3.3.54 \(\int (e^{2 e^x+8 x+2 e^x x-2 x \log (x^2 \log (2))} (4+e^x (4+2 x)-2 \log (x^2 \log (2)))+e^{e^x+4 x+e^x x-x \log (x^2 \log (2))} (-4 \log (3)+e^x (-4-2 x) \log (3)+2 \log (3) \log (x^2 \log (2)))) \, dx\)

Optimal. Leaf size=29 \[ \left (e^{e^x+x \left (4+e^x-\log \left (x^2 \log (2)\right )\right )}-\log (3)\right )^2 \]

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Rubi [B]  time = 0.59, antiderivative size = 63, normalized size of antiderivative = 2.17, number of steps used = 3, number of rules used = 1, integrand size = 101, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.010, Rules used = {6706} \begin {gather*} e^{2 e^x x+8 x+2 e^x} \left (x^2\right )^{-2 x} \log ^{-2 x}(2)-e^{e^x x+4 x+e^x} \left (x^2\right )^{-x} \log (9) \log ^{-x}(2) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(2*E^x + 8*x + 2*E^x*x - 2*x*Log[x^2*Log[2]])*(4 + E^x*(4 + 2*x) - 2*Log[x^2*Log[2]]) + E^(E^x + 4*x + E
^x*x - x*Log[x^2*Log[2]])*(-4*Log[3] + E^x*(-4 - 2*x)*Log[3] + 2*Log[3]*Log[x^2*Log[2]]),x]

[Out]

E^(2*E^x + 8*x + 2*E^x*x)/((x^2)^(2*x)*Log[2]^(2*x)) - (E^(E^x + 4*x + E^x*x)*Log[9])/((x^2)^x*Log[2]^x)

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int e^{2 e^x+8 x+2 e^x x-2 x \log \left (x^2 \log (2)\right )} \left (4+e^x (4+2 x)-2 \log \left (x^2 \log (2)\right )\right ) \, dx+\int e^{e^x+4 x+e^x x-x \log \left (x^2 \log (2)\right )} \left (-4 \log (3)+e^x (-4-2 x) \log (3)+2 \log (3) \log \left (x^2 \log (2)\right )\right ) \, dx\\ &=e^{2 e^x+8 x+2 e^x x} \left (x^2\right )^{-2 x} \log ^{-2 x}(2)-e^{e^x+4 x+e^x x} \left (x^2\right )^{-x} \log ^{-x}(2) \log (9)\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 5.80, size = 0, normalized size = 0.00 \begin {gather*} \int \left (e^{2 e^x+8 x+2 e^x x-2 x \log \left (x^2 \log (2)\right )} \left (4+e^x (4+2 x)-2 \log \left (x^2 \log (2)\right )\right )+e^{e^x+4 x+e^x x-x \log \left (x^2 \log (2)\right )} \left (-4 \log (3)+e^x (-4-2 x) \log (3)+2 \log (3) \log \left (x^2 \log (2)\right )\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[E^(2*E^x + 8*x + 2*E^x*x - 2*x*Log[x^2*Log[2]])*(4 + E^x*(4 + 2*x) - 2*Log[x^2*Log[2]]) + E^(E^x + 4
*x + E^x*x - x*Log[x^2*Log[2]])*(-4*Log[3] + E^x*(-4 - 2*x)*Log[3] + 2*Log[3]*Log[x^2*Log[2]]),x]

[Out]

Integrate[E^(2*E^x + 8*x + 2*E^x*x - 2*x*Log[x^2*Log[2]])*(4 + E^x*(4 + 2*x) - 2*Log[x^2*Log[2]]) + E^(E^x + 4
*x + E^x*x - x*Log[x^2*Log[2]])*(-4*Log[3] + E^x*(-4 - 2*x)*Log[3] + 2*Log[3]*Log[x^2*Log[2]]), x]

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fricas [A]  time = 0.79, size = 48, normalized size = 1.66 \begin {gather*} -2 \, e^{\left ({\left (x + 1\right )} e^{x} - x \log \left (x^{2} \log \relax (2)\right ) + 4 \, x\right )} \log \relax (3) + e^{\left (2 \, {\left (x + 1\right )} e^{x} - 2 \, x \log \left (x^{2} \log \relax (2)\right ) + 8 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*log(x^2*log(2))+(2*x+4)*exp(x)+4)*exp(-x*log(x^2*log(2))+exp(x)*x+4*x)^2*exp(exp(x))^2+(2*log(3)
*log(x^2*log(2))+(-2*x-4)*log(3)*exp(x)-4*log(3))*exp(-x*log(x^2*log(2))+exp(x)*x+4*x)*exp(exp(x)),x, algorith
m="fricas")

[Out]

-2*e^((x + 1)*e^x - x*log(x^2*log(2)) + 4*x)*log(3) + e^(2*(x + 1)*e^x - 2*x*log(x^2*log(2)) + 8*x)

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giac [A]  time = 0.97, size = 50, normalized size = 1.72 \begin {gather*} -2 \, e^{\left (x e^{x} - x \log \left (x^{2} \log \relax (2)\right ) + 4 \, x + e^{x}\right )} \log \relax (3) + e^{\left (2 \, x e^{x} - 2 \, x \log \left (x^{2} \log \relax (2)\right ) + 8 \, x + 2 \, e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*log(x^2*log(2))+(2*x+4)*exp(x)+4)*exp(-x*log(x^2*log(2))+exp(x)*x+4*x)^2*exp(exp(x))^2+(2*log(3)
*log(x^2*log(2))+(-2*x-4)*log(3)*exp(x)-4*log(3))*exp(-x*log(x^2*log(2))+exp(x)*x+4*x)*exp(exp(x)),x, algorith
m="giac")

[Out]

-2*e^(x*e^x - x*log(x^2*log(2)) + 4*x + e^x)*log(3) + e^(2*x*e^x - 2*x*log(x^2*log(2)) + 8*x + 2*e^x)

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maple [A]  time = 1.77, size = 51, normalized size = 1.76

method	result	size
default	\(-2 \ln \relax (3) {\mathrm e}^{-x \ln \left (x^{2} \ln \relax (2)\right )+{\mathrm e}^{x} x +4 x +{\mathrm e}^{x}}+{\mathrm e}^{-2 x \ln \left (x^{2} \ln \relax (2)\right )+2 \,{\mathrm e}^{x} x +8 x +2 \,{\mathrm e}^{x}}\)	\(51\)
risch	\({\mathrm e}^{-4 x \ln \relax (x )+2 \,{\mathrm e}^{x} x -2 x \ln \left (\ln \relax (2)\right )+2 \,{\mathrm e}^{x}+8 x} {\mathrm e}^{i \pi x \mathrm {csgn}\left (i x^{2}\right )^{3}} {\mathrm e}^{-2 i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}} {\mathrm e}^{i \pi x \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )}-2 \ln \relax (3) {\mathrm e}^{-2 x \ln \relax (x )+{\mathrm e}^{x} x -x \ln \left (\ln \relax (2)\right )+4 x +{\mathrm e}^{x}} {\mathrm e}^{\frac {i \pi x \mathrm {csgn}\left (i x^{2}\right )^{3}}{2}} {\mathrm e}^{-i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}} {\mathrm e}^{\frac {i \pi x \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )}{2}}\)	\(164\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*ln(3)*exp(-x*ln(x^2*ln(2))+exp(x)*x+4*x+exp(x))+exp(-2*x*ln(x^2*ln(2))+2*exp(x)*x+8*x+2*exp(x))

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maxima [A]  time = 0.70, size = 52, normalized size = 1.79 \begin {gather*} -2 \, e^{\left (x e^{x} - 2 \, x \log \relax (x) - x \log \left (\log \relax (2)\right ) + 4 \, x + e^{x}\right )} \log \relax (3) + e^{\left (2 \, x e^{x} - 4 \, x \log \relax (x) - 2 \, x \log \left (\log \relax (2)\right ) + 8 \, x + 2 \, e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*log(x^2*log(2))+(2*x+4)*exp(x)+4)*exp(-x*log(x^2*log(2))+exp(x)*x+4*x)^2*exp(exp(x))^2+(2*log(3)
*log(x^2*log(2))+(-2*x-4)*log(3)*exp(x)-4*log(3))*exp(-x*log(x^2*log(2))+exp(x)*x+4*x)*exp(exp(x)),x, algorith
m="maxima")

[Out]

-2*e^(x*e^x - 2*x*log(x) - x*log(log(2)) + 4*x + e^x)*log(3) + e^(2*x*e^x - 4*x*log(x) - 2*x*log(log(2)) + 8*x
+ 2*e^x)

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mupad [B]  time = 0.66, size = 59, normalized size = 2.03 \begin {gather*} -\frac {{\mathrm {e}}^{x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^{{\mathrm {e}}^x}\,\left (2\,{\ln \relax (2)}^x\,\ln \relax (3)\,{\left (x^2\right )}^x-{\mathrm {e}}^{x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^{{\mathrm {e}}^x}\right )}{{\ln \relax (2)}^{2\,x}\,{\left (x^2\right )}^{2\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(8*x + 2*x*exp(x) - 2*x*log(x^2*log(2)))*exp(2*exp(x))*(exp(x)*(2*x + 4) - 2*log(x^2*log(2)) + 4) - exp
(4*x + x*exp(x) - x*log(x^2*log(2)))*exp(exp(x))*(4*log(3) - 2*log(x^2*log(2))*log(3) + exp(x)*log(3)*(2*x + 4
)),x)

[Out]

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

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sympy [B]  time = 3.58, size = 60, normalized size = 2.07 \begin {gather*} - 2 e^{x e^{x} - x \log {\left (x^{2} \log {\relax (2 )} \right )} + 4 x} e^{e^{x}} \log {\relax (3 )} + e^{2 x e^{x} - 2 x \log {\left (x^{2} \log {\relax (2 )} \right )} + 8 x} e^{2 e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*ln(x**2*ln(2))+(2*x+4)*exp(x)+4)*exp(-x*ln(x**2*ln(2))+exp(x)*x+4*x)**2*exp(exp(x))**2+(2*ln(3)*
ln(x**2*ln(2))+(-2*x-4)*ln(3)*exp(x)-4*ln(3))*exp(-x*ln(x**2*ln(2))+exp(x)*x+4*x)*exp(exp(x)),x)

[Out]

-2*exp(x*exp(x) - x*log(x**2*log(2)) + 4*x)*exp(exp(x))*log(3) + exp(2*x*exp(x) - 2*x*log(x**2*log(2)) + 8*x)*
exp(2*exp(x))

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