2+\log (x))}{e^2}-\frac {86 \text {Ei}(2 (2+\log (x)))}{e^4}-\frac {238 \text {Ei}(3 (2+\log (x)))}{e^6}-\frac {40 \text {Ei}(4 (2+\log (x)))}{e^8}+\frac {100 \text {Ei}(5 (2+\log (x)))}{e^{10}}+x \log (x)+2 x^2 \log (x)+x^3 \log (x)+(2+\log (x)) \log (2+\log (x))-2 \int \frac {1+3 x-40 x^2-113 x^3-20 x^4+50 x^5}{x (2+\log (x))} \, dx+4 \text {Subst}\left (\int \frac {e^x}{2+x} \, dx,x,\log (x)\right )+12 \text {Subst}\left (\int \frac {e^{3 x}}{2+x} \, dx,x,\log (x)\right )+16 \text {Subst}\left (\int \frac {e^{2 x}}{2+x} \, dx,x,\log (x)\right )+20 \int x \log (2+\log (x)) \, dx-25 \int \log (2+\log (x)) \, dx\\ &=2 x-21 x^2-38 x^3-5 x^4+10 x^5-2 x (1+x)^2-\frac {19 \text {Ei}(2+\log (x))}{e^2}-\frac {70 \text {Ei}(2 (2+\log (x)))}{e^4}-\frac {226 \text {Ei}(3 (2+\log (x)))}{e^6}-\frac {40 \text {Ei}(4 (2+\log (x)))}{e^8}+\frac {100 \text {Ei}(5 (2+\log (x)))}{e^{10}}+x \log (x)+2 x^2 \log (x)+x^3 \log (x)+(2+\log (x)) \log (2+\log (x))-2 \int \frac {1+3 x-40 x^2-113 x^3-20 x^4+50 x^5}{x (2+\log (x))} \, dx+20 \int x \log (2+\log (x)) \, dx-25 \int \log (2+\log (x)) \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.04, size = 25, normalized size = 0.83 \begin {gather*} (5 x (-5+2 x)+\log (x)) \left (x (1+x)^2+\log (2+\log (x))\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 7.82, size = 56, normalized size = 1.87
method | result | size |
default | \(x \ln \left (x \right )+2 x^{2} \ln \left (x \right )-25 x^{2}+x^{3} \ln \left (x \right )-40 x^{3}-5 x^{4}+10 x^{5}+\left (10 x^{2}+\ln \left (x \right )-25 x \right ) \ln \left (\ln \left (x \right )+2\right )\) | \(56\) |
risch | \(x \ln \left (x \right )+2 x^{2} \ln \left (x \right )-25 x^{2}+x^{3} \ln \left (x \right )-40 x^{3}-5 x^{4}+10 x^{5}+\left (10 x^{2}+\ln \left (x \right )-25 x \right ) \ln \left (\ln \left (x \right )+2\right )\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 51, normalized size = 1.70 \begin {gather*} 10 \, x^{5} - 5 \, x^{4} - 40 \, x^{3} - 25 \, x^{2} + {\left (x^{3} + 2 \, x^{2} + x\right )} \log \left (x\right ) + {\left (10 \, x^{2} - 25 \, x + \log \left (x\right )\right )} \log \left (\log \left (x\right ) + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 51, normalized size = 1.70 \begin {gather*} 10 \, x^{5} - 5 \, x^{4} - 40 \, x^{3} - 25 \, x^{2} + {\left (x^{3} + 2 \, x^{2} + x\right )} \log \left (x\right ) + {\left (10 \, x^{2} - 25 \, x + \log \left (x\right )\right )} \log \left (\log \left (x\right ) + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.14, size = 51, normalized size = 1.70 \begin {gather*} 10 x^{5} - 5 x^{4} - 40 x^{3} - 25 x^{2} + \left (10 x^{2} - 25 x + \log {\left (x \right )}\right ) \log {\left (\log {\left (x \right )} + 2 \right )} + \left (x^{3} + 2 x^{2} + x\right ) \log {\left (x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 51, normalized size = 1.70 \begin {gather*} 10 \, x^{5} - 5 \, x^{4} - 40 \, x^{3} - 25 \, x^{2} + {\left (x^{3} + 2 \, x^{2} + x\right )} \log \left (x\right ) + {\left (10 \, x^{2} - 25 \, x + \log \left (x\right )\right )} \log \left (\log \left (x\right ) + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.62, size = 51, normalized size = 1.70 \begin {gather*} \ln \left (x\right )\,\left (x^3+2\,x^2+x\right )+\ln \left (\ln \left (x\right )+2\right )\,\left (\ln \left (x\right )-25\,x+10\,x^2\right )-25\,x^2-40\,x^3-5\,x^4+10\,x^5 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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