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

Optimal. Leaf size=24 \[ \frac {1}{2} e^{e^{\frac {1}{4}+x}} \left (-5+(3-x+\log (x))^2\right ) \]

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Rubi [F]  time = 5.69, 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^{e^{\frac {1}{4} (1+4 x)}} \left (6-8 x+2 x^2+e^{\frac {1}{4} (1+4 x)} \left (4 x-6 x^2+x^3\right )+\left (2-2 x+e^{\frac {1}{4} (1+4 x)} \left (6 x-2 x^2\right )\right ) \log (x)+e^{\frac {1}{4} (1+4 x)} x \log ^2(x)\right )}{2 x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.08, size = 33, normalized size = 1.38 \begin {gather*} \frac {1}{2} e^{e^{\frac {1}{4}+x}} \left (4-6 x+x^2+(6-2 x) \log (x)+\log ^2(x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(E^E^(1/4 + x)*(4 - 6*x + x^2 + (6 - 2*x)*Log[x] + Log[x]^2))/2

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fricas [A]  time = 0.77, size = 26, normalized size = 1.08 \begin {gather*} \frac {1}{2} \, {\left (x^{2} - 2 \, {\left (x - 3\right )} \log \relax (x) + \log \relax (x)^{2} - 6 \, x + 4\right )} e^{\left (e^{\left (x + \frac {1}{4}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(x*exp(x+1/4)*log(x)^2+((-2*x^2+6*x)*exp(x+1/4)-2*x+2)*log(x)+(x^3-6*x^2+4*x)*exp(x+1/4)+2*x^2-8
*x+6)*exp(exp(x+1/4))/x,x, algorithm="fricas")

[Out]

1/2*(x^2 - 2*(x - 3)*log(x) + log(x)^2 - 6*x + 4)*e^(e^(x + 1/4))

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giac [B]  time = 0.22, size = 56, normalized size = 2.33 \begin {gather*} \frac {1}{2} \, x^{2} e^{\left (e^{\left (x + \frac {1}{4}\right )}\right )} - x e^{\left (e^{\left (x + \frac {1}{4}\right )}\right )} \log \relax (x) + \frac {1}{2} \, e^{\left (e^{\left (x + \frac {1}{4}\right )}\right )} \log \relax (x)^{2} - 3 \, x e^{\left (e^{\left (x + \frac {1}{4}\right )}\right )} + 3 \, e^{\left (e^{\left (x + \frac {1}{4}\right )}\right )} \log \relax (x) + 2 \, e^{\left (e^{\left (x + \frac {1}{4}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(x*exp(x+1/4)*log(x)^2+((-2*x^2+6*x)*exp(x+1/4)-2*x+2)*log(x)+(x^3-6*x^2+4*x)*exp(x+1/4)+2*x^2-8
*x+6)*exp(exp(x+1/4))/x,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.04, size = 29, normalized size = 1.21

method	result	size
risch	\(\frac {\left (x^{2}-2 x \ln \relax (x )+\ln \relax (x )^{2}-6 x +6 \ln \relax (x )+4\right ) {\mathrm e}^{{\mathrm e}^{x +\frac {1}{4}}}}{2}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(x^2-2*x*ln(x)+ln(x)^2-6*x+6*ln(x)+4)*exp(exp(x+1/4))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {1}{2} \, {\left (x^{2} - 2 \, {\left (x - 3\right )} \log \relax (x) + \log \relax (x)^{2} - 6 \, x + 4\right )} e^{\left (e^{\left (x + \frac {1}{4}\right )}\right )} - 4 \, {\rm Ei}\left (e^{\left (x + \frac {1}{4}\right )}\right ) + 4 \, \int e^{\left (e^{\left (x + \frac {1}{4}\right )}\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(x*exp(x+1/4)*log(x)^2+((-2*x^2+6*x)*exp(x+1/4)-2*x+2)*log(x)+(x^3-6*x^2+4*x)*exp(x+1/4)+2*x^2-8
*x+6)*exp(exp(x+1/4))/x,x, algorithm="maxima")

[Out]

1/2*(x^2 - 2*(x - 3)*log(x) + log(x)^2 - 6*x + 4)*e^(e^(x + 1/4)) - 4*Ei(e^(x + 1/4)) + 4*integrate(e^(e^(x +
1/4)), x)

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mupad [B]  time = 6.51, size = 32, normalized size = 1.33 \begin {gather*} {\mathrm {e}}^{{\mathrm {e}}^{1/4}\,{\mathrm {e}}^x}\,\left (\frac {x^2}{2}-x\,\ln \relax (x)-3\,x+\frac {{\ln \relax (x)}^2}{2}+3\,\ln \relax (x)+2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 64.36, size = 34, normalized size = 1.42 \begin {gather*} \frac {\left (x^{2} - 2 x \log {\relax (x )} - 6 x + \log {\relax (x )}^{2} + 6 \log {\relax (x )} + 4\right ) e^{e^{x + \frac {1}{4}}}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x**2 - 2*x*log(x) - 6*x + log(x)**2 + 6*log(x) + 4)*exp(exp(x + 1/4))/2

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