3.82.13 \(\int \frac {e^{\frac {2 x^2+4 x^3+2 x^4+4 x \log (4)+(2 x+4 x^2+2 x^3) \log (x)}{x+2 x^2+x^3+(1+2 x+x^2) \log (x)}} (2 x^2+6 x^3+6 x^4+2 x^5+(-4-4 x-8 x^2) \log (4)+(4 x+12 x^2+12 x^3+4 x^4+(4-4 x) \log (4)) \log (x)+(2+6 x+6 x^2+2 x^3) \log ^2(x))}{x^2+3 x^3+3 x^4+x^5+(2 x+6 x^2+6 x^3+2 x^4) \log (x)+(1+3 x+3 x^2+x^3) \log ^2(x)} \, dx\)

Optimal. Leaf size=25 \[ e^{4 \left (\frac {x}{2}+\frac {x \log (4)}{(1+x)^2 (x+\log (x))}\right )} \]

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Rubi [F]  time = 53.74, 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 {\exp \left (\frac {2 x^2+4 x^3+2 x^4+4 x \log (4)+\left (2 x+4 x^2+2 x^3\right ) \log (x)}{x+2 x^2+x^3+\left (1+2 x+x^2\right ) \log (x)}\right ) \left (2 x^2+6 x^3+6 x^4+2 x^5+\left (-4-4 x-8 x^2\right ) \log (4)+\left (4 x+12 x^2+12 x^3+4 x^4+(4-4 x) \log (4)\right ) \log (x)+\left (2+6 x+6 x^2+2 x^3\right ) \log ^2(x)\right )}{x^2+3 x^3+3 x^4+x^5+\left (2 x+6 x^2+6 x^3+2 x^4\right ) \log (x)+\left (1+3 x+3 x^2+x^3\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((2*x^2 + 4*x^3 + 2*x^4 + 4*x*Log[4] + (2*x + 4*x^2 + 2*x^3)*Log[x])/(x + 2*x^2 + x^3 + (1 + 2*x + x^2)
*Log[x]))*(2*x^2 + 6*x^3 + 6*x^4 + 2*x^5 + (-4 - 4*x - 8*x^2)*Log[4] + (4*x + 12*x^2 + 12*x^3 + 4*x^4 + (4 - 4
*x)*Log[4])*Log[x] + (2 + 6*x + 6*x^2 + 2*x^3)*Log[x]^2))/(x^2 + 3*x^3 + 3*x^4 + x^5 + (2*x + 6*x^2 + 6*x^3 +
2*x^4)*Log[x] + (1 + 3*x + 3*x^2 + x^3)*Log[x]^2),x]

[Out]

Defer[Int][2^(1 + (8*x)/((1 + x)^2*(x + Log[x])))*E^((2*x^2)/(x + Log[x]))*x^((2*x)/(x + Log[x])), x] - Log[4]
*Defer[Int][(2^(2 + (8*x)/((1 + x)^2*(x + Log[x])))*E^((2*x^2)/(x + Log[x]))*x^((2*x)/(x + Log[x])))/((1 + x)*
(x + Log[x])^2), x] + Log[4]*Defer[Int][(2^(3 + (8*x)/((1 + x)^2*(x + Log[x])))*E^((2*x^2)/(x + Log[x]))*x^((2
*x)/(x + Log[x])))/((1 + x)^3*(x + Log[x])), x] - Log[4]*Defer[Int][(2^(2 + (8*x)/((1 + x)^2*(x + Log[x])))*E^
((2*x^2)/(x + Log[x]))*x^((2*x)/(x + Log[x])))/((1 + x)^2*(x + Log[x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2^{1+\frac {8 x}{(1+x)^2 (x+\log (x))}} e^{\frac {2 x^2}{x+\log (x)}} x^{\frac {2 x}{x+\log (x)}} \left (3 x^3+3 x^4+x^5+x^2 (1-4 \log (4))-2 \log (4)-2 x \log (4)+2 \left (x+3 x^2+3 x^3+x^4+\log (4)-x \log (4)\right ) \log (x)+(1+x)^3 \log ^2(x)\right )}{(1+x)^3 (x+\log (x))^2} \, dx\\ &=\int \left (2^{1+\frac {8 x}{(1+x)^2 (x+\log (x))}} e^{\frac {2 x^2}{x+\log (x)}} x^{\frac {2 x}{x+\log (x)}}-\frac {2^{2+\frac {8 x}{(1+x)^2 (x+\log (x))}} e^{\frac {2 x^2}{x+\log (x)}} x^{\frac {2 x}{x+\log (x)}} \log (4)}{(1+x) (x+\log (x))^2}-\frac {2^{2+\frac {8 x}{(1+x)^2 (x+\log (x))}} e^{\frac {2 x^2}{x+\log (x)}} (-1+x) x^{\frac {2 x}{x+\log (x)}} \log (4)}{(1+x)^3 (x+\log (x))}\right ) \, dx\\ &=-\left (\log (4) \int \frac {2^{2+\frac {8 x}{(1+x)^2 (x+\log (x))}} e^{\frac {2 x^2}{x+\log (x)}} x^{\frac {2 x}{x+\log (x)}}}{(1+x) (x+\log (x))^2} \, dx\right )-\log (4) \int \frac {2^{2+\frac {8 x}{(1+x)^2 (x+\log (x))}} e^{\frac {2 x^2}{x+\log (x)}} (-1+x) x^{\frac {2 x}{x+\log (x)}}}{(1+x)^3 (x+\log (x))} \, dx+\int 2^{1+\frac {8 x}{(1+x)^2 (x+\log (x))}} e^{\frac {2 x^2}{x+\log (x)}} x^{\frac {2 x}{x+\log (x)}} \, dx\\ &=-\left (\log (4) \int \frac {2^{2+\frac {8 x}{(1+x)^2 (x+\log (x))}} e^{\frac {2 x^2}{x+\log (x)}} x^{\frac {2 x}{x+\log (x)}}}{(1+x) (x+\log (x))^2} \, dx\right )-\log (4) \int \left (-\frac {2^{3+\frac {8 x}{(1+x)^2 (x+\log (x))}} e^{\frac {2 x^2}{x+\log (x)}} x^{\frac {2 x}{x+\log (x)}}}{(1+x)^3 (x+\log (x))}+\frac {2^{2+\frac {8 x}{(1+x)^2 (x+\log (x))}} e^{\frac {2 x^2}{x+\log (x)}} x^{\frac {2 x}{x+\log (x)}}}{(1+x)^2 (x+\log (x))}\right ) \, dx+\int 2^{1+\frac {8 x}{(1+x)^2 (x+\log (x))}} e^{\frac {2 x^2}{x+\log (x)}} x^{\frac {2 x}{x+\log (x)}} \, dx\\ &=-\left (\log (4) \int \frac {2^{2+\frac {8 x}{(1+x)^2 (x+\log (x))}} e^{\frac {2 x^2}{x+\log (x)}} x^{\frac {2 x}{x+\log (x)}}}{(1+x) (x+\log (x))^2} \, dx\right )+\log (4) \int \frac {2^{3+\frac {8 x}{(1+x)^2 (x+\log (x))}} e^{\frac {2 x^2}{x+\log (x)}} x^{\frac {2 x}{x+\log (x)}}}{(1+x)^3 (x+\log (x))} \, dx-\log (4) \int \frac {2^{2+\frac {8 x}{(1+x)^2 (x+\log (x))}} e^{\frac {2 x^2}{x+\log (x)}} x^{\frac {2 x}{x+\log (x)}}}{(1+x)^2 (x+\log (x))} \, dx+\int 2^{1+\frac {8 x}{(1+x)^2 (x+\log (x))}} e^{\frac {2 x^2}{x+\log (x)}} x^{\frac {2 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 {e^{\frac {2 x^2+4 x^3+2 x^4+4 x \log (4)+\left (2 x+4 x^2+2 x^3\right ) \log (x)}{x+2 x^2+x^3+\left (1+2 x+x^2\right ) \log (x)}} \left (2 x^2+6 x^3+6 x^4+2 x^5+\left (-4-4 x-8 x^2\right ) \log (4)+\left (4 x+12 x^2+12 x^3+4 x^4+(4-4 x) \log (4)\right ) \log (x)+\left (2+6 x+6 x^2+2 x^3\right ) \log ^2(x)\right )}{x^2+3 x^3+3 x^4+x^5+\left (2 x+6 x^2+6 x^3+2 x^4\right ) \log (x)+\left (1+3 x+3 x^2+x^3\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(E^((2*x^2 + 4*x^3 + 2*x^4 + 4*x*Log[4] + (2*x + 4*x^2 + 2*x^3)*Log[x])/(x + 2*x^2 + x^3 + (1 + 2*x
+ x^2)*Log[x]))*(2*x^2 + 6*x^3 + 6*x^4 + 2*x^5 + (-4 - 4*x - 8*x^2)*Log[4] + (4*x + 12*x^2 + 12*x^3 + 4*x^4 +
(4 - 4*x)*Log[4])*Log[x] + (2 + 6*x + 6*x^2 + 2*x^3)*Log[x]^2))/(x^2 + 3*x^3 + 3*x^4 + x^5 + (2*x + 6*x^2 + 6*
x^3 + 2*x^4)*Log[x] + (1 + 3*x + 3*x^2 + x^3)*Log[x]^2),x]

[Out]

Integrate[(E^((2*x^2 + 4*x^3 + 2*x^4 + 4*x*Log[4] + (2*x + 4*x^2 + 2*x^3)*Log[x])/(x + 2*x^2 + x^3 + (1 + 2*x
+ x^2)*Log[x]))*(2*x^2 + 6*x^3 + 6*x^4 + 2*x^5 + (-4 - 4*x - 8*x^2)*Log[4] + (4*x + 12*x^2 + 12*x^3 + 4*x^4 +
(4 - 4*x)*Log[4])*Log[x] + (2 + 6*x + 6*x^2 + 2*x^3)*Log[x]^2))/(x^2 + 3*x^3 + 3*x^4 + x^5 + (2*x + 6*x^2 + 6*
x^3 + 2*x^4)*Log[x] + (1 + 3*x + 3*x^2 + x^3)*Log[x]^2), x]

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fricas [B]  time = 0.59, size = 56, normalized size = 2.24 \begin {gather*} e^{\left (\frac {2 \, {\left (x^{4} + 2 \, x^{3} + x^{2} + 4 \, x \log \relax (2) + {\left (x^{3} + 2 \, x^{2} + x\right )} \log \relax (x)\right )}}{x^{3} + 2 \, x^{2} + {\left (x^{2} + 2 \, x + 1\right )} \log \relax (x) + x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3+6*x^2+6*x+2)*log(x)^2+(2*(-4*x+4)*log(2)+4*x^4+12*x^3+12*x^2+4*x)*log(x)+2*(-8*x^2-4*x-4)*lo
g(2)+2*x^5+6*x^4+6*x^3+2*x^2)*exp(((2*x^3+4*x^2+2*x)*log(x)+8*x*log(2)+2*x^4+4*x^3+2*x^2)/((x^2+2*x+1)*log(x)+
x^3+2*x^2+x))/((x^3+3*x^2+3*x+1)*log(x)^2+(2*x^4+6*x^3+6*x^2+2*x)*log(x)+x^5+3*x^4+3*x^3+x^2),x, algorithm="fr
icas")

[Out]

e^(2*(x^4 + 2*x^3 + x^2 + 4*x*log(2) + (x^3 + 2*x^2 + x)*log(x))/(x^3 + 2*x^2 + (x^2 + 2*x + 1)*log(x) + x))

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giac [B]  time = 0.83, size = 216, normalized size = 8.64 \begin {gather*} e^{\left (\frac {2 \, x^{4}}{x^{3} + x^{2} \log \relax (x) + 2 \, x^{2} + 2 \, x \log \relax (x) + x + \log \relax (x)} + \frac {2 \, x^{3} \log \relax (x)}{x^{3} + x^{2} \log \relax (x) + 2 \, x^{2} + 2 \, x \log \relax (x) + x + \log \relax (x)} + \frac {4 \, x^{3}}{x^{3} + x^{2} \log \relax (x) + 2 \, x^{2} + 2 \, x \log \relax (x) + x + \log \relax (x)} + \frac {4 \, x^{2} \log \relax (x)}{x^{3} + x^{2} \log \relax (x) + 2 \, x^{2} + 2 \, x \log \relax (x) + x + \log \relax (x)} + \frac {2 \, x^{2}}{x^{3} + x^{2} \log \relax (x) + 2 \, x^{2} + 2 \, x \log \relax (x) + x + \log \relax (x)} + \frac {8 \, x \log \relax (2)}{x^{3} + x^{2} \log \relax (x) + 2 \, x^{2} + 2 \, x \log \relax (x) + x + \log \relax (x)} + \frac {2 \, x \log \relax (x)}{x^{3} + x^{2} \log \relax (x) + 2 \, x^{2} + 2 \, x \log \relax (x) + x + \log \relax (x)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3+6*x^2+6*x+2)*log(x)^2+(2*(-4*x+4)*log(2)+4*x^4+12*x^3+12*x^2+4*x)*log(x)+2*(-8*x^2-4*x-4)*lo
g(2)+2*x^5+6*x^4+6*x^3+2*x^2)*exp(((2*x^3+4*x^2+2*x)*log(x)+8*x*log(2)+2*x^4+4*x^3+2*x^2)/((x^2+2*x+1)*log(x)+
x^3+2*x^2+x))/((x^3+3*x^2+3*x+1)*log(x)^2+(2*x^4+6*x^3+6*x^2+2*x)*log(x)+x^5+3*x^4+3*x^3+x^2),x, algorithm="gi
ac")

[Out]

e^(2*x^4/(x^3 + x^2*log(x) + 2*x^2 + 2*x*log(x) + x + log(x)) + 2*x^3*log(x)/(x^3 + x^2*log(x) + 2*x^2 + 2*x*l
og(x) + x + log(x)) + 4*x^3/(x^3 + x^2*log(x) + 2*x^2 + 2*x*log(x) + x + log(x)) + 4*x^2*log(x)/(x^3 + x^2*log
(x) + 2*x^2 + 2*x*log(x) + x + log(x)) + 2*x^2/(x^3 + x^2*log(x) + 2*x^2 + 2*x*log(x) + x + log(x)) + 8*x*log(
2)/(x^3 + x^2*log(x) + 2*x^2 + 2*x*log(x) + x + log(x)) + 2*x*log(x)/(x^3 + x^2*log(x) + 2*x^2 + 2*x*log(x) +
x + log(x)))

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maple [A]  time = 0.06, size = 43, normalized size = 1.72

method	result	size
risch	\({\mathrm e}^{\frac {2 x \left (x^{2} \ln \relax (x )+x^{3}+2 x \ln \relax (x )+2 x^{2}+4 \ln \relax (2)+\ln \relax (x )+x \right )}{\left (x +1\right )^{2} \left (x +\ln \relax (x )\right )}}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^3+6*x^2+6*x+2)*ln(x)^2+(2*(-4*x+4)*ln(2)+4*x^4+12*x^3+12*x^2+4*x)*ln(x)+2*(-8*x^2-4*x-4)*ln(2)+2*x^5
+6*x^4+6*x^3+2*x^2)*exp(((2*x^3+4*x^2+2*x)*ln(x)+8*x*ln(2)+2*x^4+4*x^3+2*x^2)/((x^2+2*x+1)*ln(x)+x^3+2*x^2+x))
/((x^3+3*x^2+3*x+1)*ln(x)^2+(2*x^4+6*x^3+6*x^2+2*x)*ln(x)+x^5+3*x^4+3*x^3+x^2),x,method=_RETURNVERBOSE)

[Out]

exp(2*x*(x^2*ln(x)+x^3+2*x*ln(x)+2*x^2+4*ln(2)+ln(x)+x)/(x+1)^2/(x+ln(x)))

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maxima [B]  time = 1.06, size = 88, normalized size = 3.52 \begin {gather*} e^{\left (2 \, x + \frac {8 \, \log \relax (2) \log \relax (x)}{{\left (x + 1\right )} \log \relax (x)^{2} - 2 \, {\left (x + 1\right )} \log \relax (x) + x + 1} - \frac {8 \, \log \relax (2) \log \relax (x)}{{\left (x - 2\right )} \log \relax (x)^{2} + \log \relax (x)^{3} - {\left (2 \, x - 1\right )} \log \relax (x) + x} + \frac {8 \, \log \relax (2)}{x^{2} - {\left (x^{2} + 2 \, x + 1\right )} \log \relax (x) + 2 \, x + 1}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3+6*x^2+6*x+2)*log(x)^2+(2*(-4*x+4)*log(2)+4*x^4+12*x^3+12*x^2+4*x)*log(x)+2*(-8*x^2-4*x-4)*lo
g(2)+2*x^5+6*x^4+6*x^3+2*x^2)*exp(((2*x^3+4*x^2+2*x)*log(x)+8*x*log(2)+2*x^4+4*x^3+2*x^2)/((x^2+2*x+1)*log(x)+
x^3+2*x^2+x))/((x^3+3*x^2+3*x+1)*log(x)^2+(2*x^4+6*x^3+6*x^2+2*x)*log(x)+x^5+3*x^4+3*x^3+x^2),x, algorithm="ma
xima")

[Out]

e^(2*x + 8*log(2)*log(x)/((x + 1)*log(x)^2 - 2*(x + 1)*log(x) + x + 1) - 8*log(2)*log(x)/((x - 2)*log(x)^2 + l
og(x)^3 - (2*x - 1)*log(x) + x) + 8*log(2)/(x^2 - (x^2 + 2*x + 1)*log(x) + 2*x + 1))

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mupad [B]  time = 5.83, size = 134, normalized size = 5.36 \begin {gather*} {256}^{\frac {x}{x+\ln \relax (x)+x^2\,\ln \relax (x)+2\,x\,\ln \relax (x)+2\,x^2+x^3}}\,x^{\frac {2\,x}{x+\ln \relax (x)}}\,{\mathrm {e}}^{\frac {2\,x^2}{x+\ln \relax (x)+x^2\,\ln \relax (x)+2\,x\,\ln \relax (x)+2\,x^2+x^3}}\,{\mathrm {e}}^{\frac {2\,x^4}{x+\ln \relax (x)+x^2\,\ln \relax (x)+2\,x\,\ln \relax (x)+2\,x^2+x^3}}\,{\mathrm {e}}^{\frac {4\,x^3}{x+\ln \relax (x)+x^2\,\ln \relax (x)+2\,x\,\ln \relax (x)+2\,x^2+x^3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((8*x*log(2) + 2*x^2 + 4*x^3 + 2*x^4 + log(x)*(2*x + 4*x^2 + 2*x^3))/(x + log(x)*(2*x + x^2 + 1) + 2*x
^2 + x^3))*(log(x)*(4*x - 2*log(2)*(4*x - 4) + 12*x^2 + 12*x^3 + 4*x^4) - 2*log(2)*(4*x + 8*x^2 + 4) + log(x)^
2*(6*x + 6*x^2 + 2*x^3 + 2) + 2*x^2 + 6*x^3 + 6*x^4 + 2*x^5))/(log(x)*(2*x + 6*x^2 + 6*x^3 + 2*x^4) + x^2 + 3*
x^3 + 3*x^4 + x^5 + log(x)^2*(3*x + 3*x^2 + x^3 + 1)),x)

[Out]

256^(x/(x + log(x) + x^2*log(x) + 2*x*log(x) + 2*x^2 + x^3))*x^((2*x)/(x + log(x)))*exp((2*x^2)/(x + log(x) +
x^2*log(x) + 2*x*log(x) + 2*x^2 + x^3))*exp((2*x^4)/(x + log(x) + x^2*log(x) + 2*x*log(x) + 2*x^2 + x^3))*exp(
(4*x^3)/(x + log(x) + x^2*log(x) + 2*x*log(x) + 2*x^2 + x^3))

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sympy [B]  time = 1.01, size = 61, normalized size = 2.44 \begin {gather*} e^{\frac {2 x^{4} + 4 x^{3} + 2 x^{2} + 8 x \log {\relax (2 )} + \left (2 x^{3} + 4 x^{2} + 2 x\right ) \log {\relax (x )}}{x^{3} + 2 x^{2} + x + \left (x^{2} + 2 x + 1\right ) \log {\relax (x )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**3+6*x**2+6*x+2)*ln(x)**2+(2*(-4*x+4)*ln(2)+4*x**4+12*x**3+12*x**2+4*x)*ln(x)+2*(-8*x**2-4*x-4
)*ln(2)+2*x**5+6*x**4+6*x**3+2*x**2)*exp(((2*x**3+4*x**2+2*x)*ln(x)+8*x*ln(2)+2*x**4+4*x**3+2*x**2)/((x**2+2*x
+1)*ln(x)+x**3+2*x**2+x))/((x**3+3*x**2+3*x+1)*ln(x)**2+(2*x**4+6*x**3+6*x**2+2*x)*ln(x)+x**5+3*x**4+3*x**3+x*
*2),x)

[Out]

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

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