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

Optimal. Leaf size=25 \[ 2+e^{e^{1+e^4-x}}+\frac {3+x+x^2}{\log (x)} \]

________________________________________________________________________________________

Rubi [F]  time = 0.97, 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 {-3-x-x^2+\left (x+2 x^2\right ) \log (x)-e^{1+e^4+e^{1+e^4-x}-x} x \log ^2(x)}{x \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 1.27, size = 26, normalized size = 1.04 \begin {gather*} \frac {3+x+x^2+e^{e^{1+e^4-x}} \log (x)}{\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 1.16, size = 48, normalized size = 1.92 \begin {gather*} \frac {{\left ({\left (x^{2} + x + 3\right )} e^{\left (-x + e^{4} + 1\right )} + e^{\left (-x + e^{4} + e^{\left (-x + e^{4} + 1\right )} + 1\right )} \log \relax (x)\right )} e^{\left (x - e^{4} - 1\right )}}{\log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [B]  time = 0.38, size = 65, normalized size = 2.60 \begin {gather*} \frac {{\left (x^{2} e^{\left (-x + e^{4} + 1\right )} + x e^{\left (-x + e^{4} + 1\right )} + e^{\left (-x + e^{4} + e^{\left (-x + e^{4} + 1\right )} + 1\right )} \log \relax (x) + 3 \, e^{\left (-x + e^{4} + 1\right )}\right )} e^{\left (x - e^{4} - 1\right )}}{\log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.02, size = 22, normalized size = 0.88

method	result	size
risch	\(\frac {x^{2}+x +3}{\ln \relax (x )}+{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{4}-x +1}}\)	\(22\)
default	\({\mathrm e}^{{\mathrm e}^{{\mathrm e}^{4}-x +1}}+\frac {x^{2}}{\ln \relax (x )}+\frac {x}{\ln \relax (x )}+\frac {3}{\ln \relax (x )}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x*exp(exp(4)-x+1)*ln(x)^2*exp(exp(exp(4)-x+1))+(2*x^2+x)*ln(x)-x^2-x-3)/x/ln(x)^2,x,method=_RETURNVERBOS
E)

[Out]

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

________________________________________________________________________________________

maxima [C]  time = 0.89, size = 42, normalized size = 1.68 \begin {gather*} \frac {3}{\log \relax (x)} + 2 \, {\rm Ei}\left (2 \, \log \relax (x)\right ) + {\rm Ei}\left (\log \relax (x)\right ) + e^{\left (e^{\left (-x + e^{4} + 1\right )}\right )} - \Gamma \left (-1, -\log \relax (x)\right ) - 2 \, \Gamma \left (-1, -2 \, \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.98, size = 39, normalized size = 1.56 \begin {gather*} x+{\mathrm {e}}^{{\mathrm {e}}^{-x}\,\mathrm {e}\,{\mathrm {e}}^{{\mathrm {e}}^4}}+\frac {x+x^2-x\,\ln \relax (x)\,\left (2\,x+1\right )+3}{\ln \relax (x)}+2\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.33, size = 19, normalized size = 0.76 \begin {gather*} \frac {x^{2} + x + 3}{\log {\relax (x )}} + e^{e^{- x + 1 + e^{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________