3.79.73 \(\int \frac {-4 \log ^2(4 x)+e^{2 e^x} (1+(1-2 e^x x) \log (4 x))}{3 x^2 \log ^2(4 x)} \, dx\)

Optimal. Leaf size=24 \[ \frac {4-\frac {e^{2 e^x}}{\log (4 x)}}{3 x} \]

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Rubi [F]  time = 1.44, 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 {-4 \log ^2(4 x)+e^{2 e^x} \left (1+\left (1-2 e^x x\right ) \log (4 x)\right )}{3 x^2 \log ^2(4 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-4*Log[4*x]^2 + E^(2*E^x)*(1 + (1 - 2*E^x*x)*Log[4*x]))/(3*x^2*Log[4*x]^2),x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.12, size = 24, normalized size = 1.00 \begin {gather*} \frac {4-\frac {e^{2 e^x}}{\log (4 x)}}{3 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4*Log[4*x]^2 + E^(2*E^x)*(1 + (1 - 2*E^x*x)*Log[4*x]))/(3*x^2*Log[4*x]^2),x]

[Out]

(4 - E^(2*E^x)/Log[4*x])/(3*x)

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fricas [A]  time = 0.61, size = 23, normalized size = 0.96 \begin {gather*} -\frac {e^{\left (2 \, e^{x}\right )} - 4 \, \log \left (4 \, x\right )}{3 \, x \log \left (4 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(e^(2*e^x) - 4*log(4*x))/(x*log(4*x))

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giac [A]  time = 0.22, size = 23, normalized size = 0.96 \begin {gather*} -\frac {e^{\left (2 \, e^{x}\right )} - 4 \, \log \left (4 \, x\right )}{3 \, x \log \left (4 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(e^(2*e^x) - 4*log(4*x))/(x*log(4*x))

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maple [A]  time = 0.05, size = 23, normalized size = 0.96

method	result	size
risch	\(\frac {4}{3 x}-\frac {{\mathrm e}^{2 \,{\mathrm e}^{x}}}{3 \ln \left (4 x \right ) x}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/3/x-1/3/ln(4*x)/x*exp(2*exp(x))

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maxima [A]  time = 0.49, size = 25, normalized size = 1.04 \begin {gather*} -\frac {e^{\left (2 \, e^{x}\right )}}{3 \, {\left (2 \, x \log \relax (2) + x \log \relax (x)\right )}} + \frac {4}{3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*e^(2*e^x)/(2*x*log(2) + x*log(x)) + 4/3/x

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mupad [B]  time = 6.35, size = 22, normalized size = 0.92 \begin {gather*} \frac {4}{3\,x}-\frac {{\mathrm {e}}^{2\,{\mathrm {e}}^x}}{3\,x\,\ln \left (4\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((exp(2*exp(x))*(log(4*x)*(2*x*exp(x) - 1) - 1))/3 + (4*log(4*x)^2)/3)/(x^2*log(4*x)^2),x)

[Out]

4/(3*x) - exp(2*exp(x))/(3*x*log(4*x))

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sympy [A]  time = 0.35, size = 19, normalized size = 0.79 \begin {gather*} - \frac {e^{2 e^{x}}}{3 x \log {\left (4 x \right )}} + \frac {4}{3 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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