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

Optimal. Leaf size=18 \[ 4+e^x+e^{x+\frac {2 x^2}{\log (x)}} \]

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Rubi [F]  time = 0.43, 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^x \log ^2(x)+e^{\frac {2 x^2+x \log (x)}{\log (x)}} \left (-2 x+4 x \log (x)+\log ^2(x)\right )}{\log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.22, size = 17, normalized size = 0.94 \begin {gather*} e^x+e^{x+\frac {2 x^2}{\log (x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^x + E^(x + (2*x^2)/Log[x])

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fricas [A]  time = 1.07, size = 19, normalized size = 1.06 \begin {gather*} e^{x} + e^{\left (\frac {2 \, x^{2} + x \log \relax (x)}{\log \relax (x)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^x + e^((2*x^2 + x*log(x))/log(x))

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giac [A]  time = 0.30, size = 19, normalized size = 1.06 \begin {gather*} e^{x} + e^{\left (\frac {2 \, x^{2} + x \log \relax (x)}{\log \relax (x)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^x + e^((2*x^2 + x*log(x))/log(x))

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maple [A]  time = 0.06, size = 17, normalized size = 0.94

method	result	size
risch	\({\mathrm e}^{\frac {x \left (2 x +\ln \relax (x )\right )}{\ln \relax (x )}}+{\mathrm e}^{x}\)	\(17\)
default	\({\mathrm e}^{\frac {x \ln \relax (x )+2 x^{2}}{\ln \relax (x )}}+{\mathrm e}^{x}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(x*(2*x+ln(x))/ln(x))+exp(x)

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maxima [A]  time = 0.42, size = 15, normalized size = 0.83 \begin {gather*} e^{\left (x + \frac {2 \, x^{2}}{\log \relax (x)}\right )} + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(x + 2*x^2/log(x)) + e^x

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mupad [B]  time = 3.71, size = 15, normalized size = 0.83 \begin {gather*} {\mathrm {e}}^x\,\left ({\mathrm {e}}^{\frac {2\,x^2}{\ln \relax (x)}}+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(x)*(exp((2*x^2)/log(x)) + 1)

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sympy [A]  time = 0.41, size = 17, normalized size = 0.94 \begin {gather*} e^{x} + e^{\frac {2 x^{2} + x \log {\relax (x )}}{\log {\relax (x )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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