3.27.62 \(\int \frac {-e^4-x+(-e^4-x) \log (x)+e^4 \log (x) \log (2 x \log (x))+((-e^8-2 e^4 x-x^2) \log (x)+x^x ((e^8+2 e^4 x+x^2) \log (x)+(e^8+2 e^4 x+x^2) \log ^2(x))) \log ^2(2 x \log (x))}{(e^8+2 e^4 x+x^2) \log (x) \log ^2(2 x \log (x))} \, dx\) [2662]

Optimal. Leaf size=25 \[ -4-x+x^x+\frac {x}{\left (e^4+x\right ) \log (2 x \log (x))} \]

[Out]

exp(x*ln(x))-4+1/(x+exp(4))*x/ln(2*x*ln(x))-x

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Rubi [F]
time = 1.64, 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 {-e^4-x+\left (-e^4-x\right ) \log (x)+e^4 \log (x) \log (2 x \log (x))+\left (\left (-e^8-2 e^4 x-x^2\right ) \log (x)+x^x \left (\left (e^8+2 e^4 x+x^2\right ) \log (x)+\left (e^8+2 e^4 x+x^2\right ) \log ^2(x)\right )\right ) \log ^2(2 x \log (x))}{\left (e^8+2 e^4 x+x^2\right ) \log (x) \log ^2(2 x \log (x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-E^4 - x + (-E^4 - x)*Log[x] + E^4*Log[x]*Log[2*x*Log[x]] + ((-E^8 - 2*E^4*x - x^2)*Log[x] + x^x*((E^8 +
2*E^4*x + x^2)*Log[x] + (E^8 + 2*E^4*x + x^2)*Log[x]^2))*Log[2*x*Log[x]]^2)/((E^8 + 2*E^4*x + x^2)*Log[x]*Log[
2*x*Log[x]]^2),x]

[Out]

-x + x^x - Defer[Int][1/((E^4 + x)*Log[2*x*Log[x]]^2), x] - Defer[Int][1/((E^4 + x)*Log[x]*Log[2*x*Log[x]]^2),
x] + E^4*Defer[Int][1/((E^4 + x)^2*Log[2*x*Log[x]]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-e^4-x+\left (-e^4-x\right ) \log (x)+e^4 \log (x) \log (2 x \log (x))+\left (\left (-e^8-2 e^4 x-x^2\right ) \log (x)+x^x \left (\left (e^8+2 e^4 x+x^2\right ) \log (x)+\left (e^8+2 e^4 x+x^2\right ) \log ^2(x)\right )\right ) \log ^2(2 x \log (x))}{\left (e^4+x\right )^2 \log (x) \log ^2(2 x \log (x))} \, dx\\ &=\int \frac {-e^4-x+x^x \left (e^4+x\right )^2 \log ^2(x) \log ^2(2 x \log (x))+\log (x) \left (-e^4-x+e^4 \log (2 x \log (x))+\left (e^4+x\right )^2 \left (-1+x^x\right ) \log ^2(2 x \log (x))\right )}{\left (e^4+x\right )^2 \log (x) \log ^2(2 x \log (x))} \, dx\\ &=\int \left (-1+x^x (1+\log (x))-\frac {e^4}{\left (e^4+x\right )^2 \log ^2(2 x \log (x))}-\frac {x}{\left (e^4+x\right )^2 \log ^2(2 x \log (x))}-\frac {e^4}{\left (e^4+x\right )^2 \log (x) \log ^2(2 x \log (x))}-\frac {x}{\left (e^4+x\right )^2 \log (x) \log ^2(2 x \log (x))}+\frac {e^4}{\left (e^4+x\right )^2 \log (2 x \log (x))}\right ) \, dx\\ &=-x-e^4 \int \frac {1}{\left (e^4+x\right )^2 \log ^2(2 x \log (x))} \, dx-e^4 \int \frac {1}{\left (e^4+x\right )^2 \log (x) \log ^2(2 x \log (x))} \, dx+e^4 \int \frac {1}{\left (e^4+x\right )^2 \log (2 x \log (x))} \, dx+\int x^x (1+\log (x)) \, dx-\int \frac {x}{\left (e^4+x\right )^2 \log ^2(2 x \log (x))} \, dx-\int \frac {x}{\left (e^4+x\right )^2 \log (x) \log ^2(2 x \log (x))} \, dx\\ &=-x-e^4 \int \frac {1}{\left (e^4+x\right )^2 \log ^2(2 x \log (x))} \, dx-e^4 \int \frac {1}{\left (e^4+x\right )^2 \log (x) \log ^2(2 x \log (x))} \, dx+e^4 \int \frac {1}{\left (e^4+x\right )^2 \log (2 x \log (x))} \, dx+\int \left (x^x+x^x \log (x)\right ) \, dx-\int \left (-\frac {e^4}{\left (e^4+x\right )^2 \log ^2(2 x \log (x))}+\frac {1}{\left (e^4+x\right ) \log ^2(2 x \log (x))}\right ) \, dx-\int \left (-\frac {e^4}{\left (e^4+x\right )^2 \log (x) \log ^2(2 x \log (x))}+\frac {1}{\left (e^4+x\right ) \log (x) \log ^2(2 x \log (x))}\right ) \, dx\\ &=-x+e^4 \int \frac {1}{\left (e^4+x\right )^2 \log (2 x \log (x))} \, dx+\int x^x \, dx+\int x^x \log (x) \, dx-\int \frac {1}{\left (e^4+x\right ) \log ^2(2 x \log (x))} \, dx-\int \frac {1}{\left (e^4+x\right ) \log (x) \log ^2(2 x \log (x))} \, dx\\ &=-x+x^x+e^4 \int \frac {1}{\left (e^4+x\right )^2 \log (2 x \log (x))} \, dx-\int \frac {1}{\left (e^4+x\right ) \log ^2(2 x \log (x))} \, dx-\int \frac {1}{\left (e^4+x\right ) \log (x) \log ^2(2 x \log (x))} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.10, size = 24, normalized size = 0.96 \begin {gather*} -x+x^x+\frac {x}{\left (e^4+x\right ) \log (2 x \log (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-E^4 - x + (-E^4 - x)*Log[x] + E^4*Log[x]*Log[2*x*Log[x]] + ((-E^8 - 2*E^4*x - x^2)*Log[x] + x^x*((
E^8 + 2*E^4*x + x^2)*Log[x] + (E^8 + 2*E^4*x + x^2)*Log[x]^2))*Log[2*x*Log[x]]^2)/((E^8 + 2*E^4*x + x^2)*Log[x
]*Log[2*x*Log[x]]^2),x]

[Out]

-x + x^x + x/((E^4 + x)*Log[2*x*Log[x]])

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 29.27, size = 105, normalized size = 4.20

method	result	size
risch	\(x^{x}-x +\frac {2 x}{\left (x +{\mathrm e}^{4}\right ) \left (2 \ln \left (2\right )+2 \ln \left (\ln \left (x \right )\right )+2 \ln \left (x \right )-i \pi \,\mathrm {csgn}\left (i \ln \left (x \right )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \left (x \right )\right )+i \pi \,\mathrm {csgn}\left (i \ln \left (x \right )\right ) \mathrm {csgn}\left (i x \ln \left (x \right )\right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \left (x \right )\right )^{2}-i \pi \mathrm {csgn}\left (i x \ln \left (x \right )\right )^{3}\right )}\)	\(105\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((((exp(4)^2+2*x*exp(4)+x^2)*ln(x)^2+(exp(4)^2+2*x*exp(4)+x^2)*ln(x))*exp(x*ln(x))+(-exp(4)^2-2*x*exp(4)-x
^2)*ln(x))*ln(2*x*ln(x))^2+exp(4)*ln(x)*ln(2*x*ln(x))+(-exp(4)-x)*ln(x)-exp(4)-x)/(exp(4)^2+2*x*exp(4)+x^2)/ln
(x)/ln(2*x*ln(x))^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (24) = 48\).
time = 0.52, size = 98, normalized size = 3.92 \begin {gather*} -\frac {x^{2} \log \left (2\right ) - {\left (x \log \left (2\right ) + e^{4} \log \left (2\right ) + {\left (x + e^{4}\right )} \log \left (x\right ) + {\left (x + e^{4}\right )} \log \left (\log \left (x\right )\right )\right )} x^{x} + {\left (e^{4} \log \left (2\right ) - 1\right )} x + {\left (x^{2} + x e^{4}\right )} \log \left (x\right ) + {\left (x^{2} + x e^{4}\right )} \log \left (\log \left (x\right )\right )}{x \log \left (2\right ) + e^{4} \log \left (2\right ) + {\left (x + e^{4}\right )} \log \left (x\right ) + {\left (x + e^{4}\right )} \log \left (\log \left (x\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((exp(4)^2+2*x*exp(4)+x^2)*log(x)^2+(exp(4)^2+2*x*exp(4)+x^2)*log(x))*exp(x*log(x))+(-exp(4)^2-2*x
*exp(4)-x^2)*log(x))*log(2*x*log(x))^2+exp(4)*log(x)*log(2*x*log(x))+(-exp(4)-x)*log(x)-exp(4)-x)/(exp(4)^2+2*
x*exp(4)+x^2)/log(x)/log(2*x*log(x))^2,x, algorithm="maxima")

[Out]

-(x^2*log(2) - (x*log(2) + e^4*log(2) + (x + e^4)*log(x) + (x + e^4)*log(log(x)))*x^x + (e^4*log(2) - 1)*x + (
x^2 + x*e^4)*log(x) + (x^2 + x*e^4)*log(log(x)))/(x*log(2) + e^4*log(2) + (x + e^4)*log(x) + (x + e^4)*log(log
(x)))

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Fricas [A]
time = 0.35, size = 43, normalized size = 1.72 \begin {gather*} \frac {{\left ({\left (x + e^{4}\right )} x^{x} - x^{2} - x e^{4}\right )} \log \left (2 \, x \log \left (x\right )\right ) + x}{{\left (x + e^{4}\right )} \log \left (2 \, x \log \left (x\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((exp(4)^2+2*x*exp(4)+x^2)*log(x)^2+(exp(4)^2+2*x*exp(4)+x^2)*log(x))*exp(x*log(x))+(-exp(4)^2-2*x
*exp(4)-x^2)*log(x))*log(2*x*log(x))^2+exp(4)*log(x)*log(2*x*log(x))+(-exp(4)-x)*log(x)-exp(4)-x)/(exp(4)^2+2*
x*exp(4)+x^2)/log(x)/log(2*x*log(x))^2,x, algorithm="fricas")

[Out]

(((x + e^4)*x^x - x^2 - x*e^4)*log(2*x*log(x)) + x)/((x + e^4)*log(2*x*log(x)))

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Sympy [A]
time = 0.28, size = 22, normalized size = 0.88 \begin {gather*} - x + \frac {x}{\left (x + e^{4}\right ) \log {\left (2 x \log {\left (x \right )} \right )}} + e^{x \log {\left (x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((exp(4)**2+2*x*exp(4)+x**2)*ln(x)**2+(exp(4)**2+2*x*exp(4)+x**2)*ln(x))*exp(x*ln(x))+(-exp(4)**2-
2*x*exp(4)-x**2)*ln(x))*ln(2*x*ln(x))**2+exp(4)*ln(x)*ln(2*x*ln(x))+(-exp(4)-x)*ln(x)-exp(4)-x)/(exp(4)**2+2*x
*exp(4)+x**2)/ln(x)/ln(2*x*ln(x))**2,x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1320 vs. \(2 (24) = 48\).
time = 3.02, size = 1320, normalized size = 52.80 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((exp(4)^2+2*x*exp(4)+x^2)*log(x)^2+(exp(4)^2+2*x*exp(4)+x^2)*log(x))*exp(x*log(x))+(-exp(4)^2-2*x
*exp(4)-x^2)*log(x))*log(2*x*log(x))^2+exp(4)*log(x)*log(2*x*log(x))+(-exp(4)-x)*log(x)-exp(4)-x)/(exp(4)^2+2*
x*exp(4)+x^2)/log(x)/log(2*x*log(x))^2,x, algorithm="giac")

[Out]

2*x^2*e^4*log(2)*log(x + e^4)/(x^2*log(2) + x*e^4*log(2) + x^2*log(x) + x*e^4*log(x) + x^2*log(log(x)) + x*e^4
*log(log(x))) + 2*x^2*e^4*log(x + e^4)*log(x)/(x^2*log(2) + x*e^4*log(2) + x^2*log(x) + x*e^4*log(x) + x^2*log
(log(x)) + x*e^4*log(log(x))) - 2*x^2*e^4*log(2)*log(-x - e^4)/(x^2*log(2) + x*e^4*log(2) + x^2*log(x) + x*e^4
*log(x) + x^2*log(log(x)) + x*e^4*log(log(x))) - 2*x^2*e^4*log(x)*log(-x - e^4)/(x^2*log(2) + x*e^4*log(2) + x
^2*log(x) + x*e^4*log(x) + x^2*log(log(x)) + x*e^4*log(log(x))) + 2*x^2*e^4*log(x + e^4)*log(log(x))/(x^2*log(
2) + x*e^4*log(2) + x^2*log(x) + x*e^4*log(x) + x^2*log(log(x)) + x*e^4*log(log(x))) - 2*x^2*e^4*log(-x - e^4)
*log(log(x))/(x^2*log(2) + x*e^4*log(2) + x^2*log(x) + x*e^4*log(x) + x^2*log(log(x)) + x*e^4*log(log(x))) + x
^2*x^x*log(2)/(x^2*log(2) + x*e^4*log(2) + x^2*log(x) + x*e^4*log(x) + x^2*log(log(x)) + x*e^4*log(log(x))) -
x^3*log(2)/(x^2*log(2) + x*e^4*log(2) + x^2*log(x) + x*e^4*log(x) + x^2*log(log(x)) + x*e^4*log(log(x))) + x*x
^x*e^4*log(2)/(x^2*log(2) + x*e^4*log(2) + x^2*log(x) + x*e^4*log(x) + x^2*log(log(x)) + x*e^4*log(log(x))) -
x^2*e^4*log(2)/(x^2*log(2) + x*e^4*log(2) + x^2*log(x) + x*e^4*log(x) + x^2*log(log(x)) + x*e^4*log(log(x))) +
2*x*e^8*log(2)*log(x + e^4)/(x^2*log(2) + x*e^4*log(2) + x^2*log(x) + x*e^4*log(x) + x^2*log(log(x)) + x*e^4*
log(log(x))) + x^2*x^x*log(x)/(x^2*log(2) + x*e^4*log(2) + x^2*log(x) + x*e^4*log(x) + x^2*log(log(x)) + x*e^4
*log(log(x))) - x^3*log(x)/(x^2*log(2) + x*e^4*log(2) + x^2*log(x) + x*e^4*log(x) + x^2*log(log(x)) + x*e^4*lo
g(log(x))) + x*x^x*e^4*log(x)/(x^2*log(2) + x*e^4*log(2) + x^2*log(x) + x*e^4*log(x) + x^2*log(log(x)) + x*e^4
*log(log(x))) - x^2*e^4*log(x)/(x^2*log(2) + x*e^4*log(2) + x^2*log(x) + x*e^4*log(x) + x^2*log(log(x)) + x*e^
4*log(log(x))) + 2*x*e^8*log(x + e^4)*log(x)/(x^2*log(2) + x*e^4*log(2) + x^2*log(x) + x*e^4*log(x) + x^2*log(
log(x)) + x*e^4*log(log(x))) - 2*x*e^8*log(2)*log(-x - e^4)/(x^2*log(2) + x*e^4*log(2) + x^2*log(x) + x*e^4*lo
g(x) + x^2*log(log(x)) + x*e^4*log(log(x))) - 2*x*e^8*log(x)*log(-x - e^4)/(x^2*log(2) + x*e^4*log(2) + x^2*lo
g(x) + x*e^4*log(x) + x^2*log(log(x)) + x*e^4*log(log(x))) + x^2*x^x*log(log(x))/(x^2*log(2) + x*e^4*log(2) +
x^2*log(x) + x*e^4*log(x) + x^2*log(log(x)) + x*e^4*log(log(x))) - x^3*log(log(x))/(x^2*log(2) + x*e^4*log(2)
+ x^2*log(x) + x*e^4*log(x) + x^2*log(log(x)) + x*e^4*log(log(x))) + x*x^x*e^4*log(log(x))/(x^2*log(2) + x*e^4
*log(2) + x^2*log(x) + x*e^4*log(x) + x^2*log(log(x)) + x*e^4*log(log(x))) - x^2*e^4*log(log(x))/(x^2*log(2) +
x*e^4*log(2) + x^2*log(x) + x*e^4*log(x) + x^2*log(log(x)) + x*e^4*log(log(x))) + 2*x*e^8*log(x + e^4)*log(lo
g(x))/(x^2*log(2) + x*e^4*log(2) + x^2*log(x) + x*e^4*log(x) + x^2*log(log(x)) + x*e^4*log(log(x))) - 2*x*e^8*
log(-x - e^4)*log(log(x))/(x^2*log(2) + x*e^4*log(2) + x^2*log(x) + x*e^4*log(x) + x^2*log(log(x)) + x*e^4*log
(log(x))) + x^2/(x^2*log(2) + x*e^4*log(2) + x^2*log(x) + x*e^4*log(x) + x^2*log(log(x)) + x*e^4*log(log(x)))

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Mupad [B]
time = 2.48, size = 118, normalized size = 4.72 \begin {gather*} \frac {\frac {x}{x+{\mathrm {e}}^4}-\frac {x\,\ln \left (2\,x\,\ln \left (x\right )\right )\,{\mathrm {e}}^4\,\ln \left (x\right )}{{\left (x+{\mathrm {e}}^4\right )}^2\,\left (\ln \left (x\right )+1\right )}}{\ln \left (2\,x\,\ln \left (x\right )\right )}-x+x^x-\frac {\frac {2\,x^2\,{\mathrm {e}}^4}{{\left (x+{\mathrm {e}}^4\right )}^3}-\frac {x\,\ln \left (x\right )\,\left ({\mathrm {e}}^8-x\,{\mathrm {e}}^4\right )}{{\left (x+{\mathrm {e}}^4\right )}^3}}{\ln \left (x\right )+1}+\frac {2\,x^2\,{\mathrm {e}}^4}{x^3+3\,{\mathrm {e}}^4\,x^2+3\,{\mathrm {e}}^8\,x+{\mathrm {e}}^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x + exp(4) + log(x)*(x + exp(4)) + log(2*x*log(x))^2*(log(x)*(exp(8) + 2*x*exp(4) + x^2) - exp(x*log(x))
*(log(x)*(exp(8) + 2*x*exp(4) + x^2) + log(x)^2*(exp(8) + 2*x*exp(4) + x^2))) - log(2*x*log(x))*exp(4)*log(x))
/(log(2*x*log(x))^2*log(x)*(exp(8) + 2*x*exp(4) + x^2)),x)

[Out]

(x/(x + exp(4)) - (x*log(2*x*log(x))*exp(4)*log(x))/((x + exp(4))^2*(log(x) + 1)))/log(2*x*log(x)) - x + x^x -
((2*x^2*exp(4))/(x + exp(4))^3 - (x*log(x)*(exp(8) - x*exp(4)))/(x + exp(4))^3)/(log(x) + 1) + (2*x^2*exp(4))
/(exp(12) + 3*x*exp(8) + 3*x^2*exp(4) + x^3)

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