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

Optimal. Leaf size=16 \[ x^{2^{\frac {2}{5} \left (-2+e^x+2 x\right )}} \]

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Rubi [F]  time = 2.17, 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}{5} e^{\frac {1}{5} \left (e^x \log (4)+(-2+2 x) \log (4)\right )} x^{-1+e^{\frac {1}{5} \left (e^x \log (4)+(-2+2 x) \log (4)\right )}} \left (5+\left (2 x \log (4)+e^x x \log (4)\right ) \log (x)\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{5} \int e^{\frac {1}{5} \left (e^x \log (4)+(-2+2 x) \log (4)\right )} x^{-1+e^{\frac {1}{5} \left (e^x \log (4)+(-2+2 x) \log (4)\right )}} \left (5+\left (2 x \log (4)+e^x x \log (4)\right ) \log (x)\right ) \, dx\\ &=\frac {1}{5} \int 2^{\frac {2}{5} \left (-2+e^x+2 x\right )} x^{-1+2^{\frac {2}{5} \left (-2+e^x+2 x\right )}} \left (5+\left (2+e^x\right ) x \log (4) \log (x)\right ) \, dx\\ &=\frac {1}{5} \int \left (5\ 2^{\frac {2}{5} \left (-2+e^x+2 x\right )} x^{-1+2^{\frac {2}{5} \left (-2+e^x+2 x\right )}}+2^{\frac {2}{5} \left (-2+e^x+2 x\right )} \left (2+e^x\right ) x^{2^{\frac {2}{5} \left (-2+e^x+2 x\right )}} \log (4) \log (x)\right ) \, dx\\ &=\frac {1}{5} \log (4) \int 2^{\frac {2}{5} \left (-2+e^x+2 x\right )} \left (2+e^x\right ) x^{2^{\frac {2}{5} \left (-2+e^x+2 x\right )}} \log (x) \, dx+\int 2^{\frac {2}{5} \left (-2+e^x+2 x\right )} x^{-1+2^{\frac {2}{5} \left (-2+e^x+2 x\right )}} \, dx\\ &=-\left (\frac {1}{5} \log (4) \int \frac {\int 2^{\frac {1}{5} \left (1+2 e^x+4 x\right )} x^{2^{\frac {2}{5} \left (-2+e^x+2 x\right )}} \, dx+\int 2^{\frac {2}{5} \left (-2+e^x+2 x\right )} e^x x^{2^{\frac {2}{5} \left (-2+e^x+2 x\right )}} \, dx}{x} \, dx\right )+\frac {1}{5} (\log (4) \log (x)) \int 2^{\frac {1}{5} \left (1+2 e^x+4 x\right )} x^{2^{\frac {2}{5} \left (-2+e^x+2 x\right )}} \, dx+\frac {1}{5} (\log (4) \log (x)) \int 2^{\frac {2}{5} \left (-2+e^x+2 x\right )} e^x x^{2^{\frac {2}{5} \left (-2+e^x+2 x\right )}} \, dx+\int 2^{\frac {2}{5} \left (-2+e^x+2 x\right )} x^{-1+2^{\frac {2}{5} \left (-2+e^x+2 x\right )}} \, dx\\ &=-\left (\frac {1}{5} \log (4) \int \left (\frac {\int 2^{\frac {1}{5} \left (1+2 e^x+4 x\right )} x^{2^{\frac {2}{5} \left (-2+e^x+2 x\right )}} \, dx}{x}+\frac {\int 2^{\frac {2}{5} \left (-2+e^x+2 x\right )} e^x x^{2^{\frac {2}{5} \left (-2+e^x+2 x\right )}} \, dx}{x}\right ) \, dx\right )+\frac {1}{5} (\log (4) \log (x)) \int 2^{\frac {1}{5} \left (1+2 e^x+4 x\right )} x^{2^{\frac {2}{5} \left (-2+e^x+2 x\right )}} \, dx+\frac {1}{5} (\log (4) \log (x)) \int 2^{\frac {2}{5} \left (-2+e^x+2 x\right )} e^x x^{2^{\frac {2}{5} \left (-2+e^x+2 x\right )}} \, dx+\int 2^{\frac {2}{5} \left (-2+e^x+2 x\right )} x^{-1+2^{\frac {2}{5} \left (-2+e^x+2 x\right )}} \, dx\\ &=-\left (\frac {1}{5} \log (4) \int \frac {\int 2^{\frac {1}{5} \left (1+2 e^x+4 x\right )} x^{2^{\frac {2}{5} \left (-2+e^x+2 x\right )}} \, dx}{x} \, dx\right )-\frac {1}{5} \log (4) \int \frac {\int 2^{\frac {2}{5} \left (-2+e^x+2 x\right )} e^x x^{2^{\frac {2}{5} \left (-2+e^x+2 x\right )}} \, dx}{x} \, dx+\frac {1}{5} (\log (4) \log (x)) \int 2^{\frac {1}{5} \left (1+2 e^x+4 x\right )} x^{2^{\frac {2}{5} \left (-2+e^x+2 x\right )}} \, dx+\frac {1}{5} (\log (4) \log (x)) \int 2^{\frac {2}{5} \left (-2+e^x+2 x\right )} e^x x^{2^{\frac {2}{5} \left (-2+e^x+2 x\right )}} \, dx+\int 2^{\frac {2}{5} \left (-2+e^x+2 x\right )} x^{-1+2^{\frac {2}{5} \left (-2+e^x+2 x\right )}} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 2.12, size = 71, normalized size = 4.44 \begin {gather*} \frac {1}{5} \int e^{\frac {1}{5} \left (e^x \log (4)+(-2+2 x) \log (4)\right )} x^{-1+e^{\frac {1}{5} \left (e^x \log (4)+(-2+2 x) \log (4)\right )}} \left (5+\left (2 x \log (4)+e^x x \log (4)\right ) \log (x)\right ) \, dx \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.66, size = 17, normalized size = 1.06 \begin {gather*} x^{e^{\left (\frac {4}{5} \, {\left (x - 1\right )} \log \relax (2) + \frac {2}{5} \, e^{x} \log \relax (2)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.20, size = 19, normalized size = 1.19 \begin {gather*} x^{e^{\left (\frac {4}{5} \, x \log \relax (2) + \frac {2}{5} \, e^{x} \log \relax (2) - \frac {4}{5} \, \log \relax (2)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.10, size = 14, normalized size = 0.88

method	result	size
risch	\(x^{2^{\frac {2 \,{\mathrm e}^{x}}{5}+\frac {4 x}{5}-\frac {4}{5}}}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^(2^(2/5*exp(x)+4/5*x-4/5))

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maxima [A]  time = 0.94, size = 20, normalized size = 1.25 \begin {gather*} x^{\frac {1}{2} \cdot 2^{\frac {1}{5}} e^{\left (\frac {4}{5} \, x \log \relax (2) + \frac {2}{5} \, e^{x} \log \relax (2)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 4.61, size = 18, normalized size = 1.12 \begin {gather*} x^{\frac {2^{\frac {4\,x}{5}}\,2^{1/5}\,{\left (2^{{\mathrm {e}}^x}\right )}^{2/5}}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^((2^((4*x)/5)*2^(1/5)*(2^exp(x))^(2/5))/2)

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sympy [A]  time = 6.06, size = 27, normalized size = 1.69 \begin {gather*} e^{e^{\left (\frac {4 x}{5} - \frac {4}{5}\right ) \log {\relax (2 )} + \frac {2 e^{x} \log {\relax (2 )}}{5}} \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(exp((4*x/5 - 4/5)*log(2) + 2*exp(x)*log(2)/5)*log(x))

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