3.91.39 \(\int \frac {2 x^2 \log ^2(4)+e^x (-6 x^4+(6 x^3-18 x^4-6 x^5) \log (4))+(-4 x^2 \log (4)+4 x \log ^2(4)+e^x (18 x^4+6 x^5+(-24 x^3-6 x^4) \log (4))) \log (x)+(2 x^2-4 x \log (4)+2 \log ^2(4)) \log ^2(x)}{3 x^2 \log ^2(4)+(-6 x^2 \log (4)+6 x \log ^2(4)) \log (x)+(3 x^2-6 x \log (4)+3 \log ^2(4)) \log ^2(x)} \, dx\)

Optimal. Leaf size=30 \[ x \left (\frac {2}{3}+\frac {2 e^x x^2}{\log (x)-\frac {\log (4) (x+\log (x))}{x}}\right ) \]

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Rubi [F]  time = 9.36, 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 {2 x^2 \log ^2(4)+e^x \left (-6 x^4+\left (6 x^3-18 x^4-6 x^5\right ) \log (4)\right )+\left (-4 x^2 \log (4)+4 x \log ^2(4)+e^x \left (18 x^4+6 x^5+\left (-24 x^3-6 x^4\right ) \log (4)\right )\right ) \log (x)+\left (2 x^2-4 x \log (4)+2 \log ^2(4)\right ) \log ^2(x)}{3 x^2 \log ^2(4)+\left (-6 x^2 \log (4)+6 x \log ^2(4)\right ) \log (x)+\left (3 x^2-6 x \log (4)+3 \log ^2(4)\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2*x^2*Log[4]^2 + E^x*(-6*x^4 + (6*x^3 - 18*x^4 - 6*x^5)*Log[4]) + (-4*x^2*Log[4] + 4*x*Log[4]^2 + E^x*(18
*x^4 + 6*x^5 + (-24*x^3 - 6*x^4)*Log[4]))*Log[x] + (2*x^2 - 4*x*Log[4] + 2*Log[4]^2)*Log[x]^2)/(3*x^2*Log[4]^2
+ (-6*x^2*Log[4] + 6*x*Log[4]^2)*Log[x] + (3*x^2 - 6*x*Log[4] + 3*Log[4]^2)*Log[x]^2),x]

[Out]

(2*x)/3 - 2*Log[4]^5*Defer[Int][E^x/(x*Log[4] - x*Log[x] + Log[4]*Log[x])^2, x] - 2*Log[4]^4*Defer[Int][(E^x*x
)/(x*Log[4] - x*Log[x] + Log[4]*Log[x])^2, x] - 2*Log[4]^3*Defer[Int][(E^x*x^2)/(x*Log[4] - x*Log[x] + Log[4]*
Log[x])^2, x] + 2*(1 - Log[4])*Log[4]*Defer[Int][(E^x*x^3)/(x*Log[4] - x*Log[x] + Log[4]*Log[x])^2, x] - 2*Def
er[Int][(E^x*x^4)/(x*Log[4] - x*Log[x] + Log[4]*Log[x])^2, x] - 2*Log[4]^6*Defer[Int][E^x/((x - Log[4])*(x*Log
[4] - x*Log[x] + Log[4]*Log[x])^2), x] + 2*Log[4]^3*Defer[Int][E^x/(x*Log[4] - x*Log[x] + Log[4]*Log[x]), x] +
2*Log[4]^2*Defer[Int][(E^x*x)/(x*Log[4] - x*Log[x] + Log[4]*Log[x]), x] + 2*Log[4]*Defer[Int][(E^x*x^2)/(x*Lo
g[4] - x*Log[x] + Log[4]*Log[x]), x] - 6*Defer[Int][(E^x*x^3)/(x*Log[4] - x*Log[x] + Log[4]*Log[x]), x] - 2*De
fer[Int][(E^x*x^4)/(x*Log[4] - x*Log[x] + Log[4]*Log[x]), x] + 2*Log[4]^4*Defer[Int][E^x/((x - Log[4])*(x*Log[
4] - x*Log[x] + Log[4]*Log[x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (x^2 \left (\log ^2(4)-3 e^x x \left (x-\log (4)+x^2 \log (4)+x \log (64)\right )\right )+x \left (2 \log (4) (-x+\log (4))+3 e^x x^2 \left (3 x+x^2-4 \log (4)-x \log (4)\right )\right ) \log (x)+(x-\log (4))^2 \log ^2(x)\right )}{3 (x \log (4)+(-x+\log (4)) \log (x))^2} \, dx\\ &=\frac {2}{3} \int \frac {x^2 \left (\log ^2(4)-3 e^x x \left (x-\log (4)+x^2 \log (4)+x \log (64)\right )\right )+x \left (2 \log (4) (-x+\log (4))+3 e^x x^2 \left (3 x+x^2-4 \log (4)-x \log (4)\right )\right ) \log (x)+(x-\log (4))^2 \log ^2(x)}{(x \log (4)+(-x+\log (4)) \log (x))^2} \, dx\\ &=\frac {2}{3} \int \left (1+\frac {3 e^x x^3 \left (\log (4)-x^2 \log (4)-x (1+\log (64))+x^2 \log (x)+3 x \left (1-\frac {2 \log (2)}{3}\right ) \log (x)-\log (256) \log (x)\right )}{(x \log (4)-x \log (x)+\log (4) \log (x))^2}\right ) \, dx\\ &=\frac {2 x}{3}+2 \int \frac {e^x x^3 \left (\log (4)-x^2 \log (4)-x (1+\log (64))+x^2 \log (x)+3 x \left (1-\frac {2 \log (2)}{3}\right ) \log (x)-\log (256) \log (x)\right )}{(x \log (4)-x \log (x)+\log (4) \log (x))^2} \, dx\\ &=\frac {2 x}{3}+2 \int \left (\frac {e^x x^3 \left (-x^2-\log ^2(4)-x \left (\log ^2(4)-\log (16)\right )\right )}{(x-\log (4)) (x \log (4)-x \log (x)+\log (4) \log (x))^2}+\frac {e^x x^3 \left (-x^2-x (3-\log (4))+\log (256)\right )}{(x-\log (4)) (x \log (4)-x \log (x)+\log (4) \log (x))}\right ) \, dx\\ &=\frac {2 x}{3}+2 \int \frac {e^x x^3 \left (-x^2-\log ^2(4)-x \left (\log ^2(4)-\log (16)\right )\right )}{(x-\log (4)) (x \log (4)-x \log (x)+\log (4) \log (x))^2} \, dx+2 \int \frac {e^x x^3 \left (-x^2-x (3-\log (4))+\log (256)\right )}{(x-\log (4)) (x \log (4)-x \log (x)+\log (4) \log (x))} \, dx\\ &=\frac {2 x}{3}+2 \int \left (-\frac {e^x x^4}{(x \log (4)-x \log (x)+\log (4) \log (x))^2}-\frac {e^x x^3 (-1+\log (4)) \log (4)}{(x \log (4)-x \log (x)+\log (4) \log (x))^2}-\frac {e^x x^2 \log ^3(4)}{(x \log (4)-x \log (x)+\log (4) \log (x))^2}-\frac {e^x x \log ^4(4)}{(x \log (4)-x \log (x)+\log (4) \log (x))^2}-\frac {e^x \log ^5(4)}{(x \log (4)-x \log (x)+\log (4) \log (x))^2}-\frac {e^x \log ^6(4)}{(x-\log (4)) (x \log (4)-x \log (x)+\log (4) \log (x))^2}\right ) \, dx+2 \int \left (-\frac {3 e^x x^3}{x \log (4)-x \log (x)+\log (4) \log (x)}-\frac {e^x x^4}{x \log (4)-x \log (x)+\log (4) \log (x)}+\frac {e^x x^2 \log (4)}{x \log (4)-x \log (x)+\log (4) \log (x)}+\frac {e^x x \log ^2(4)}{x \log (4)-x \log (x)+\log (4) \log (x)}+\frac {e^x \log ^3(4)}{x \log (4)-x \log (x)+\log (4) \log (x)}+\frac {e^x \log ^4(4)}{(x-\log (4)) (x \log (4)-x \log (x)+\log (4) \log (x))}\right ) \, dx\\ &=\frac {2 x}{3}-2 \int \frac {e^x x^4}{(x \log (4)-x \log (x)+\log (4) \log (x))^2} \, dx-2 \int \frac {e^x x^4}{x \log (4)-x \log (x)+\log (4) \log (x)} \, dx-6 \int \frac {e^x x^3}{x \log (4)-x \log (x)+\log (4) \log (x)} \, dx+(2 \log (4)) \int \frac {e^x x^2}{x \log (4)-x \log (x)+\log (4) \log (x)} \, dx+(2 (1-\log (4)) \log (4)) \int \frac {e^x x^3}{(x \log (4)-x \log (x)+\log (4) \log (x))^2} \, dx+\left (2 \log ^2(4)\right ) \int \frac {e^x x}{x \log (4)-x \log (x)+\log (4) \log (x)} \, dx-\left (2 \log ^3(4)\right ) \int \frac {e^x x^2}{(x \log (4)-x \log (x)+\log (4) \log (x))^2} \, dx+\left (2 \log ^3(4)\right ) \int \frac {e^x}{x \log (4)-x \log (x)+\log (4) \log (x)} \, dx-\left (2 \log ^4(4)\right ) \int \frac {e^x x}{(x \log (4)-x \log (x)+\log (4) \log (x))^2} \, dx+\left (2 \log ^4(4)\right ) \int \frac {e^x}{(x-\log (4)) (x \log (4)-x \log (x)+\log (4) \log (x))} \, dx-\left (2 \log ^5(4)\right ) \int \frac {e^x}{(x \log (4)-x \log (x)+\log (4) \log (x))^2} \, dx-\left (2 \log ^6(4)\right ) \int \frac {e^x}{(x-\log (4)) (x \log (4)-x \log (x)+\log (4) \log (x))^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.28, size = 75, normalized size = 2.50 \begin {gather*} \frac {2}{3} \left (x-\frac {3 e^x x^4 \left (x^2+\log ^2(4)+x \left (4 \log ^2(4)-\log (16)-\log (4) \log (64)\right )\right )}{\left (x^2+x (-2+\log (4)) \log (4)+\log ^2(4)\right ) (x \log (4)+(-x+\log (4)) \log (x))}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*x^2*Log[4]^2 + E^x*(-6*x^4 + (6*x^3 - 18*x^4 - 6*x^5)*Log[4]) + (-4*x^2*Log[4] + 4*x*Log[4]^2 + E
^x*(18*x^4 + 6*x^5 + (-24*x^3 - 6*x^4)*Log[4]))*Log[x] + (2*x^2 - 4*x*Log[4] + 2*Log[4]^2)*Log[x]^2)/(3*x^2*Lo
g[4]^2 + (-6*x^2*Log[4] + 6*x*Log[4]^2)*Log[x] + (3*x^2 - 6*x*Log[4] + 3*Log[4]^2)*Log[x]^2),x]

[Out]

(2*(x - (3*E^x*x^4*(x^2 + Log[4]^2 + x*(4*Log[4]^2 - Log[16] - Log[4]*Log[64])))/((x^2 + x*(-2 + Log[4])*Log[4
] + Log[4]^2)*(x*Log[4] + (-x + Log[4])*Log[x]))))/3

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fricas [A]  time = 0.45, size = 47, normalized size = 1.57 \begin {gather*} -\frac {2 \, {\left (3 \, x^{4} e^{x} - 2 \, x^{2} \log \relax (2) + {\left (x^{2} - 2 \, x \log \relax (2)\right )} \log \relax (x)\right )}}{3 \, {\left (2 \, x \log \relax (2) - {\left (x - 2 \, \log \relax (2)\right )} \log \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*log(2)^2-8*x*log(2)+2*x^2)*log(x)^2+((2*(-6*x^4-24*x^3)*log(2)+6*x^5+18*x^4)*exp(x)+16*x*log(2)^
2-8*x^2*log(2))*log(x)+(2*(-6*x^5-18*x^4+6*x^3)*log(2)-6*x^4)*exp(x)+8*x^2*log(2)^2)/((12*log(2)^2-12*x*log(2)
+3*x^2)*log(x)^2+(24*x*log(2)^2-12*x^2*log(2))*log(x)+12*x^2*log(2)^2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.16, size = 49, normalized size = 1.63 \begin {gather*} -\frac {2 \, {\left (3 \, x^{4} e^{x} - 2 \, x^{2} \log \relax (2) + x^{2} \log \relax (x) - 2 \, x \log \relax (2) \log \relax (x)\right )}}{3 \, {\left (2 \, x \log \relax (2) - x \log \relax (x) + 2 \, \log \relax (2) \log \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*log(2)^2-8*x*log(2)+2*x^2)*log(x)^2+((2*(-6*x^4-24*x^3)*log(2)+6*x^5+18*x^4)*exp(x)+16*x*log(2)^
2-8*x^2*log(2))*log(x)+(2*(-6*x^5-18*x^4+6*x^3)*log(2)-6*x^4)*exp(x)+8*x^2*log(2)^2)/((12*log(2)^2-12*x*log(2)
+3*x^2)*log(x)^2+(24*x*log(2)^2-12*x^2*log(2))*log(x)+12*x^2*log(2)^2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.07, size = 31, normalized size = 1.03

method	result	size
risch	\(\frac {2 x}{3}-\frac {2 x^{4} {\mathrm e}^{x}}{2 \ln \relax (2) \ln \relax (x )-x \ln \relax (x )+2 x \ln \relax (2)}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*ln(2)^2-8*x*ln(2)+2*x^2)*ln(x)^2+((2*(-6*x^4-24*x^3)*ln(2)+6*x^5+18*x^4)*exp(x)+16*x*ln(2)^2-8*x^2*ln(
2))*ln(x)+(2*(-6*x^5-18*x^4+6*x^3)*ln(2)-6*x^4)*exp(x)+8*x^2*ln(2)^2)/((12*ln(2)^2-12*x*ln(2)+3*x^2)*ln(x)^2+(
24*x*ln(2)^2-12*x^2*ln(2))*ln(x)+12*x^2*ln(2)^2),x,method=_RETURNVERBOSE)

[Out]

2/3*x-2*x^4*exp(x)/(2*ln(2)*ln(x)-x*ln(x)+2*x*ln(2))

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maxima [A]  time = 0.47, size = 47, normalized size = 1.57 \begin {gather*} -\frac {2 \, {\left (3 \, x^{4} e^{x} - 2 \, x^{2} \log \relax (2) + {\left (x^{2} - 2 \, x \log \relax (2)\right )} \log \relax (x)\right )}}{3 \, {\left (2 \, x \log \relax (2) - {\left (x - 2 \, \log \relax (2)\right )} \log \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*log(2)^2-8*x*log(2)+2*x^2)*log(x)^2+((2*(-6*x^4-24*x^3)*log(2)+6*x^5+18*x^4)*exp(x)+16*x*log(2)^
2-8*x^2*log(2))*log(x)+(2*(-6*x^5-18*x^4+6*x^3)*log(2)-6*x^4)*exp(x)+8*x^2*log(2)^2)/((12*log(2)^2-12*x*log(2)
+3*x^2)*log(x)^2+(24*x*log(2)^2-12*x^2*log(2))*log(x)+12*x^2*log(2)^2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {8\,x^2\,{\ln \relax (2)}^2+\ln \relax (x)\,\left ({\mathrm {e}}^x\,\left (18\,x^4-2\,\ln \relax (2)\,\left (6\,x^4+24\,x^3\right )+6\,x^5\right )+16\,x\,{\ln \relax (2)}^2-8\,x^2\,\ln \relax (2)\right )-{\mathrm {e}}^x\,\left (2\,\ln \relax (2)\,\left (6\,x^5+18\,x^4-6\,x^3\right )+6\,x^4\right )+{\ln \relax (x)}^2\,\left (2\,x^2-8\,\ln \relax (2)\,x+8\,{\ln \relax (2)}^2\right )}{12\,x^2\,{\ln \relax (2)}^2+{\ln \relax (x)}^2\,\left (3\,x^2-12\,\ln \relax (2)\,x+12\,{\ln \relax (2)}^2\right )+\ln \relax (x)\,\left (24\,x\,{\ln \relax (2)}^2-12\,x^2\,\ln \relax (2)\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x^2*log(2)^2 + log(x)*(exp(x)*(18*x^4 - 2*log(2)*(24*x^3 + 6*x^4) + 6*x^5) + 16*x*log(2)^2 - 8*x^2*log(
2)) - exp(x)*(2*log(2)*(18*x^4 - 6*x^3 + 6*x^5) + 6*x^4) + log(x)^2*(8*log(2)^2 - 8*x*log(2) + 2*x^2))/(12*x^2
*log(2)^2 + log(x)^2*(12*log(2)^2 - 12*x*log(2) + 3*x^2) + log(x)*(24*x*log(2)^2 - 12*x^2*log(2))),x)

[Out]

int((8*x^2*log(2)^2 + log(x)*(exp(x)*(18*x^4 - 2*log(2)*(24*x^3 + 6*x^4) + 6*x^5) + 16*x*log(2)^2 - 8*x^2*log(
2)) - exp(x)*(2*log(2)*(18*x^4 - 6*x^3 + 6*x^5) + 6*x^4) + log(x)^2*(8*log(2)^2 - 8*x*log(2) + 2*x^2))/(12*x^2
*log(2)^2 + log(x)^2*(12*log(2)^2 - 12*x*log(2) + 3*x^2) + log(x)*(24*x*log(2)^2 - 12*x^2*log(2))), x)

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sympy [A]  time = 0.55, size = 32, normalized size = 1.07 \begin {gather*} \frac {2 x^{4} e^{x}}{x \log {\relax (x )} - 2 x \log {\relax (2 )} - 2 \log {\relax (2 )} \log {\relax (x )}} + \frac {2 x}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*ln(2)**2-8*x*ln(2)+2*x**2)*ln(x)**2+((2*(-6*x**4-24*x**3)*ln(2)+6*x**5+18*x**4)*exp(x)+16*x*ln(2
)**2-8*x**2*ln(2))*ln(x)+(2*(-6*x**5-18*x**4+6*x**3)*ln(2)-6*x**4)*exp(x)+8*x**2*ln(2)**2)/((12*ln(2)**2-12*x*
ln(2)+3*x**2)*ln(x)**2+(24*x*ln(2)**2-12*x**2*ln(2))*ln(x)+12*x**2*ln(2)**2),x)

[Out]

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

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