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

Optimal. Leaf size=29 \[ \frac {\log (4) \left (e^x-\frac {1+\log \left ((x+\log (4))^2\right )}{x}\right )}{\log \left (\frac {x}{4}\right )} \]

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Rubi [F]  time = 3.72, 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 {x \log (4)+\log ^2(4)+e^x \left (-x^2 \log (4)-x \log ^2(4)\right )+\left (-x \log (4)+\log ^2(4)+e^x \left (x^3 \log (4)+x^2 \log ^2(4)\right )\right ) \log \left (\frac {x}{4}\right )+\left (x \log (4)+\log ^2(4)+\left (x \log (4)+\log ^2(4)\right ) \log \left (\frac {x}{4}\right )\right ) \log \left (x^2+2 x \log (4)+\log ^2(4)\right )}{\left (x^3+x^2 \log (4)\right ) \log ^2\left (\frac {x}{4}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.14, size = 30, normalized size = 1.03 \begin {gather*} \frac {\log (4) \left (-1+e^x x-\log \left ((x+\log (4))^2\right )\right )}{x \log \left (\frac {x}{4}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.72, size = 42, normalized size = 1.45 \begin {gather*} \frac {2 \, {\left (x e^{x} \log \relax (2) - \log \relax (2) \log \left (x^{2} + 4 \, x \log \relax (2) + 4 \, \log \relax (2)^{2}\right ) - \log \relax (2)\right )}}{x \log \left (\frac {1}{4} \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.30, size = 46, normalized size = 1.59 \begin {gather*} -\frac {2 \, {\left (x e^{x} \log \relax (2) - \log \relax (2) \log \left (x^{2} + 4 \, x \log \relax (2) + 4 \, \log \relax (2)^{2}\right ) - \log \relax (2)\right )}}{2 \, x \log \relax (2) - x \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.48, size = 115, normalized size = 3.97

method	result	size
risch	\(-\frac {4 \ln \relax (2) \ln \left (\ln \relax (2)+\frac {x}{2}\right )}{x \ln \left (\frac {x}{4}\right )}+\frac {\ln \relax (2) \left (i \pi \mathrm {csgn}\left (i \left (\ln \relax (2)+\frac {x}{2}\right )\right )^{2} \mathrm {csgn}\left (i \left (\ln \relax (2)+\frac {x}{2}\right )^{2}\right )-2 i \pi \,\mathrm {csgn}\left (i \left (\ln \relax (2)+\frac {x}{2}\right )\right ) \mathrm {csgn}\left (i \left (\ln \relax (2)+\frac {x}{2}\right )^{2}\right )^{2}+i \pi \mathrm {csgn}\left (i \left (\ln \relax (2)+\frac {x}{2}\right )^{2}\right )^{3}+2 \,{\mathrm e}^{x} x -2\right )}{x \ln \left (\frac {x}{4}\right )}\)	\(115\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*ln(2)/x/ln(1/4*x)*ln(ln(2)+1/2*x)+1/x*ln(2)*(I*Pi*csgn(I*(ln(2)+1/2*x))^2*csgn(I*(ln(2)+1/2*x)^2)-2*I*Pi*cs
gn(I*(ln(2)+1/2*x))*csgn(I*(ln(2)+1/2*x)^2)^2+I*Pi*csgn(I*(ln(2)+1/2*x)^2)^3+2*exp(x)*x-2)/ln(1/4*x)

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maxima [A]  time = 0.48, size = 37, normalized size = 1.28 \begin {gather*} -\frac {2 \, {\left (x e^{x} \log \relax (2) - 2 \, \log \relax (2) \log \left (x + 2 \, \log \relax (2)\right ) - \log \relax (2)\right )}}{2 \, x \log \relax (2) - x \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {2\,x\,\ln \relax (2)+\ln \left (x^2+4\,\ln \relax (2)\,x+4\,{\ln \relax (2)}^2\right )\,\left (2\,x\,\ln \relax (2)+\ln \left (\frac {x}{4}\right )\,\left (2\,x\,\ln \relax (2)+4\,{\ln \relax (2)}^2\right )+4\,{\ln \relax (2)}^2\right )+\ln \left (\frac {x}{4}\right )\,\left ({\mathrm {e}}^x\,\left (2\,\ln \relax (2)\,x^3+4\,{\ln \relax (2)}^2\,x^2\right )-2\,x\,\ln \relax (2)+4\,{\ln \relax (2)}^2\right )+4\,{\ln \relax (2)}^2-{\mathrm {e}}^x\,\left (2\,\ln \relax (2)\,x^2+4\,{\ln \relax (2)}^2\,x\right )}{{\ln \left (\frac {x}{4}\right )}^2\,\left (x^3+2\,\ln \relax (2)\,x^2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 0.63, size = 54, normalized size = 1.86 \begin {gather*} \frac {2 e^{x} \log {\relax (2 )}}{\log {\left (\frac {x}{4} \right )}} - \frac {2 \log {\relax (2 )} \log {\left (x^{2} + 4 x \log {\relax (2 )} + 4 \log {\relax (2 )}^{2} \right )}}{x \log {\left (\frac {x}{4} \right )}} - \frac {2 \log {\relax (2 )}}{x \log {\left (\frac {x}{4} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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