3.7.76 \(\int \frac {-e^{3 x}+e^{2 x} (-1-2 \log (4))-3 x \log (4)-\log ^2(4)+e^x (-3 x-2 \log (4)-\log ^2(4))+(-e^{3 x} x-3 x \log (4)-2 e^{2 x} x \log (4)+e^x (-3 x+3 x^2-x \log ^2(4))) \log (x)}{3 (e^{4 x} x+9 x^3+6 x^2 \log (4)+x \log ^2(4)+e^{3 x} (2 x+2 x \log (4))+e^{2 x} (x+6 x^2+4 x \log (4)+x \log ^2(4))+e^x (6 x^2+(2 x+6 x^2) \log (4)+2 x \log ^2(4))) \log ^2(x)} \, dx\)

Optimal. Leaf size=26 \[ \frac {1}{3 \left (1+e^x+\frac {3 x}{e^x+\log (4)}\right ) \log (x)} \]

________________________________________________________________________________________

Rubi [F]  time = 6.79, 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 {-e^{3 x}+e^{2 x} (-1-2 \log (4))-3 x \log (4)-\log ^2(4)+e^x \left (-3 x-2 \log (4)-\log ^2(4)\right )+\left (-e^{3 x} x-3 x \log (4)-2 e^{2 x} x \log (4)+e^x \left (-3 x+3 x^2-x \log ^2(4)\right )\right ) \log (x)}{3 \left (e^{4 x} x+9 x^3+6 x^2 \log (4)+x \log ^2(4)+e^{3 x} (2 x+2 x \log (4))+e^{2 x} \left (x+6 x^2+4 x \log (4)+x \log ^2(4)\right )+e^x \left (6 x^2+\left (2 x+6 x^2\right ) \log (4)+2 x \log ^2(4)\right )\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-E^(3*x) + E^(2*x)*(-1 - 2*Log[4]) - 3*x*Log[4] - Log[4]^2 + E^x*(-3*x - 2*Log[4] - Log[4]^2) + (-(E^(3*x
)*x) - 3*x*Log[4] - 2*E^(2*x)*x*Log[4] + E^x*(-3*x + 3*x^2 - x*Log[4]^2))*Log[x])/(3*(E^(4*x)*x + 9*x^3 + 6*x^
2*Log[4] + x*Log[4]^2 + E^(3*x)*(2*x + 2*x*Log[4]) + E^(2*x)*(x + 6*x^2 + 4*x*Log[4] + x*Log[4]^2) + E^x*(6*x^
2 + (2*x + 6*x^2)*Log[4] + 2*x*Log[4]^2))*Log[x]^2),x]

[Out]

(Log[4]*Defer[Int][1/(x*(-E^(2*x) - 3*x - Log[4] - E^x*(1 + Log[4]))*Log[x]^2), x])/3 + Defer[Int][E^x/(x*(-E^
(2*x) - 3*x - Log[4] - E^x*(1 + Log[4]))*Log[x]^2), x]/3 + Defer[Int][E^x/((-E^(2*x) - 3*x - Log[4] - E^x*(1 +
Log[4]))*Log[x]), x]/3 - ((Log[4]^2 - Log[4]*Log[16] + Log[256])*Defer[Int][1/((E^(2*x) + 3*x + Log[4] + E^x*
(1 + Log[4]))^2*Log[x]), x])/3 - (2*(2 - Log[2])*Defer[Int][E^x/((E^(2*x) + 3*x + Log[4] + E^x*(1 + Log[4]))^2
*Log[x]), x])/3 - (1 - Log[4])*Defer[Int][x/((E^(2*x) + 3*x + Log[4] + E^x*(1 + Log[4]))^2*Log[x]), x] + 2*Def
er[Int][(E^x*x)/((E^(2*x) + 3*x + Log[4] + E^x*(1 + Log[4]))^2*Log[x]), x] + ((1 - Log[4])*Defer[Int][1/((E^(2
*x) + 3*x + Log[4] + E^x*(1 + Log[4]))*Log[x]), x])/3

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \frac {-e^{3 x}+e^{2 x} (-1-2 \log (4))-3 x \log (4)-\log ^2(4)+e^x \left (-3 x-2 \log (4)-\log ^2(4)\right )+\left (-e^{3 x} x-3 x \log (4)-2 e^{2 x} x \log (4)+e^x \left (-3 x+3 x^2-x \log ^2(4)\right )\right ) \log (x)}{\left (e^{4 x} x+9 x^3+6 x^2 \log (4)+x \log ^2(4)+e^{3 x} (2 x+2 x \log (4))+e^{2 x} \left (x+6 x^2+4 x \log (4)+x \log ^2(4)\right )+e^x \left (6 x^2+\left (2 x+6 x^2\right ) \log (4)+2 x \log ^2(4)\right )\right ) \log ^2(x)} \, dx\\ &=\frac {1}{3} \int \frac {-\left (\left (e^x+\log (4)\right ) \left (e^{2 x}+3 x+\log (4)+e^x (1+\log (4))\right )\right )-x \left (e^{3 x}+e^x \left (3-3 x+\log ^2(4)\right )+e^{2 x} \log (16)+\log (64)\right ) \log (x)}{x \left (e^{2 x}+3 x+\log (4)+e^x (1+\log (4))\right )^2 \log ^2(x)} \, dx\\ &=\frac {1}{3} \int \left (\frac {6 e^x x-4 e^x \left (1-\frac {\log (2)}{2}\right )-3 x (1-\log (4))-\log ^2(4) \left (1+\frac {-\log (4) \log (16)+\log (256)}{\log ^2(4)}\right )}{\left (e^{2 x}+3 x+\log (4)+e^x (1+\log (4))\right )^2 \log (x)}+\frac {-e^x-\log (4)-e^x x \log (x)+x (1-\log (4)) \log (x)}{x \left (e^{2 x}+3 x+\log (4)+e^x (1+\log (4))\right ) \log ^2(x)}\right ) \, dx\\ &=\frac {1}{3} \int \frac {6 e^x x-4 e^x \left (1-\frac {\log (2)}{2}\right )-3 x (1-\log (4))-\log ^2(4) \left (1+\frac {-\log (4) \log (16)+\log (256)}{\log ^2(4)}\right )}{\left (e^{2 x}+3 x+\log (4)+e^x (1+\log (4))\right )^2 \log (x)} \, dx+\frac {1}{3} \int \frac {-e^x-\log (4)-e^x x \log (x)+x (1-\log (4)) \log (x)}{x \left (e^{2 x}+3 x+\log (4)+e^x (1+\log (4))\right ) \log ^2(x)} \, dx\\ &=\frac {1}{3} \int \left (\frac {e^x}{x \left (-e^{2 x}-3 x-\log (4)-e^x (1+\log (4))\right ) \log ^2(x)}+\frac {\log (4)}{x \left (-e^{2 x}-3 x-\log (4)-e^x (1+\log (4))\right ) \log ^2(x)}+\frac {e^x}{\left (-e^{2 x}-3 x-\log (4)-e^x (1+\log (4))\right ) \log (x)}+\frac {1-\log (4)}{\left (e^{2 x}+3 x+\log (4)+e^x (1+\log (4))\right ) \log (x)}\right ) \, dx+\frac {1}{3} \int \left (\frac {6 e^x x}{\left (e^{2 x}+3 x+\log (4)+e^x (1+\log (4))\right )^2 \log (x)}+\frac {2 e^x (-2+\log (2))}{\left (e^{2 x}+3 x+\log (4)+e^x (1+\log (4))\right )^2 \log (x)}+\frac {3 x (-1+\log (4))}{\left (e^{2 x}+3 x+\log (4)+e^x (1+\log (4))\right )^2 \log (x)}+\frac {-\log ^2(4)+\log (4) \log (16)-\log (256)}{\left (e^{2 x}+3 x+\log (4)+e^x (1+\log (4))\right )^2 \log (x)}\right ) \, dx\\ &=\frac {1}{3} \int \frac {e^x}{x \left (-e^{2 x}-3 x-\log (4)-e^x (1+\log (4))\right ) \log ^2(x)} \, dx+\frac {1}{3} \int \frac {e^x}{\left (-e^{2 x}-3 x-\log (4)-e^x (1+\log (4))\right ) \log (x)} \, dx+2 \int \frac {e^x x}{\left (e^{2 x}+3 x+\log (4)+e^x (1+\log (4))\right )^2 \log (x)} \, dx-\frac {1}{3} (2 (2-\log (2))) \int \frac {e^x}{\left (e^{2 x}+3 x+\log (4)+e^x (1+\log (4))\right )^2 \log (x)} \, dx+\frac {1}{3} (1-\log (4)) \int \frac {1}{\left (e^{2 x}+3 x+\log (4)+e^x (1+\log (4))\right ) \log (x)} \, dx+(-1+\log (4)) \int \frac {x}{\left (e^{2 x}+3 x+\log (4)+e^x (1+\log (4))\right )^2 \log (x)} \, dx+\frac {1}{3} \log (4) \int \frac {1}{x \left (-e^{2 x}-3 x-\log (4)-e^x (1+\log (4))\right ) \log ^2(x)} \, dx+\frac {1}{3} \left (-\log ^2(4)+\log (4) \log (16)-\log (256)\right ) \int \frac {1}{\left (e^{2 x}+3 x+\log (4)+e^x (1+\log (4))\right )^2 \log (x)} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.15, size = 36, normalized size = 1.38 \begin {gather*} \frac {e^x+\log (4)}{3 \left (e^x+e^{2 x}+3 x+\log (4)+e^x \log (4)\right ) \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-E^(3*x) + E^(2*x)*(-1 - 2*Log[4]) - 3*x*Log[4] - Log[4]^2 + E^x*(-3*x - 2*Log[4] - Log[4]^2) + (-(
E^(3*x)*x) - 3*x*Log[4] - 2*E^(2*x)*x*Log[4] + E^x*(-3*x + 3*x^2 - x*Log[4]^2))*Log[x])/(3*(E^(4*x)*x + 9*x^3
+ 6*x^2*Log[4] + x*Log[4]^2 + E^(3*x)*(2*x + 2*x*Log[4]) + E^(2*x)*(x + 6*x^2 + 4*x*Log[4] + x*Log[4]^2) + E^x
*(6*x^2 + (2*x + 6*x^2)*Log[4] + 2*x*Log[4]^2))*Log[x]^2),x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.49, size = 36, normalized size = 1.38 \begin {gather*} \frac {e^{x} + 2 \, \log \relax (2)}{3 \, {\left ({\left (2 \, \log \relax (2) + 1\right )} e^{x} + 3 \, x + e^{\left (2 \, x\right )} + 2 \, \log \relax (2)\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((-x*exp(x)^3-4*x*log(2)*exp(x)^2+(-4*x*log(2)^2+3*x^2-3*x)*exp(x)-6*x*log(2))*log(x)-exp(x)^3+(
-4*log(2)-1)*exp(x)^2+(-4*log(2)^2-4*log(2)-3*x)*exp(x)-4*log(2)^2-6*x*log(2))/log(x)^2/(x*exp(x)^4+(4*x*log(2
)+2*x)*exp(x)^3+(4*x*log(2)^2+8*x*log(2)+6*x^2+x)*exp(x)^2+(8*x*log(2)^2+2*(6*x^2+2*x)*log(2)+6*x^2)*exp(x)+4*
x*log(2)^2+12*x^2*log(2)+9*x^3),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((-x*exp(x)^3-4*x*log(2)*exp(x)^2+(-4*x*log(2)^2+3*x^2-3*x)*exp(x)-6*x*log(2))*log(x)-exp(x)^3+(
-4*log(2)-1)*exp(x)^2+(-4*log(2)^2-4*log(2)-3*x)*exp(x)-4*log(2)^2-6*x*log(2))/log(x)^2/(x*exp(x)^4+(4*x*log(2
)+2*x)*exp(x)^3+(4*x*log(2)^2+8*x*log(2)+6*x^2+x)*exp(x)^2+(8*x*log(2)^2+2*(6*x^2+2*x)*log(2)+6*x^2)*exp(x)+4*
x*log(2)^2+12*x^2*log(2)+9*x^3),x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

maple [A]  time = 0.07, size = 36, normalized size = 1.38

method	result	size
risch	\(\frac {{\mathrm e}^{x}+2 \ln \relax (2)}{3 \left ({\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} \ln \relax (2)+{\mathrm e}^{x}+2 \ln \relax (2)+3 x \right ) \ln \relax (x )}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*((-x*exp(x)^3-4*x*ln(2)*exp(x)^2+(-4*x*ln(2)^2+3*x^2-3*x)*exp(x)-6*x*ln(2))*ln(x)-exp(x)^3+(-4*ln(2)-1
)*exp(x)^2+(-4*ln(2)^2-4*ln(2)-3*x)*exp(x)-4*ln(2)^2-6*x*ln(2))/ln(x)^2/(x*exp(x)^4+(4*x*ln(2)+2*x)*exp(x)^3+(
4*x*ln(2)^2+8*x*ln(2)+6*x^2+x)*exp(x)^2+(8*x*ln(2)^2+2*(6*x^2+2*x)*ln(2)+6*x^2)*exp(x)+4*x*ln(2)^2+12*x^2*ln(2
)+9*x^3),x,method=_RETURNVERBOSE)

[Out]

1/3*(exp(x)+2*ln(2))/(exp(2*x)+2*exp(x)*ln(2)+exp(x)+2*ln(2)+3*x)/ln(x)

________________________________________________________________________________________

maxima [A]  time = 0.83, size = 41, normalized size = 1.58 \begin {gather*} \frac {e^{x} + 2 \, \log \relax (2)}{3 \, {\left ({\left (2 \, \log \relax (2) + 1\right )} e^{x} \log \relax (x) + {\left (3 \, x + 2 \, \log \relax (2)\right )} \log \relax (x) + e^{\left (2 \, x\right )} \log \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((-x*exp(x)^3-4*x*log(2)*exp(x)^2+(-4*x*log(2)^2+3*x^2-3*x)*exp(x)-6*x*log(2))*log(x)-exp(x)^3+(
-4*log(2)-1)*exp(x)^2+(-4*log(2)^2-4*log(2)-3*x)*exp(x)-4*log(2)^2-6*x*log(2))/log(x)^2/(x*exp(x)^4+(4*x*log(2
)+2*x)*exp(x)^3+(4*x*log(2)^2+8*x*log(2)+6*x^2+x)*exp(x)^2+(8*x*log(2)^2+2*(6*x^2+2*x)*log(2)+6*x^2)*exp(x)+4*
x*log(2)^2+12*x^2*log(2)+9*x^3),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {\frac {{\mathrm {e}}^{3\,x}}{3}+2\,x\,\ln \relax (2)+\frac {{\mathrm {e}}^x\,\left (3\,x+4\,\ln \relax (2)+4\,{\ln \relax (2)}^2\right )}{3}+\frac {{\mathrm {e}}^{2\,x}\,\left (4\,\ln \relax (2)+1\right )}{3}+\frac {\ln \relax (x)\,\left (x\,{\mathrm {e}}^{3\,x}+6\,x\,\ln \relax (2)+{\mathrm {e}}^x\,\left (3\,x+4\,x\,{\ln \relax (2)}^2-3\,x^2\right )+4\,x\,{\mathrm {e}}^{2\,x}\,\ln \relax (2)\right )}{3}+\frac {4\,{\ln \relax (2)}^2}{3}}{{\ln \relax (x)}^2\,\left (x\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{3\,x}\,\left (2\,x+4\,x\,\ln \relax (2)\right )+{\mathrm {e}}^{2\,x}\,\left (x+8\,x\,\ln \relax (2)+4\,x\,{\ln \relax (2)}^2+6\,x^2\right )+4\,x\,{\ln \relax (2)}^2+12\,x^2\,\ln \relax (2)+{\mathrm {e}}^x\,\left (2\,\ln \relax (2)\,\left (6\,x^2+2\,x\right )+8\,x\,{\ln \relax (2)}^2+6\,x^2\right )+9\,x^3\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(3*x)/3 + 2*x*log(2) + (exp(x)*(3*x + 4*log(2) + 4*log(2)^2))/3 + (exp(2*x)*(4*log(2) + 1))/3 + (log(
x)*(x*exp(3*x) + 6*x*log(2) + exp(x)*(3*x + 4*x*log(2)^2 - 3*x^2) + 4*x*exp(2*x)*log(2)))/3 + (4*log(2)^2)/3)/
(log(x)^2*(x*exp(4*x) + exp(3*x)*(2*x + 4*x*log(2)) + exp(2*x)*(x + 8*x*log(2) + 4*x*log(2)^2 + 6*x^2) + 4*x*l
og(2)^2 + 12*x^2*log(2) + exp(x)*(2*log(2)*(2*x + 6*x^2) + 8*x*log(2)^2 + 6*x^2) + 9*x^3)),x)

[Out]

int(-(exp(3*x)/3 + 2*x*log(2) + (exp(x)*(3*x + 4*log(2) + 4*log(2)^2))/3 + (exp(2*x)*(4*log(2) + 1))/3 + (log(
x)*(x*exp(3*x) + 6*x*log(2) + exp(x)*(3*x + 4*x*log(2)^2 - 3*x^2) + 4*x*exp(2*x)*log(2)))/3 + (4*log(2)^2)/3)/
(log(x)^2*(x*exp(4*x) + exp(3*x)*(2*x + 4*x*log(2)) + exp(2*x)*(x + 8*x*log(2) + 4*x*log(2)^2 + 6*x^2) + 4*x*l
og(2)^2 + 12*x^2*log(2) + exp(x)*(2*log(2)*(2*x + 6*x^2) + 8*x*log(2)^2 + 6*x^2) + 9*x^3)), x)

________________________________________________________________________________________

sympy [B]  time = 0.46, size = 49, normalized size = 1.88 \begin {gather*} \frac {e^{x} + 2 \log {\relax (2 )}}{9 x \log {\relax (x )} + \left (3 \log {\relax (x )} + 6 \log {\relax (2 )} \log {\relax (x )}\right ) e^{x} + 3 e^{2 x} \log {\relax (x )} + 6 \log {\relax (2 )} \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((-x*exp(x)**3-4*x*ln(2)*exp(x)**2+(-4*x*ln(2)**2+3*x**2-3*x)*exp(x)-6*x*ln(2))*ln(x)-exp(x)**3+
(-4*ln(2)-1)*exp(x)**2+(-4*ln(2)**2-4*ln(2)-3*x)*exp(x)-4*ln(2)**2-6*x*ln(2))/ln(x)**2/(x*exp(x)**4+(4*x*ln(2)
+2*x)*exp(x)**3+(4*x*ln(2)**2+8*x*ln(2)+6*x**2+x)*exp(x)**2+(8*x*ln(2)**2+2*(6*x**2+2*x)*ln(2)+6*x**2)*exp(x)+
4*x*ln(2)**2+12*x**2*ln(2)+9*x**3),x)

[Out]

(exp(x) + 2*log(2))/(9*x*log(x) + (3*log(x) + 6*log(2)*log(x))*exp(x) + 3*exp(2*x)*log(x) + 6*log(2)*log(x))

________________________________________________________________________________________