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

Optimal. Leaf size=21 \[ -e^{-x+x^{\frac {e^3+4 x}{x}}} \]

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Rubi [F]  time = 1.34, 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+x^{\frac {e^3+4 x}{x}}} \left (x^2+x^{\frac {e^3+4 x}{x}} \left (-e^3-4 x+e^3 \log (x)\right )\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 1.32, size = 19, normalized size = 0.90 \begin {gather*} -e^{-x+x^{4+\frac {e^3}{x}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-E^(-x + x^(4 + E^3/x))

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fricas [A]  time = 0.54, size = 19, normalized size = 0.90 \begin {gather*} -e^{\left (x^{\frac {4 \, x + e^{3}}{x}} - x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^(x^((4*x + e^3)/x) - x)

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giac [A]  time = 0.28, size = 17, normalized size = 0.81 \begin {gather*} -e^{\left (x^{\frac {e^{3}}{x} + 4} - x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^(x^(e^3/x + 4) - x)

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maple [A]  time = 0.06, size = 20, normalized size = 0.95

method	result	size
risch	\(-{\mathrm e}^{x^{\frac {{\mathrm e}^{3}+4 x}{x}}-x}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((ln(x)*exp(3)-exp(3)-4*x)*exp((exp(3)+4*x)*ln(x)/x)+x^2)*exp(exp((exp(3)+4*x)*ln(x)/x)-x)/x^2,x,method=_R
ETURNVERBOSE)

[Out]

-exp(x^((exp(3)+4*x)/x)-x)

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maxima [A]  time = 0.50, size = 19, normalized size = 0.90 \begin {gather*} -e^{\left (x^{4} x^{\frac {e^{3}}{x}} - x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^(x^4*x^(e^3/x) - x)

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mupad [B]  time = 4.48, size = 17, normalized size = 0.81 \begin {gather*} -{\mathrm {e}}^{x^{\frac {{\mathrm {e}}^3}{x}+4}}\,{\mathrm {e}}^{-x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-exp(x^(exp(3)/x + 4))*exp(-x)

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sympy [A]  time = 0.72, size = 17, normalized size = 0.81 \begin {gather*} - e^{- x + e^{\frac {\left (4 x + e^{3}\right ) \log {\relax (x )}}{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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