3.72.79 \(\int \frac {-1+x^{e^4 x+4 e^{5-x} x} (e^4 x+4 e^{5-x} x+(e^4 x+4 e^{5-x} (x-x^2)) \log (x))}{x} \, dx\)

Optimal. Leaf size=22 \[ x^{\left (e^4+4 e^{5-x}\right ) x}-\log (x) \]

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Rubi [F]  time = 1.47, 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 {-1+x^{e^4 x+4 e^{5-x} x} \left (e^4 x+4 e^{5-x} x+\left (e^4 x+4 e^{5-x} \left (x-x^2\right )\right ) \log (x)\right )}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-1 + x^(E^4*x + 4*E^(5 - x)*x)*(E^4*x + 4*E^(5 - x)*x + (E^4*x + 4*E^(5 - x)*(x - x^2))*Log[x]))/x,x]

[Out]

-Log[x] + E^4*Defer[Int][x^(E^(4 - x)*(4*E + E^x)*x), x] + E^4*Log[x]*Defer[Int][x^(E^(4 - x)*(4*E + E^x)*x),
x] + 4*Defer[Int][E^(5 - x)*x^(E^(4 - x)*(4*E + E^x)*x), x] + 4*Log[x]*Defer[Int][E^(5 - x)*x^(E^(4 - x)*(4*E
+ E^x)*x), x] - 4*Log[x]*Defer[Int][E^(5 - x)*x^(1 + E^(4 - x)*(4*E + E^x)*x), x] - E^4*Defer[Int][Defer[Int][
x^(E^(4 - x)*(4*E + E^x)*x), x]/x, x] - 4*Defer[Int][Defer[Int][E^(5 - x)*x^(E^(4 - x)*(4*E + E^x)*x), x]/x, x
] + 4*Defer[Int][Defer[Int][E^(5 - x)*x^(1 + E^4*x + 4*E^(5 - x)*x), x]/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {1}{x}+e^{4-x} x^{e^{4-x} \left (4 e+e^x\right ) x} \left (4 e+e^x+4 e \log (x)+e^x \log (x)-4 e x \log (x)\right )\right ) \, dx\\ &=-\log (x)+\int e^{4-x} x^{e^{4-x} \left (4 e+e^x\right ) x} \left (4 e+e^x+4 e \log (x)+e^x \log (x)-4 e x \log (x)\right ) \, dx\\ &=-\log (x)+\int \left (e^4 x^{e^{4-x} \left (4 e+e^x\right ) x}+4 e^{5-x} x^{e^{4-x} \left (4 e+e^x\right ) x}+e^4 x^{e^{4-x} \left (4 e+e^x\right ) x} \log (x)+4 e^{5-x} x^{e^{4-x} \left (4 e+e^x\right ) x} \log (x)-4 e^{5-x} x^{1+e^{4-x} \left (4 e+e^x\right ) x} \log (x)\right ) \, dx\\ &=-\log (x)+4 \int e^{5-x} x^{e^{4-x} \left (4 e+e^x\right ) x} \, dx+4 \int e^{5-x} x^{e^{4-x} \left (4 e+e^x\right ) x} \log (x) \, dx-4 \int e^{5-x} x^{1+e^{4-x} \left (4 e+e^x\right ) x} \log (x) \, dx+e^4 \int x^{e^{4-x} \left (4 e+e^x\right ) x} \, dx+e^4 \int x^{e^{4-x} \left (4 e+e^x\right ) x} \log (x) \, dx\\ &=-\log (x)+4 \int e^{5-x} x^{e^{4-x} \left (4 e+e^x\right ) x} \, dx-4 \int \frac {\int e^{5-x} x^{e^{4-x} \left (4 e+e^x\right ) x} \, dx}{x} \, dx+4 \int \frac {\int e^{5-x} x^{1+e^4 x+4 e^{5-x} x} \, dx}{x} \, dx+e^4 \int x^{e^{4-x} \left (4 e+e^x\right ) x} \, dx-e^4 \int \frac {\int x^{e^{4-x} \left (4 e+e^x\right ) x} \, dx}{x} \, dx+(4 \log (x)) \int e^{5-x} x^{e^{4-x} \left (4 e+e^x\right ) x} \, dx-(4 \log (x)) \int e^{5-x} x^{1+e^{4-x} \left (4 e+e^x\right ) x} \, dx+\left (e^4 \log (x)\right ) \int x^{e^{4-x} \left (4 e+e^x\right ) x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.51, size = 23, normalized size = 1.05 \begin {gather*} x^{e^4 \left (1+4 e^{1-x}\right ) x}-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + x^(E^4*x + 4*E^(5 - x)*x)*(E^4*x + 4*E^(5 - x)*x + (E^4*x + 4*E^(5 - x)*(x - x^2))*Log[x]))/x,
x]

[Out]

x^(E^4*(1 + 4*E^(1 - x))*x) - Log[x]

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fricas [A]  time = 0.70, size = 24, normalized size = 1.09 \begin {gather*} x^{x e^{4} + x e^{\left (-x + 2 \, \log \relax (2) + 5\right )}} - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-x^2+x)*exp(2*log(2)+5-x)+x*exp(4))*log(x)+x*exp(2*log(2)+5-x)+x*exp(4))*exp((x*exp(2*log(2)+5-x
)+x*exp(4))*log(x))-1)/x,x, algorithm="fricas")

[Out]

x^(x*e^4 + x*e^(-x + 2*log(2) + 5)) - log(x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x e^{4} + x e^{\left (-x + 2 \, \log \relax (2) + 5\right )} + {\left (x e^{4} - {\left (x^{2} - x\right )} e^{\left (-x + 2 \, \log \relax (2) + 5\right )}\right )} \log \relax (x)\right )} x^{x e^{4} + x e^{\left (-x + 2 \, \log \relax (2) + 5\right )}} - 1}{x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-x^2+x)*exp(2*log(2)+5-x)+x*exp(4))*log(x)+x*exp(2*log(2)+5-x)+x*exp(4))*exp((x*exp(2*log(2)+5-x
)+x*exp(4))*log(x))-1)/x,x, algorithm="giac")

[Out]

integrate(((x*e^4 + x*e^(-x + 2*log(2) + 5) + (x*e^4 - (x^2 - x)*e^(-x + 2*log(2) + 5))*log(x))*x^(x*e^4 + x*e
^(-x + 2*log(2) + 5)) - 1)/x, x)

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maple [A]  time = 0.16, size = 21, normalized size = 0.95

method	result	size
risch	\(x^{\left ({\mathrm e}^{4}+4 \,{\mathrm e}^{5-x}\right ) x}-\ln \relax (x )\)	\(21\)
default	\({\mathrm e}^{\left (x \,{\mathrm e}^{2 \ln \relax (2)+5-x}+x \,{\mathrm e}^{4}\right ) \ln \relax (x )}-\ln \relax (x )\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((((-x^2+x)*exp(2*ln(2)+5-x)+x*exp(4))*ln(x)+x*exp(2*ln(2)+5-x)+x*exp(4))*exp((x*exp(2*ln(2)+5-x)+x*exp(4)
)*ln(x))-1)/x,x,method=_RETURNVERBOSE)

[Out]

x^((exp(4)+4*exp(5-x))*x)-ln(x)

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maxima [A]  time = 0.40, size = 24, normalized size = 1.09 \begin {gather*} e^{\left (x e^{4} \log \relax (x) + 4 \, x e^{\left (-x + 5\right )} \log \relax (x)\right )} - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-x^2+x)*exp(2*log(2)+5-x)+x*exp(4))*log(x)+x*exp(2*log(2)+5-x)+x*exp(4))*exp((x*exp(2*log(2)+5-x
)+x*exp(4))*log(x))-1)/x,x, algorithm="maxima")

[Out]

e^(x*e^4*log(x) + 4*x*e^(-x + 5)*log(x)) - log(x)

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mupad [B]  time = 4.53, size = 23, normalized size = 1.05 \begin {gather*} x^{x\,{\mathrm {e}}^4}\,x^{4\,x\,{\mathrm {e}}^{5-x}}-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(log(x)*(x*exp(4) + x*exp(2*log(2) - x + 5)))*(x*exp(4) + x*exp(2*log(2) - x + 5) + log(x)*(x*exp(4) +
exp(2*log(2) - x + 5)*(x - x^2))) - 1)/x,x)

[Out]

x^(x*exp(4))*x^(4*x*exp(5 - x)) - log(x)

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sympy [A]  time = 0.54, size = 20, normalized size = 0.91 \begin {gather*} e^{\left (4 x e^{5 - x} + x e^{4}\right ) \log {\relax (x )}} - \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-x**2+x)*exp(2*ln(2)+5-x)+x*exp(4))*ln(x)+x*exp(2*ln(2)+5-x)+x*exp(4))*exp((x*exp(2*ln(2)+5-x)+x
*exp(4))*ln(x))-1)/x,x)

[Out]

exp((4*x*exp(5 - x) + x*exp(4))*log(x)) - log(x)

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