3.59.63 \(\int \frac {e^{e^{2 x}} (-2 x^2+(-9-x^2+e^{2 x} (108-18 x+12 x^2-2 x^3)) \log (9+x^2))+e^{e^{2 x}} (9+x^2+e^{2 x} (-216+18 x-24 x^2+2 x^3)) \log (9+x^2) \log (\log (9+x^2))+e^{e^{2 x}+2 x} (108+12 x^2) \log (9+x^2) \log ^2(\log (9+x^2))}{(27+3 x^2) \log (9+x^2)+(-54-6 x^2) \log (9+x^2) \log (\log (9+x^2))+(27+3 x^2) \log (9+x^2) \log ^2(\log (9+x^2))} \, dx\)

Optimal. Leaf size=25 \[ e^{e^{2 x}} \left (2+\frac {x}{-3+3 \log \left (\log \left (9+x^2\right )\right )}\right ) \]

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Rubi [F]  time = 5.62, 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^{e^{2 x}} \left (-2 x^2+\left (-9-x^2+e^{2 x} \left (108-18 x+12 x^2-2 x^3\right )\right ) \log \left (9+x^2\right )\right )+e^{e^{2 x}} \left (9+x^2+e^{2 x} \left (-216+18 x-24 x^2+2 x^3\right )\right ) \log \left (9+x^2\right ) \log \left (\log \left (9+x^2\right )\right )+e^{e^{2 x}+2 x} \left (108+12 x^2\right ) \log \left (9+x^2\right ) \log ^2\left (\log \left (9+x^2\right )\right )}{\left (27+3 x^2\right ) \log \left (9+x^2\right )+\left (-54-6 x^2\right ) \log \left (9+x^2\right ) \log \left (\log \left (9+x^2\right )\right )+\left (27+3 x^2\right ) \log \left (9+x^2\right ) \log ^2\left (\log \left (9+x^2\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^E^(2*x)*(-2*x^2 + (-9 - x^2 + E^(2*x)*(108 - 18*x + 12*x^2 - 2*x^3))*Log[9 + x^2]) + E^E^(2*x)*(9 + x^2
+ E^(2*x)*(-216 + 18*x - 24*x^2 + 2*x^3))*Log[9 + x^2]*Log[Log[9 + x^2]] + E^(E^(2*x) + 2*x)*(108 + 12*x^2)*L
og[9 + x^2]*Log[Log[9 + x^2]]^2)/((27 + 3*x^2)*Log[9 + x^2] + (-54 - 6*x^2)*Log[9 + x^2]*Log[Log[9 + x^2]] + (
27 + 3*x^2)*Log[9 + x^2]*Log[Log[9 + x^2]]^2),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{e^{2 x}} \left (-2 x^2+\left (9+x^2\right ) \log \left (9+x^2\right ) \left (-1+\log \left (\log \left (9+x^2\right )\right )\right ) \left (1+2 e^{2 x} (-6+x)+12 e^{2 x} \log \left (\log \left (9+x^2\right )\right )\right )\right )}{3 \left (9+x^2\right ) \log \left (9+x^2\right ) \left (1-\log \left (\log \left (9+x^2\right )\right )\right )^2} \, dx\\ &=\frac {1}{3} \int \frac {e^{e^{2 x}} \left (-2 x^2+\left (9+x^2\right ) \log \left (9+x^2\right ) \left (-1+\log \left (\log \left (9+x^2\right )\right )\right ) \left (1+2 e^{2 x} (-6+x)+12 e^{2 x} \log \left (\log \left (9+x^2\right )\right )\right )\right )}{\left (9+x^2\right ) \log \left (9+x^2\right ) \left (1-\log \left (\log \left (9+x^2\right )\right )\right )^2} \, dx\\ &=\frac {1}{3} \int \left (\frac {2 e^{e^{2 x}+2 x} \left (-6+x+6 \log \left (\log \left (9+x^2\right )\right )\right )}{-1+\log \left (\log \left (9+x^2\right )\right )}+\frac {e^{e^{2 x}} \left (-2 x^2-9 \log \left (9+x^2\right )-x^2 \log \left (9+x^2\right )+9 \log \left (9+x^2\right ) \log \left (\log \left (9+x^2\right )\right )+x^2 \log \left (9+x^2\right ) \log \left (\log \left (9+x^2\right )\right )\right )}{\left (9+x^2\right ) \log \left (9+x^2\right ) \left (-1+\log \left (\log \left (9+x^2\right )\right )\right )^2}\right ) \, dx\\ &=\frac {1}{3} \int \frac {e^{e^{2 x}} \left (-2 x^2-9 \log \left (9+x^2\right )-x^2 \log \left (9+x^2\right )+9 \log \left (9+x^2\right ) \log \left (\log \left (9+x^2\right )\right )+x^2 \log \left (9+x^2\right ) \log \left (\log \left (9+x^2\right )\right )\right )}{\left (9+x^2\right ) \log \left (9+x^2\right ) \left (-1+\log \left (\log \left (9+x^2\right )\right )\right )^2} \, dx+\frac {2}{3} \int \frac {e^{e^{2 x}+2 x} \left (-6+x+6 \log \left (\log \left (9+x^2\right )\right )\right )}{-1+\log \left (\log \left (9+x^2\right )\right )} \, dx\\ &=\frac {1}{3} \int \frac {e^{e^{2 x}} \left (-1-\frac {2 x^2}{\left (9+x^2\right ) \log \left (9+x^2\right )}+\log \left (\log \left (9+x^2\right )\right )\right )}{\left (1-\log \left (\log \left (9+x^2\right )\right )\right )^2} \, dx+\frac {2}{3} \int \left (6 e^{e^{2 x}+2 x}+\frac {e^{e^{2 x}+2 x} x}{-1+\log \left (\log \left (9+x^2\right )\right )}\right ) \, dx\\ &=\frac {1}{3} \int \left (-\frac {2 e^{e^{2 x}} x^2}{\left (9+x^2\right ) \log \left (9+x^2\right ) \left (-1+\log \left (\log \left (9+x^2\right )\right )\right )^2}+\frac {e^{e^{2 x}}}{-1+\log \left (\log \left (9+x^2\right )\right )}\right ) \, dx+\frac {2}{3} \int \frac {e^{e^{2 x}+2 x} x}{-1+\log \left (\log \left (9+x^2\right )\right )} \, dx+4 \int e^{e^{2 x}+2 x} \, dx\\ &=\frac {1}{3} \int \frac {e^{e^{2 x}}}{-1+\log \left (\log \left (9+x^2\right )\right )} \, dx-\frac {2}{3} \int \frac {e^{e^{2 x}} x^2}{\left (9+x^2\right ) \log \left (9+x^2\right ) \left (-1+\log \left (\log \left (9+x^2\right )\right )\right )^2} \, dx+\frac {2}{3} \int \frac {e^{e^{2 x}+2 x} x}{-1+\log \left (\log \left (9+x^2\right )\right )} \, dx+2 \operatorname {Subst}\left (\int e^x \, dx,x,e^{2 x}\right )\\ &=2 e^{e^{2 x}}+\frac {1}{3} \int \frac {e^{e^{2 x}}}{-1+\log \left (\log \left (9+x^2\right )\right )} \, dx-\frac {2}{3} \int \left (\frac {e^{e^{2 x}}}{\log \left (9+x^2\right ) \left (-1+\log \left (\log \left (9+x^2\right )\right )\right )^2}-\frac {9 e^{e^{2 x}}}{\left (9+x^2\right ) \log \left (9+x^2\right ) \left (-1+\log \left (\log \left (9+x^2\right )\right )\right )^2}\right ) \, dx+\frac {2}{3} \int \frac {e^{e^{2 x}+2 x} x}{-1+\log \left (\log \left (9+x^2\right )\right )} \, dx\\ &=2 e^{e^{2 x}}+\frac {1}{3} \int \frac {e^{e^{2 x}}}{-1+\log \left (\log \left (9+x^2\right )\right )} \, dx-\frac {2}{3} \int \frac {e^{e^{2 x}}}{\log \left (9+x^2\right ) \left (-1+\log \left (\log \left (9+x^2\right )\right )\right )^2} \, dx+\frac {2}{3} \int \frac {e^{e^{2 x}+2 x} x}{-1+\log \left (\log \left (9+x^2\right )\right )} \, dx+6 \int \frac {e^{e^{2 x}}}{\left (9+x^2\right ) \log \left (9+x^2\right ) \left (-1+\log \left (\log \left (9+x^2\right )\right )\right )^2} \, dx\\ &=2 e^{e^{2 x}}+\frac {1}{3} \int \frac {e^{e^{2 x}}}{-1+\log \left (\log \left (9+x^2\right )\right )} \, dx-\frac {2}{3} \int \frac {e^{e^{2 x}}}{\log \left (9+x^2\right ) \left (-1+\log \left (\log \left (9+x^2\right )\right )\right )^2} \, dx+\frac {2}{3} \int \frac {e^{e^{2 x}+2 x} x}{-1+\log \left (\log \left (9+x^2\right )\right )} \, dx+6 \int \left (\frac {i e^{e^{2 x}}}{6 (3 i-x) \log \left (9+x^2\right ) \left (-1+\log \left (\log \left (9+x^2\right )\right )\right )^2}+\frac {i e^{e^{2 x}}}{6 (3 i+x) \log \left (9+x^2\right ) \left (-1+\log \left (\log \left (9+x^2\right )\right )\right )^2}\right ) \, dx\\ &=2 e^{e^{2 x}}+i \int \frac {e^{e^{2 x}}}{(3 i-x) \log \left (9+x^2\right ) \left (-1+\log \left (\log \left (9+x^2\right )\right )\right )^2} \, dx+i \int \frac {e^{e^{2 x}}}{(3 i+x) \log \left (9+x^2\right ) \left (-1+\log \left (\log \left (9+x^2\right )\right )\right )^2} \, dx+\frac {1}{3} \int \frac {e^{e^{2 x}}}{-1+\log \left (\log \left (9+x^2\right )\right )} \, dx-\frac {2}{3} \int \frac {e^{e^{2 x}}}{\log \left (9+x^2\right ) \left (-1+\log \left (\log \left (9+x^2\right )\right )\right )^2} \, dx+\frac {2}{3} \int \frac {e^{e^{2 x}+2 x} x}{-1+\log \left (\log \left (9+x^2\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 26, normalized size = 1.04 \begin {gather*} \frac {1}{3} e^{e^{2 x}} \left (6+\frac {x}{-1+\log \left (\log \left (9+x^2\right )\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^E^(2*x)*(-2*x^2 + (-9 - x^2 + E^(2*x)*(108 - 18*x + 12*x^2 - 2*x^3))*Log[9 + x^2]) + E^E^(2*x)*(9
+ x^2 + E^(2*x)*(-216 + 18*x - 24*x^2 + 2*x^3))*Log[9 + x^2]*Log[Log[9 + x^2]] + E^(E^(2*x) + 2*x)*(108 + 12*
x^2)*Log[9 + x^2]*Log[Log[9 + x^2]]^2)/((27 + 3*x^2)*Log[9 + x^2] + (-54 - 6*x^2)*Log[9 + x^2]*Log[Log[9 + x^2
]] + (27 + 3*x^2)*Log[9 + x^2]*Log[Log[9 + x^2]]^2),x]

[Out]

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

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fricas [B]  time = 0.74, size = 55, normalized size = 2.20 \begin {gather*} \frac {{\left (x - 6\right )} e^{\left (2 \, x + e^{\left (2 \, x\right )}\right )} + 6 \, e^{\left (2 \, x + e^{\left (2 \, x\right )}\right )} \log \left (\log \left (x^{2} + 9\right )\right )}{3 \, {\left (e^{\left (2 \, x\right )} \log \left (\log \left (x^{2} + 9\right )\right ) - e^{\left (2 \, x\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x^2+108)*exp(2*x)*log(x^2+9)*exp(exp(2*x))*log(log(x^2+9))^2+((2*x^3-24*x^2+18*x-216)*exp(2*x)+
x^2+9)*log(x^2+9)*exp(exp(2*x))*log(log(x^2+9))+(((-2*x^3+12*x^2-18*x+108)*exp(2*x)-x^2-9)*log(x^2+9)-2*x^2)*e
xp(exp(2*x)))/((3*x^2+27)*log(x^2+9)*log(log(x^2+9))^2+(-6*x^2-54)*log(x^2+9)*log(log(x^2+9))+(3*x^2+27)*log(x
^2+9)),x, algorithm="fricas")

[Out]

1/3*((x - 6)*e^(2*x + e^(2*x)) + 6*e^(2*x + e^(2*x))*log(log(x^2 + 9)))/(e^(2*x)*log(log(x^2 + 9)) - e^(2*x))

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giac [B]  time = 0.22, size = 64, normalized size = 2.56 \begin {gather*} \frac {x e^{\left (2 \, x + e^{\left (2 \, x\right )}\right )} + 6 \, e^{\left (2 \, x + e^{\left (2 \, x\right )}\right )} \log \left (\log \left (x^{2} + 9\right )\right ) - 6 \, e^{\left (2 \, x + e^{\left (2 \, x\right )}\right )}}{3 \, {\left (e^{\left (2 \, x\right )} \log \left (\log \left (x^{2} + 9\right )\right ) - e^{\left (2 \, x\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x^2+108)*exp(2*x)*log(x^2+9)*exp(exp(2*x))*log(log(x^2+9))^2+((2*x^3-24*x^2+18*x-216)*exp(2*x)+
x^2+9)*log(x^2+9)*exp(exp(2*x))*log(log(x^2+9))+(((-2*x^3+12*x^2-18*x+108)*exp(2*x)-x^2-9)*log(x^2+9)-2*x^2)*e
xp(exp(2*x)))/((3*x^2+27)*log(x^2+9)*log(log(x^2+9))^2+(-6*x^2-54)*log(x^2+9)*log(log(x^2+9))+(3*x^2+27)*log(x
^2+9)),x, algorithm="giac")

[Out]

1/3*(x*e^(2*x + e^(2*x)) + 6*e^(2*x + e^(2*x))*log(log(x^2 + 9)) - 6*e^(2*x + e^(2*x)))/(e^(2*x)*log(log(x^2 +
9)) - e^(2*x))

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maple [A]  time = 0.07, size = 28, normalized size = 1.12

method	result	size
risch	\(2 \,{\mathrm e}^{{\mathrm e}^{2 x}}+\frac {x \,{\mathrm e}^{{\mathrm e}^{2 x}}}{3 \ln \left (\ln \left (x^{2}+9\right )\right )-3}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((12*x^2+108)*exp(2*x)*ln(x^2+9)*exp(exp(2*x))*ln(ln(x^2+9))^2+((2*x^3-24*x^2+18*x-216)*exp(2*x)+x^2+9)*ln
(x^2+9)*exp(exp(2*x))*ln(ln(x^2+9))+(((-2*x^3+12*x^2-18*x+108)*exp(2*x)-x^2-9)*ln(x^2+9)-2*x^2)*exp(exp(2*x)))
/((3*x^2+27)*ln(x^2+9)*ln(ln(x^2+9))^2+(-6*x^2-54)*ln(x^2+9)*ln(ln(x^2+9))+(3*x^2+27)*ln(x^2+9)),x,method=_RET
URNVERBOSE)

[Out]

2*exp(exp(2*x))+1/3*x*exp(exp(2*x))/(ln(ln(x^2+9))-1)

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maxima [A]  time = 0.58, size = 37, normalized size = 1.48 \begin {gather*} \frac {{\left (x - 6\right )} e^{\left (e^{\left (2 \, x\right )}\right )} + 6 \, e^{\left (e^{\left (2 \, x\right )}\right )} \log \left (\log \left (x^{2} + 9\right )\right )}{3 \, {\left (\log \left (\log \left (x^{2} + 9\right )\right ) - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x^2+108)*exp(2*x)*log(x^2+9)*exp(exp(2*x))*log(log(x^2+9))^2+((2*x^3-24*x^2+18*x-216)*exp(2*x)+
x^2+9)*log(x^2+9)*exp(exp(2*x))*log(log(x^2+9))+(((-2*x^3+12*x^2-18*x+108)*exp(2*x)-x^2-9)*log(x^2+9)-2*x^2)*e
xp(exp(2*x)))/((3*x^2+27)*log(x^2+9)*log(log(x^2+9))^2+(-6*x^2-54)*log(x^2+9)*log(log(x^2+9))+(3*x^2+27)*log(x
^2+9)),x, algorithm="maxima")

[Out]

1/3*((x - 6)*e^(e^(2*x)) + 6*e^(e^(2*x))*log(log(x^2 + 9)))/(log(log(x^2 + 9)) - 1)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\mathrm {e}}^{2\,x+{\mathrm {e}}^{2\,x}}\,\ln \left (x^2+9\right )\,\left (12\,x^2+108\right )\,{\ln \left (\ln \left (x^2+9\right )\right )}^2+{\mathrm {e}}^{{\mathrm {e}}^{2\,x}}\,\ln \left (x^2+9\right )\,\left ({\mathrm {e}}^{2\,x}\,\left (2\,x^3-24\,x^2+18\,x-216\right )+x^2+9\right )\,\ln \left (\ln \left (x^2+9\right )\right )-{\mathrm {e}}^{{\mathrm {e}}^{2\,x}}\,\left (\ln \left (x^2+9\right )\,\left ({\mathrm {e}}^{2\,x}\,\left (2\,x^3-12\,x^2+18\,x-108\right )+x^2+9\right )+2\,x^2\right )}{\ln \left (x^2+9\right )\,\left (3\,x^2+27\right )\,{\ln \left (\ln \left (x^2+9\right )\right )}^2-\ln \left (x^2+9\right )\,\left (6\,x^2+54\right )\,\ln \left (\ln \left (x^2+9\right )\right )+\ln \left (x^2+9\right )\,\left (3\,x^2+27\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(log(x^2 + 9))*exp(exp(2*x))*log(x^2 + 9)*(exp(2*x)*(18*x - 24*x^2 + 2*x^3 - 216) + x^2 + 9) - exp(exp
(2*x))*(log(x^2 + 9)*(exp(2*x)*(18*x - 12*x^2 + 2*x^3 - 108) + x^2 + 9) + 2*x^2) + log(log(x^2 + 9))^2*exp(2*x
)*exp(exp(2*x))*log(x^2 + 9)*(12*x^2 + 108))/(log(x^2 + 9)*(3*x^2 + 27) + log(log(x^2 + 9))^2*log(x^2 + 9)*(3*
x^2 + 27) - log(log(x^2 + 9))*log(x^2 + 9)*(6*x^2 + 54)),x)

[Out]

int((log(log(x^2 + 9))*exp(exp(2*x))*log(x^2 + 9)*(exp(2*x)*(18*x - 24*x^2 + 2*x^3 - 216) + x^2 + 9) - exp(exp
(2*x))*(log(x^2 + 9)*(exp(2*x)*(18*x - 12*x^2 + 2*x^3 - 108) + x^2 + 9) + 2*x^2) + log(log(x^2 + 9))^2*exp(2*x
+ exp(2*x))*log(x^2 + 9)*(12*x^2 + 108))/(log(x^2 + 9)*(3*x^2 + 27) + log(log(x^2 + 9))^2*log(x^2 + 9)*(3*x^2
+ 27) - log(log(x^2 + 9))*log(x^2 + 9)*(6*x^2 + 54)), x)

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sympy [A]  time = 1.03, size = 31, normalized size = 1.24 \begin {gather*} \frac {\left (x + 6 \log {\left (\log {\left (x^{2} + 9 \right )} \right )} - 6\right ) e^{e^{2 x}}}{3 \log {\left (\log {\left (x^{2} + 9 \right )} \right )} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x**2+108)*exp(2*x)*ln(x**2+9)*exp(exp(2*x))*ln(ln(x**2+9))**2+((2*x**3-24*x**2+18*x-216)*exp(2*
x)+x**2+9)*ln(x**2+9)*exp(exp(2*x))*ln(ln(x**2+9))+(((-2*x**3+12*x**2-18*x+108)*exp(2*x)-x**2-9)*ln(x**2+9)-2*
x**2)*exp(exp(2*x)))/((3*x**2+27)*ln(x**2+9)*ln(ln(x**2+9))**2+(-6*x**2-54)*ln(x**2+9)*ln(ln(x**2+9))+(3*x**2+
27)*ln(x**2+9)),x)

[Out]

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

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