3.11.90 \(\int \frac {(x \log (x))^{x^2} (x+x \log (x)+2 x \log (x) \log (x \log (x)))}{4 \log (x)} \, dx\)

Optimal. Leaf size=20 \[ e^{\sqrt [3]{2}}+\frac {1}{4} (x \log (x))^{x^2} \]

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Rubi [F]  time = 0.38, 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 {(x \log (x))^{x^2} (x+x \log (x)+2 x \log (x) \log (x \log (x)))}{4 \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((x*Log[x])^x^2*(x + x*Log[x] + 2*x*Log[x]*Log[x*Log[x]]))/(4*Log[x]),x]

[Out]

Defer[Int][x*(x*Log[x])^x^2, x]/4 + Defer[Int][(x*(x*Log[x])^x^2)/Log[x], x]/4 + Defer[Int][x*(x*Log[x])^x^2*L
og[x*Log[x]], x]/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \frac {(x \log (x))^{x^2} (x+x \log (x)+2 x \log (x) \log (x \log (x)))}{\log (x)} \, dx\\ &=\frac {1}{4} \int \left (\frac {x (x \log (x))^{x^2} (1+\log (x))}{\log (x)}+2 x (x \log (x))^{x^2} \log (x \log (x))\right ) \, dx\\ &=\frac {1}{4} \int \frac {x (x \log (x))^{x^2} (1+\log (x))}{\log (x)} \, dx+\frac {1}{2} \int x (x \log (x))^{x^2} \log (x \log (x)) \, dx\\ &=\frac {1}{4} \int \left (x (x \log (x))^{x^2}+\frac {x (x \log (x))^{x^2}}{\log (x)}\right ) \, dx+\frac {1}{2} \int x (x \log (x))^{x^2} \log (x \log (x)) \, dx\\ &=\frac {1}{4} \int x (x \log (x))^{x^2} \, dx+\frac {1}{4} \int \frac {x (x \log (x))^{x^2}}{\log (x)} \, dx+\frac {1}{2} \int x (x \log (x))^{x^2} \log (x \log (x)) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 12, normalized size = 0.60 \begin {gather*} \frac {1}{4} (x \log (x))^{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((x*Log[x])^x^2*(x + x*Log[x] + 2*x*Log[x]*Log[x*Log[x]]))/(4*Log[x]),x]

[Out]

(x*Log[x])^x^2/4

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fricas [A]  time = 0.51, size = 15, normalized size = 0.75 \begin {gather*} e^{\left (x^{2} \log \left (x \log \relax (x)\right ) - 2 \, \log \relax (2)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*log(x)*log(x*log(x))+x*log(x)+x)*exp(x^2*log(x*log(x))-2*log(2))/log(x),x, algorithm="fricas")

[Out]

e^(x^2*log(x*log(x)) - 2*log(2))

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giac [A]  time = 0.35, size = 15, normalized size = 0.75 \begin {gather*} e^{\left (x^{2} \log \left (x \log \relax (x)\right ) - 2 \, \log \relax (2)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*log(x)*log(x*log(x))+x*log(x)+x)*exp(x^2*log(x*log(x))-2*log(2))/log(x),x, algorithm="giac")

[Out]

e^(x^2*log(x*log(x)) - 2*log(2))

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maple [C]  time = 0.06, size = 91, normalized size = 4.55

method	result	size
risch	\(\frac {{\mathrm e}^{\frac {x^{2} \left (-i \pi \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3}+i \pi \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i x \right )+i \pi \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i \ln \relax (x )\right )-i \pi \,\mathrm {csgn}\left (i x \ln \relax (x )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right )+2 \ln \relax (x )+2 \ln \left (\ln \relax (x )\right )\right )}{2}}}{4}\)	\(91\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x*ln(x)*ln(x*ln(x))+x*ln(x)+x)*exp(x^2*ln(x*ln(x))-2*ln(2))/ln(x),x,method=_RETURNVERBOSE)

[Out]

1/4*exp(1/2*x^2*(-I*Pi*csgn(I*x*ln(x))^3+I*Pi*csgn(I*x*ln(x))^2*csgn(I*x)+I*Pi*csgn(I*x*ln(x))^2*csgn(I*ln(x))
-I*Pi*csgn(I*x*ln(x))*csgn(I*x)*csgn(I*ln(x))+2*ln(x)+2*ln(ln(x))))

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maxima [A]  time = 0.54, size = 17, normalized size = 0.85 \begin {gather*} \frac {1}{4} \, e^{\left (x^{2} \log \relax (x) + x^{2} \log \left (\log \relax (x)\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*log(x)*log(x*log(x))+x*log(x)+x)*exp(x^2*log(x*log(x))-2*log(2))/log(x),x, algorithm="maxima")

[Out]

1/4*e^(x^2*log(x) + x^2*log(log(x)))

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mupad [B]  time = 0.81, size = 10, normalized size = 0.50 \begin {gather*} \frac {{\left (x\,\ln \relax (x)\right )}^{x^2}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x^2*log(x*log(x)) - 2*log(2))*(x + x*log(x) + 2*x*log(x*log(x))*log(x)))/log(x),x)

[Out]

(x*log(x))^(x^2)/4

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sympy [A]  time = 0.46, size = 12, normalized size = 0.60 \begin {gather*} \frac {e^{x^{2} \log {\left (x \log {\relax (x )} \right )}}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*ln(x)*ln(x*ln(x))+x*ln(x)+x)*exp(x**2*ln(x*ln(x))-2*ln(2))/ln(x),x)

[Out]

exp(x**2*log(x*log(x)))/4

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