3.21.75 \(\int (2 e^{2 e^x+x}+e^{2 x} (4 x^3+2 x^4)-72 x^2 \log (4)+96 x^5 \log ^2(4)+e^x (12 x+6 x^2+(-40 x^4-8 x^5) \log (4))+e^{e^x} (-2 e^{2 x} x^2+24 x^2 \log (4)+e^x (-6-4 x-2 x^2+8 x^3 \log (4)))) \, dx\)

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

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Rubi [F]  time = 0.88, 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 \left (2 e^{2 e^x+x}+e^{2 x} \left (4 x^3+2 x^4\right )-72 x^2 \log (4)+96 x^5 \log ^2(4)+e^x \left (12 x+6 x^2+\left (-40 x^4-8 x^5\right ) \log (4)\right )+e^{e^x} \left (-2 e^{2 x} x^2+24 x^2 \log (4)+e^x \left (-6-4 x-2 x^2+8 x^3 \log (4)\right )\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[2*E^(2*E^x + x) + E^(2*x)*(4*x^3 + 2*x^4) - 72*x^2*Log[4] + 96*x^5*Log[4]^2 + E^x*(12*x + 6*x^2 + (-40*x^4
- 8*x^5)*Log[4]) + E^E^x*(-2*E^(2*x)*x^2 + 24*x^2*Log[4] + E^x*(-6 - 4*x - 2*x^2 + 8*x^3*Log[4])),x]

[Out]

-6*E^E^x + E^(2*E^x) + 6*E^x*x^2 + E^(2*x)*x^4 - 24*x^3*Log[4] - 8*E^x*x^5*Log[4] + 16*x^6*Log[4]^2 - 4*Defer[
Int][E^(E^x + x)*x, x] + 24*Log[4]*Defer[Int][E^E^x*x^2, x] - 2*Defer[Int][E^(E^x + x)*x^2, x] - 2*Defer[Int][
E^(E^x + 2*x)*x^2, x] + 8*Log[4]*Defer[Int][E^(E^x + x)*x^3, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-24 x^3 \log (4)+16 x^6 \log ^2(4)+2 \int e^{2 e^x+x} \, dx+\int e^{2 x} \left (4 x^3+2 x^4\right ) \, dx+\int e^x \left (12 x+6 x^2+\left (-40 x^4-8 x^5\right ) \log (4)\right ) \, dx+\int e^{e^x} \left (-2 e^{2 x} x^2+24 x^2 \log (4)+e^x \left (-6-4 x-2 x^2+8 x^3 \log (4)\right )\right ) \, dx\\ &=-24 x^3 \log (4)+16 x^6 \log ^2(4)+2 \operatorname {Subst}\left (\int e^{2 x} \, dx,x,e^x\right )+\int e^{2 x} x^3 (4+2 x) \, dx+\int \left (12 e^x x+6 e^x x^2-8 e^x x^4 (5+x) \log (4)\right ) \, dx+\int \left (-2 e^{e^x+2 x} x^2+24 e^{e^x} x^2 \log (4)+2 e^{e^x+x} \left (-3-2 x-x^2+4 x^3 \log (4)\right )\right ) \, dx\\ &=e^{2 e^x}-24 x^3 \log (4)+16 x^6 \log ^2(4)-2 \int e^{e^x+2 x} x^2 \, dx+2 \int e^{e^x+x} \left (-3-2 x-x^2+4 x^3 \log (4)\right ) \, dx+6 \int e^x x^2 \, dx+12 \int e^x x \, dx-(8 \log (4)) \int e^x x^4 (5+x) \, dx+(24 \log (4)) \int e^{e^x} x^2 \, dx+\int \left (4 e^{2 x} x^3+2 e^{2 x} x^4\right ) \, dx\\ &=e^{2 e^x}+12 e^x x+6 e^x x^2-24 x^3 \log (4)+16 x^6 \log ^2(4)-2 \int e^{e^x+2 x} x^2 \, dx+2 \int e^{2 x} x^4 \, dx+2 \int \left (-3 e^{e^x+x}-2 e^{e^x+x} x-e^{e^x+x} x^2+4 e^{e^x+x} x^3 \log (4)\right ) \, dx+4 \int e^{2 x} x^3 \, dx-12 \int e^x \, dx-12 \int e^x x \, dx-(8 \log (4)) \int \left (5 e^x x^4+e^x x^5\right ) \, dx+(24 \log (4)) \int e^{e^x} x^2 \, dx\\ &=e^{2 e^x}-12 e^x+6 e^x x^2+2 e^{2 x} x^3+e^{2 x} x^4-24 x^3 \log (4)+16 x^6 \log ^2(4)-2 \int e^{e^x+x} x^2 \, dx-2 \int e^{e^x+2 x} x^2 \, dx-4 \int e^{e^x+x} x \, dx-4 \int e^{2 x} x^3 \, dx-6 \int e^{e^x+x} \, dx-6 \int e^{2 x} x^2 \, dx+12 \int e^x \, dx+(8 \log (4)) \int e^{e^x+x} x^3 \, dx-(8 \log (4)) \int e^x x^5 \, dx+(24 \log (4)) \int e^{e^x} x^2 \, dx-(40 \log (4)) \int e^x x^4 \, dx\\ &=e^{2 e^x}+6 e^x x^2-3 e^{2 x} x^2+e^{2 x} x^4-24 x^3 \log (4)-40 e^x x^4 \log (4)-8 e^x x^5 \log (4)+16 x^6 \log ^2(4)-2 \int e^{e^x+x} x^2 \, dx-2 \int e^{e^x+2 x} x^2 \, dx-4 \int e^{e^x+x} x \, dx+6 \int e^{2 x} x \, dx+6 \int e^{2 x} x^2 \, dx-6 \operatorname {Subst}\left (\int e^x \, dx,x,e^x\right )+(8 \log (4)) \int e^{e^x+x} x^3 \, dx+(24 \log (4)) \int e^{e^x} x^2 \, dx+(40 \log (4)) \int e^x x^4 \, dx+(160 \log (4)) \int e^x x^3 \, dx\\ &=-6 e^{e^x}+e^{2 e^x}+3 e^{2 x} x+6 e^x x^2+e^{2 x} x^4-24 x^3 \log (4)+160 e^x x^3 \log (4)-8 e^x x^5 \log (4)+16 x^6 \log ^2(4)-2 \int e^{e^x+x} x^2 \, dx-2 \int e^{e^x+2 x} x^2 \, dx-3 \int e^{2 x} \, dx-4 \int e^{e^x+x} x \, dx-6 \int e^{2 x} x \, dx+(8 \log (4)) \int e^{e^x+x} x^3 \, dx+(24 \log (4)) \int e^{e^x} x^2 \, dx-(160 \log (4)) \int e^x x^3 \, dx-(480 \log (4)) \int e^x x^2 \, dx\\ &=-6 e^{e^x}+e^{2 e^x}-\frac {3 e^{2 x}}{2}+6 e^x x^2+e^{2 x} x^4-480 e^x x^2 \log (4)-24 x^3 \log (4)-8 e^x x^5 \log (4)+16 x^6 \log ^2(4)-2 \int e^{e^x+x} x^2 \, dx-2 \int e^{e^x+2 x} x^2 \, dx+3 \int e^{2 x} \, dx-4 \int e^{e^x+x} x \, dx+(8 \log (4)) \int e^{e^x+x} x^3 \, dx+(24 \log (4)) \int e^{e^x} x^2 \, dx+(480 \log (4)) \int e^x x^2 \, dx+(960 \log (4)) \int e^x x \, dx\\ &=-6 e^{e^x}+e^{2 e^x}+6 e^x x^2+e^{2 x} x^4+960 e^x x \log (4)-24 x^3 \log (4)-8 e^x x^5 \log (4)+16 x^6 \log ^2(4)-2 \int e^{e^x+x} x^2 \, dx-2 \int e^{e^x+2 x} x^2 \, dx-4 \int e^{e^x+x} x \, dx+(8 \log (4)) \int e^{e^x+x} x^3 \, dx+(24 \log (4)) \int e^{e^x} x^2 \, dx-(960 \log (4)) \int e^x \, dx-(960 \log (4)) \int e^x x \, dx\\ &=-6 e^{e^x}+e^{2 e^x}+6 e^x x^2+e^{2 x} x^4-960 e^x \log (4)-24 x^3 \log (4)-8 e^x x^5 \log (4)+16 x^6 \log ^2(4)-2 \int e^{e^x+x} x^2 \, dx-2 \int e^{e^x+2 x} x^2 \, dx-4 \int e^{e^x+x} x \, dx+(8 \log (4)) \int e^{e^x+x} x^3 \, dx+(24 \log (4)) \int e^{e^x} x^2 \, dx+(960 \log (4)) \int e^x \, dx\\ &=-6 e^{e^x}+e^{2 e^x}+6 e^x x^2+e^{2 x} x^4-24 x^3 \log (4)-8 e^x x^5 \log (4)+16 x^6 \log ^2(4)-2 \int e^{e^x+x} x^2 \, dx-2 \int e^{e^x+2 x} x^2 \, dx-4 \int e^{e^x+x} x \, dx+(8 \log (4)) \int e^{e^x+x} x^3 \, dx+(24 \log (4)) \int e^{e^x} x^2 \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.34, size = 73, normalized size = 3.04 \begin {gather*} e^{2 e^x}+e^{2 x} x^4-24 x^3 \log (4)+16 x^6 \log ^2(4)+e^{e^x} \left (-6-2 e^x x^2+8 x^3 \log (4)\right )-2 e^x x \left (-3 x+4 x^4 \log (4)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[2*E^(2*E^x + x) + E^(2*x)*(4*x^3 + 2*x^4) - 72*x^2*Log[4] + 96*x^5*Log[4]^2 + E^x*(12*x + 6*x^2 + (-
40*x^4 - 8*x^5)*Log[4]) + E^E^x*(-2*E^(2*x)*x^2 + 24*x^2*Log[4] + E^x*(-6 - 4*x - 2*x^2 + 8*x^3*Log[4])),x]

[Out]

E^(2*E^x) + E^(2*x)*x^4 - 24*x^3*Log[4] + 16*x^6*Log[4]^2 + E^E^x*(-6 - 2*E^x*x^2 + 8*x^3*Log[4]) - 2*E^x*x*(-
3*x + 4*x^4*Log[4])

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fricas [B]  time = 1.02, size = 68, normalized size = 2.83 \begin {gather*} 64 \, x^{6} \log \relax (2)^{2} + x^{4} e^{\left (2 \, x\right )} - 48 \, x^{3} \log \relax (2) - 2 \, {\left (8 \, x^{5} \log \relax (2) - 3 \, x^{2}\right )} e^{x} + 2 \, {\left (8 \, x^{3} \log \relax (2) - x^{2} e^{x} - 3\right )} e^{\left (e^{x}\right )} + e^{\left (2 \, e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*exp(x)*exp(exp(x))^2+(-2*exp(x)^2*x^2+(16*x^3*log(2)-2*x^2-4*x-6)*exp(x)+48*x^2*log(2))*exp(exp(x)
)+(2*x^4+4*x^3)*exp(x)^2+(2*(-8*x^5-40*x^4)*log(2)+6*x^2+12*x)*exp(x)+384*x^5*log(2)^2-144*x^2*log(2),x, algor
ithm="fricas")

[Out]

64*x^6*log(2)^2 + x^4*e^(2*x) - 48*x^3*log(2) - 2*(8*x^5*log(2) - 3*x^2)*e^x + 2*(8*x^3*log(2) - x^2*e^x - 3)*
e^(e^x) + e^(2*e^x)

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giac [B]  time = 0.21, size = 85, normalized size = 3.54 \begin {gather*} 64 \, x^{6} \log \relax (2)^{2} + x^{4} e^{\left (2 \, x\right )} - 48 \, x^{3} \log \relax (2) + 2 \, {\left (8 \, x^{3} e^{\left (x + e^{x}\right )} \log \relax (2) - x^{2} e^{\left (2 \, x + e^{x}\right )} - 3 \, e^{\left (x + e^{x}\right )}\right )} e^{\left (-x\right )} - 2 \, {\left (8 \, x^{5} \log \relax (2) - 3 \, x^{2}\right )} e^{x} + e^{\left (2 \, e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*exp(x)*exp(exp(x))^2+(-2*exp(x)^2*x^2+(16*x^3*log(2)-2*x^2-4*x-6)*exp(x)+48*x^2*log(2))*exp(exp(x)
)+(2*x^4+4*x^3)*exp(x)^2+(2*(-8*x^5-40*x^4)*log(2)+6*x^2+12*x)*exp(x)+384*x^5*log(2)^2-144*x^2*log(2),x, algor
ithm="giac")

[Out]

64*x^6*log(2)^2 + x^4*e^(2*x) - 48*x^3*log(2) + 2*(8*x^3*e^(x + e^x)*log(2) - x^2*e^(2*x + e^x) - 3*e^(x + e^x
))*e^(-x) - 2*(8*x^5*log(2) - 3*x^2)*e^x + e^(2*e^x)

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maple [B]  time = 0.09, size = 67, normalized size = 2.79

method	result	size
risch	\({\mathrm e}^{2 \,{\mathrm e}^{x}}+\left (16 x^{3} \ln \relax (2)-2 \,{\mathrm e}^{x} x^{2}-6\right ) {\mathrm e}^{{\mathrm e}^{x}}+{\mathrm e}^{2 x} x^{4}+\left (-16 x^{5} \ln \relax (2)+6 x^{2}\right ) {\mathrm e}^{x}+64 x^{6} \ln \relax (2)^{2}-48 x^{3} \ln \relax (2)\)	\(67\)
default	\(6 \,{\mathrm e}^{x} x^{2}-16 \,{\mathrm e}^{x} \ln \relax (2) x^{5}+16 x^{3} \ln \relax (2) {\mathrm e}^{{\mathrm e}^{x}}-2 x^{2} {\mathrm e}^{x} {\mathrm e}^{{\mathrm e}^{x}}-6 \,{\mathrm e}^{{\mathrm e}^{x}}+{\mathrm e}^{2 x} x^{4}-48 x^{3} \ln \relax (2)+64 x^{6} \ln \relax (2)^{2}+{\mathrm e}^{2 \,{\mathrm e}^{x}}\)	\(72\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*exp(x)*exp(exp(x))^2+(-2*exp(x)^2*x^2+(16*x^3*ln(2)-2*x^2-4*x-6)*exp(x)+48*x^2*ln(2))*exp(exp(x))+(2*x^4
+4*x^3)*exp(x)^2+(2*(-8*x^5-40*x^4)*ln(2)+6*x^2+12*x)*exp(x)+384*x^5*ln(2)^2-144*x^2*ln(2),x,method=_RETURNVER
BOSE)

[Out]

exp(2*exp(x))+(16*x^3*ln(2)-2*exp(x)*x^2-6)*exp(exp(x))+exp(2*x)*x^4+(-16*x^5*ln(2)+6*x^2)*exp(x)+64*x^6*ln(2)
^2-48*x^3*ln(2)

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maxima [B]  time = 0.57, size = 68, normalized size = 2.83 \begin {gather*} 64 \, x^{6} \log \relax (2)^{2} + x^{4} e^{\left (2 \, x\right )} - 48 \, x^{3} \log \relax (2) - 2 \, {\left (8 \, x^{5} \log \relax (2) - 3 \, x^{2}\right )} e^{x} + 2 \, {\left (8 \, x^{3} \log \relax (2) - x^{2} e^{x} - 3\right )} e^{\left (e^{x}\right )} + e^{\left (2 \, e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*exp(x)*exp(exp(x))^2+(-2*exp(x)^2*x^2+(16*x^3*log(2)-2*x^2-4*x-6)*exp(x)+48*x^2*log(2))*exp(exp(x)
)+(2*x^4+4*x^3)*exp(x)^2+(2*(-8*x^5-40*x^4)*log(2)+6*x^2+12*x)*exp(x)+384*x^5*log(2)^2-144*x^2*log(2),x, algor
ithm="maxima")

[Out]

64*x^6*log(2)^2 + x^4*e^(2*x) - 48*x^3*log(2) - 2*(8*x^5*log(2) - 3*x^2)*e^x + 2*(8*x^3*log(2) - x^2*e^x - 3)*
e^(e^x) + e^(2*e^x)

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mupad [B]  time = 0.12, size = 71, normalized size = 2.96 \begin {gather*} {\mathrm {e}}^{2\,{\mathrm {e}}^x}-6\,{\mathrm {e}}^{{\mathrm {e}}^x}+64\,x^6\,{\ln \relax (2)}^2+6\,x^2\,{\mathrm {e}}^x-2\,x^2\,{\mathrm {e}}^{x+{\mathrm {e}}^x}+x^4\,{\mathrm {e}}^{2\,x}-48\,x^3\,\ln \relax (2)-16\,x^5\,{\mathrm {e}}^x\,\ln \relax (2)+16\,x^3\,{\mathrm {e}}^{{\mathrm {e}}^x}\,\ln \relax (2) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(384*x^5*log(2)^2 + exp(x)*(12*x - 2*log(2)*(40*x^4 + 8*x^5) + 6*x^2) - exp(exp(x))*(exp(x)*(4*x - 16*x^3*l
og(2) + 2*x^2 + 6) + 2*x^2*exp(2*x) - 48*x^2*log(2)) + exp(2*x)*(4*x^3 + 2*x^4) - 144*x^2*log(2) + 2*exp(2*exp
(x))*exp(x),x)

[Out]

exp(2*exp(x)) - 6*exp(exp(x)) + 64*x^6*log(2)^2 + 6*x^2*exp(x) - 2*x^2*exp(x + exp(x)) + x^4*exp(2*x) - 48*x^3
*log(2) - 16*x^5*exp(x)*log(2) + 16*x^3*exp(exp(x))*log(2)

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sympy [B]  time = 0.33, size = 73, normalized size = 3.04 \begin {gather*} 64 x^{6} \log {\relax (2 )}^{2} + x^{4} e^{2 x} - 48 x^{3} \log {\relax (2 )} + \left (- 16 x^{5} \log {\relax (2 )} + 6 x^{2}\right ) e^{x} + \left (16 x^{3} \log {\relax (2 )} - 2 x^{2} e^{x} - 6\right ) e^{e^{x}} + e^{2 e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*exp(x)*exp(exp(x))**2+(-2*exp(x)**2*x**2+(16*x**3*ln(2)-2*x**2-4*x-6)*exp(x)+48*x**2*ln(2))*exp(ex
p(x))+(2*x**4+4*x**3)*exp(x)**2+(2*(-8*x**5-40*x**4)*ln(2)+6*x**2+12*x)*exp(x)+384*x**5*ln(2)**2-144*x**2*ln(2
),x)

[Out]

64*x**6*log(2)**2 + x**4*exp(2*x) - 48*x**3*log(2) + (-16*x**5*log(2) + 6*x**2)*exp(x) + (16*x**3*log(2) - 2*x
**2*exp(x) - 6)*exp(exp(x)) + exp(2*exp(x))

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