3.46.66 \(\int \frac {e^{-4+\frac {e^{20+\frac {x}{4}} x+e^{x/4} \log (x)}{e^4 x^2}} (e^{x/4} (4+e^{20} (-4 x+x^2))+e^{x/4} (-8+x) \log (x))}{4 x^3} \, dx\)

Optimal. Leaf size=25 \[ e^{\frac {e^{-4+\frac {x}{4}} \left (e^{20}+\frac {\log (x)}{x}\right )}{x}} \]

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Rubi [F]  time = 5.14, 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^{-4+\frac {e^{20+\frac {x}{4}} x+e^{x/4} \log (x)}{e^4 x^2}} \left (e^{x/4} \left (4+e^{20} \left (-4 x+x^2\right )\right )+e^{x/4} (-8+x) \log (x)\right )}{4 x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-4 + (E^(20 + x/4)*x + E^(x/4)*Log[x])/(E^4*x^2))*(E^(x/4)*(4 + E^20*(-4*x + x^2)) + E^(x/4)*(-8 + x)*
Log[x]))/(4*x^3),x]

[Out]

Defer[Int][E^(-4 + x/4 + (E^(20 + x/4)*x + E^(x/4)*Log[x])/(E^4*x^2))/x^3, x] - Defer[Int][E^(16 + x/4 + (E^(2
0 + x/4)*x + E^(x/4)*Log[x])/(E^4*x^2))/x^2, x] + Defer[Int][E^(16 + x/4 + (E^(20 + x/4)*x + E^(x/4)*Log[x])/(
E^4*x^2))/x, x]/4 - 2*Defer[Int][(E^(-4 + x/4 + (E^(20 + x/4)*x + E^(x/4)*Log[x])/(E^4*x^2))*Log[x])/x^3, x] +
Defer[Int][(E^(-4 + x/4 + (E^(20 + x/4)*x + E^(x/4)*Log[x])/(E^4*x^2))*Log[x])/x^2, x]/4

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \frac {e^{-4+\frac {e^{20+\frac {x}{4}} x+e^{x/4} \log (x)}{e^4 x^2}} \left (e^{x/4} \left (4+e^{20} \left (-4 x+x^2\right )\right )+e^{x/4} (-8+x) \log (x)\right )}{x^3} \, dx\\ &=\frac {1}{4} \int \frac {\exp \left (-4+\frac {x}{4}+\frac {e^{20+\frac {x}{4}} x+e^{x/4} \log (x)}{e^4 x^2}\right ) \left (4-4 e^{20} x+e^{20} x^2-8 \log (x)+x \log (x)\right )}{x^3} \, dx\\ &=\frac {1}{4} \int \left (\frac {\exp \left (-4+\frac {x}{4}+\frac {e^{20+\frac {x}{4}} x+e^{x/4} \log (x)}{e^4 x^2}\right ) \left (4-4 e^{20} x+e^{20} x^2\right )}{x^3}+\frac {\exp \left (-4+\frac {x}{4}+\frac {e^{20+\frac {x}{4}} x+e^{x/4} \log (x)}{e^4 x^2}\right ) (-8+x) \log (x)}{x^3}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\exp \left (-4+\frac {x}{4}+\frac {e^{20+\frac {x}{4}} x+e^{x/4} \log (x)}{e^4 x^2}\right ) \left (4-4 e^{20} x+e^{20} x^2\right )}{x^3} \, dx+\frac {1}{4} \int \frac {\exp \left (-4+\frac {x}{4}+\frac {e^{20+\frac {x}{4}} x+e^{x/4} \log (x)}{e^4 x^2}\right ) (-8+x) \log (x)}{x^3} \, dx\\ &=\frac {1}{4} \int \left (\frac {4 \exp \left (-4+\frac {x}{4}+\frac {e^{20+\frac {x}{4}} x+e^{x/4} \log (x)}{e^4 x^2}\right )}{x^3}-\frac {4 e^{16+\frac {x}{4}+\frac {e^{20+\frac {x}{4}} x+e^{x/4} \log (x)}{e^4 x^2}}}{x^2}+\frac {e^{16+\frac {x}{4}+\frac {e^{20+\frac {x}{4}} x+e^{x/4} \log (x)}{e^4 x^2}}}{x}\right ) \, dx+\frac {1}{4} \int \left (-\frac {8 \exp \left (-4+\frac {x}{4}+\frac {e^{20+\frac {x}{4}} x+e^{x/4} \log (x)}{e^4 x^2}\right ) \log (x)}{x^3}+\frac {\exp \left (-4+\frac {x}{4}+\frac {e^{20+\frac {x}{4}} x+e^{x/4} \log (x)}{e^4 x^2}\right ) \log (x)}{x^2}\right ) \, dx\\ &=\frac {1}{4} \int \frac {e^{16+\frac {x}{4}+\frac {e^{20+\frac {x}{4}} x+e^{x/4} \log (x)}{e^4 x^2}}}{x} \, dx+\frac {1}{4} \int \frac {\exp \left (-4+\frac {x}{4}+\frac {e^{20+\frac {x}{4}} x+e^{x/4} \log (x)}{e^4 x^2}\right ) \log (x)}{x^2} \, dx-2 \int \frac {\exp \left (-4+\frac {x}{4}+\frac {e^{20+\frac {x}{4}} x+e^{x/4} \log (x)}{e^4 x^2}\right ) \log (x)}{x^3} \, dx+\int \frac {\exp \left (-4+\frac {x}{4}+\frac {e^{20+\frac {x}{4}} x+e^{x/4} \log (x)}{e^4 x^2}\right )}{x^3} \, dx-\int \frac {e^{16+\frac {x}{4}+\frac {e^{20+\frac {x}{4}} x+e^{x/4} \log (x)}{e^4 x^2}}}{x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.95, size = 31, normalized size = 1.24 \begin {gather*} e^{\frac {e^{16+\frac {x}{4}}}{x}} x^{\frac {e^{-4+\frac {x}{4}}}{x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-4 + (E^(20 + x/4)*x + E^(x/4)*Log[x])/(E^4*x^2))*(E^(x/4)*(4 + E^20*(-4*x + x^2)) + E^(x/4)*(-8
+ x)*Log[x]))/(4*x^3),x]

[Out]

E^(E^(16 + x/4)/x)*x^(E^(-4 + x/4)/x^2)

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fricas [A]  time = 0.57, size = 37, normalized size = 1.48 \begin {gather*} e^{\left (-\frac {{\left (4 \, x^{2} e^{24} - x e^{\left (\frac {1}{4} \, x + 40\right )} - e^{\left (\frac {1}{4} \, x + 20\right )} \log \relax (x)\right )} e^{\left (-24\right )}}{x^{2}} + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((-8+x)*exp(1/4*x)*log(x)+((x^2-4*x)*exp(20)+4)*exp(1/4*x))*exp((exp(1/4*x)*log(x)+x*exp(20)*exp
(1/4*x))/x^2/exp(4))/x^3/exp(4),x, algorithm="fricas")

[Out]

e^(-(4*x^2*e^24 - x*e^(1/4*x + 40) - e^(1/4*x + 20)*log(x))*e^(-24)/x^2 + 4)

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giac [A]  time = 0.22, size = 24, normalized size = 0.96 \begin {gather*} e^{\left (\frac {e^{\left (\frac {1}{4} \, x + 16\right )}}{x} + \frac {e^{\left (\frac {1}{4} \, x - 4\right )} \log \relax (x)}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((-8+x)*exp(1/4*x)*log(x)+((x^2-4*x)*exp(20)+4)*exp(1/4*x))*exp((exp(1/4*x)*log(x)+x*exp(20)*exp
(1/4*x))/x^2/exp(4))/x^3/exp(4),x, algorithm="giac")

[Out]

e^(e^(1/4*x + 16)/x + e^(1/4*x - 4)*log(x)/x^2)

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maple [A]  time = 0.04, size = 25, normalized size = 1.00

method	result	size
risch	\(x^{\frac {{\mathrm e}^{-4+\frac {x}{4}}}{x^{2}}} {\mathrm e}^{\frac {{\mathrm e}^{\frac {x}{4}+16}}{x}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4*((-8+x)*exp(1/4*x)*ln(x)+((x^2-4*x)*exp(20)+4)*exp(1/4*x))*exp((exp(1/4*x)*ln(x)+x*exp(20)*exp(1/4*x))
/x^2/exp(4))/x^3/exp(4),x,method=_RETURNVERBOSE)

[Out]

x^(1/x^2*exp(-4+1/4*x))*exp(1/x*exp(1/4*x+16))

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maxima [A]  time = 0.57, size = 24, normalized size = 0.96 \begin {gather*} e^{\left (\frac {e^{\left (\frac {1}{4} \, x + 16\right )}}{x} + \frac {e^{\left (\frac {1}{4} \, x - 4\right )} \log \relax (x)}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((-8+x)*exp(1/4*x)*log(x)+((x^2-4*x)*exp(20)+4)*exp(1/4*x))*exp((exp(1/4*x)*log(x)+x*exp(20)*exp
(1/4*x))/x^2/exp(4))/x^3/exp(4),x, algorithm="maxima")

[Out]

e^(e^(1/4*x + 16)/x + e^(1/4*x - 4)*log(x)/x^2)

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mupad [B]  time = 3.26, size = 24, normalized size = 0.96 \begin {gather*} x^{\frac {{\mathrm {e}}^{\frac {x}{4}-4}}{x^2}}\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{x/4}\,{\mathrm {e}}^{16}}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((exp(-4)*(exp(x/4)*log(x) + x*exp(x/4)*exp(20)))/x^2)*exp(-4)*(exp(x/4)*(exp(20)*(4*x - x^2) - 4) -
exp(x/4)*log(x)*(x - 8)))/(4*x^3),x)

[Out]

x^(exp(x/4 - 4)/x^2)*exp((exp(x/4)*exp(16))/x)

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sympy [A]  time = 0.60, size = 26, normalized size = 1.04 \begin {gather*} e^{\frac {x e^{20} e^{\frac {x}{4}} + e^{\frac {x}{4}} \log {\relax (x )}}{x^{2} e^{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((-8+x)*exp(1/4*x)*ln(x)+((x**2-4*x)*exp(20)+4)*exp(1/4*x))*exp((exp(1/4*x)*ln(x)+x*exp(20)*exp(
1/4*x))/x**2/exp(4))/x**3/exp(4),x)

[Out]

exp((x*exp(20)*exp(x/4) + exp(x/4)*log(x))*exp(-4)/x**2)

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