3.46.4 \(\int \frac {e^{2+4 e^{e^x}+\frac {e^{2+4 e^{e^x}} (2+2 x)^2}{(3+x)^2}} (2+2 x)^2 (4+e^{e^x+x} (12+16 x+4 x^2))}{(3+x)^2 (3+4 x+x^2)} \, dx\)

Optimal. Leaf size=27 \[ -5+e^{\frac {4 e^{2+4 e^{e^x}} (1+x)^2}{(3+x)^2}} \]

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Rubi [F]  time = 10.26, 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 {\exp \left (2+4 e^{e^x}+\frac {e^{2+4 e^{e^x}} (2+2 x)^2}{(3+x)^2}\right ) (2+2 x)^2 \left (4+e^{e^x+x} \left (12+16 x+4 x^2\right )\right )}{(3+x)^2 \left (3+4 x+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(2 + 4*E^E^x + (E^(2 + 4*E^E^x)*(2 + 2*x)^2)/(3 + x)^2)*(2 + 2*x)^2*(4 + E^(E^x + x)*(12 + 16*x + 4*x^2
)))/((3 + x)^2*(3 + 4*x + x^2)),x]

[Out]

16*Defer[Int][E^(2 + 4*E^E^x + E^x + x + (E^(2 + 4*E^E^x)*(2 + 2*x)^2)/(3 + x)^2), x] - 32*Defer[Int][E^(2 + 4
*E^E^x + (E^(2 + 4*E^E^x)*(2 + 2*x)^2)/(3 + x)^2)/(3 + x)^3, x] + 16*Defer[Int][E^(2 + 4*E^E^x + (E^(2 + 4*E^E
^x)*(2 + 2*x)^2)/(3 + x)^2)/(3 + x)^2, x] + 64*Defer[Int][E^(2 + 4*E^E^x + E^x + x + (E^(2 + 4*E^E^x)*(2 + 2*x
)^2)/(3 + x)^2)/(3 + x)^2, x] - 64*Defer[Int][E^(2 + 4*E^E^x + E^x + x + (E^(2 + 4*E^E^x)*(2 + 2*x)^2)/(3 + x)
^2)/(3 + x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (2+4 e^{e^x}+\frac {e^{2+4 e^{e^x}} (2+2 x)^2}{(3+x)^2}\right ) (2+2 x) \left (4+e^{e^x+x} \left (12+16 x+4 x^2\right )\right )}{\left (\frac {3}{2}+\frac {x}{2}\right ) (3+x)^2} \, dx\\ &=2 \int \frac {\exp \left (2+4 e^{e^x}+\frac {e^{2+4 e^{e^x}} (2+2 x)^2}{(3+x)^2}\right ) (2+2 x) \left (4+e^{e^x+x} \left (12+16 x+4 x^2\right )\right )}{(3+x)^3} \, dx\\ &=2 \int \left (\frac {8 \exp \left (2+4 e^{e^x}+\frac {e^{2+4 e^{e^x}} (2+2 x)^2}{(3+x)^2}\right ) (1+x)}{(3+x)^3}+\frac {8 \exp \left (2+4 e^{e^x}+e^x+x+\frac {e^{2+4 e^{e^x}} (2+2 x)^2}{(3+x)^2}\right ) (1+x)^2}{(3+x)^2}\right ) \, dx\\ &=16 \int \frac {\exp \left (2+4 e^{e^x}+\frac {e^{2+4 e^{e^x}} (2+2 x)^2}{(3+x)^2}\right ) (1+x)}{(3+x)^3} \, dx+16 \int \frac {\exp \left (2+4 e^{e^x}+e^x+x+\frac {e^{2+4 e^{e^x}} (2+2 x)^2}{(3+x)^2}\right ) (1+x)^2}{(3+x)^2} \, dx\\ &=16 \int \left (-\frac {2 \exp \left (2+4 e^{e^x}+\frac {e^{2+4 e^{e^x}} (2+2 x)^2}{(3+x)^2}\right )}{(3+x)^3}+\frac {\exp \left (2+4 e^{e^x}+\frac {e^{2+4 e^{e^x}} (2+2 x)^2}{(3+x)^2}\right )}{(3+x)^2}\right ) \, dx+16 \int \left (\exp \left (2+4 e^{e^x}+e^x+x+\frac {e^{2+4 e^{e^x}} (2+2 x)^2}{(3+x)^2}\right )+\frac {4 \exp \left (2+4 e^{e^x}+e^x+x+\frac {e^{2+4 e^{e^x}} (2+2 x)^2}{(3+x)^2}\right )}{(3+x)^2}-\frac {4 \exp \left (2+4 e^{e^x}+e^x+x+\frac {e^{2+4 e^{e^x}} (2+2 x)^2}{(3+x)^2}\right )}{3+x}\right ) \, dx\\ &=16 \int \exp \left (2+4 e^{e^x}+e^x+x+\frac {e^{2+4 e^{e^x}} (2+2 x)^2}{(3+x)^2}\right ) \, dx+16 \int \frac {\exp \left (2+4 e^{e^x}+\frac {e^{2+4 e^{e^x}} (2+2 x)^2}{(3+x)^2}\right )}{(3+x)^2} \, dx-32 \int \frac {\exp \left (2+4 e^{e^x}+\frac {e^{2+4 e^{e^x}} (2+2 x)^2}{(3+x)^2}\right )}{(3+x)^3} \, dx+64 \int \frac {\exp \left (2+4 e^{e^x}+e^x+x+\frac {e^{2+4 e^{e^x}} (2+2 x)^2}{(3+x)^2}\right )}{(3+x)^2} \, dx-64 \int \frac {\exp \left (2+4 e^{e^x}+e^x+x+\frac {e^{2+4 e^{e^x}} (2+2 x)^2}{(3+x)^2}\right )}{3+x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.79, size = 25, normalized size = 0.93 \begin {gather*} e^{\frac {4 e^{2+4 e^{e^x}} (1+x)^2}{(3+x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2 + 4*E^E^x + (E^(2 + 4*E^E^x)*(2 + 2*x)^2)/(3 + x)^2)*(2 + 2*x)^2*(4 + E^(E^x + x)*(12 + 16*x +
4*x^2)))/((3 + x)^2*(3 + 4*x + x^2)),x]

[Out]

E^((4*E^(2 + 4*E^E^x)*(1 + x)^2)/(3 + x)^2)

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fricas [B]  time = 0.55, size = 97, normalized size = 3.59 \begin {gather*} e^{\left ({\left (2 \, e^{x} \log \left (\frac {2 \, {\left (x + 1\right )}}{x + 3}\right ) + e^{\left (2 \, {\left (e^{x} \log \left (\frac {2 \, {\left (x + 1\right )}}{x + 3}\right ) + 2 \, e^{\left (x + e^{x}\right )} + e^{x}\right )} e^{\left (-x\right )} + x\right )} + 4 \, e^{\left (x + e^{x}\right )} + 2 \, e^{x}\right )} e^{\left (-x\right )} - 2 \, {\left (e^{x} \log \left (\frac {2 \, {\left (x + 1\right )}}{x + 3}\right ) + 2 \, e^{\left (x + e^{x}\right )} + e^{x}\right )} e^{\left (-x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^2+16*x+12)*exp(x)*exp(exp(x))+4)*exp(2*exp(exp(x))+log((2*x+2)/(3+x))+1)^2*exp(exp(2*exp(exp(x
))+log((2*x+2)/(3+x))+1)^2)/(x^2+4*x+3),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {4 \, {\left ({\left (x^{2} + 4 \, x + 3\right )} e^{\left (x + e^{x}\right )} + 1\right )} e^{\left (e^{\left (4 \, e^{\left (e^{x}\right )} + 2 \, \log \left (\frac {2 \, {\left (x + 1\right )}}{x + 3}\right ) + 2\right )} + 4 \, e^{\left (e^{x}\right )} + 2 \, \log \left (\frac {2 \, {\left (x + 1\right )}}{x + 3}\right ) + 2\right )}}{x^{2} + 4 \, x + 3}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^2+16*x+12)*exp(x)*exp(exp(x))+4)*exp(2*exp(exp(x))+log((2*x+2)/(3+x))+1)^2*exp(exp(2*exp(exp(x
))+log((2*x+2)/(3+x))+1)^2)/(x^2+4*x+3),x, algorithm="giac")

[Out]

integrate(4*((x^2 + 4*x + 3)*e^(x + e^x) + 1)*e^(e^(4*e^(e^x) + 2*log(2*(x + 1)/(x + 3)) + 2) + 4*e^(e^x) + 2*
log(2*(x + 1)/(x + 3)) + 2)/(x^2 + 4*x + 3), x)

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maple [C]  time = 0.44, size = 124, normalized size = 4.59

method	result	size
risch	\({\mathrm e}^{\frac {4 \left (x +1\right )^{2} {\mathrm e}^{4 \,{\mathrm e}^{{\mathrm e}^{x}}+2-i \pi \mathrm {csgn}\left (\frac {i \left (x +1\right )}{3+x}\right )^{3}+i \pi \mathrm {csgn}\left (\frac {i \left (x +1\right )}{3+x}\right )^{2} \mathrm {csgn}\left (\frac {i}{3+x}\right )+i \pi \mathrm {csgn}\left (\frac {i \left (x +1\right )}{3+x}\right )^{2} \mathrm {csgn}\left (i \left (x +1\right )\right )-i \pi \,\mathrm {csgn}\left (\frac {i \left (x +1\right )}{3+x}\right ) \mathrm {csgn}\left (\frac {i}{3+x}\right ) \mathrm {csgn}\left (i \left (x +1\right )\right )}}{\left (3+x \right )^{2}}}\)	\(124\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^2+16*x+12)*exp(x)*exp(exp(x))+4)*exp(2*exp(exp(x))+ln((2*x+2)/(3+x))+1)^2*exp(exp(2*exp(exp(x))+ln((
2*x+2)/(3+x))+1)^2)/(x^2+4*x+3),x,method=_RETURNVERBOSE)

[Out]

exp(4/(3+x)^2*(x+1)^2*exp(4*exp(exp(x))+2-I*Pi*csgn(I/(3+x)*(x+1))^3+I*Pi*csgn(I/(3+x)*(x+1))^2*csgn(I/(3+x))+
I*Pi*csgn(I/(3+x)*(x+1))^2*csgn(I*(x+1))-I*Pi*csgn(I/(3+x)*(x+1))*csgn(I/(3+x))*csgn(I*(x+1))))

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maxima [A]  time = 0.74, size = 47, normalized size = 1.74 \begin {gather*} e^{\left (\frac {16 \, e^{\left (4 \, e^{\left (e^{x}\right )} + 2\right )}}{x^{2} + 6 \, x + 9} - \frac {16 \, e^{\left (4 \, e^{\left (e^{x}\right )} + 2\right )}}{x + 3} + 4 \, e^{\left (4 \, e^{\left (e^{x}\right )} + 2\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^2+16*x+12)*exp(x)*exp(exp(x))+4)*exp(2*exp(exp(x))+log((2*x+2)/(3+x))+1)^2*exp(exp(2*exp(exp(x
))+log((2*x+2)/(3+x))+1)^2)/(x^2+4*x+3),x, algorithm="maxima")

[Out]

e^(16*e^(4*e^(e^x) + 2)/(x^2 + 6*x + 9) - 16*e^(4*e^(e^x) + 2)/(x + 3) + 4*e^(4*e^(e^x) + 2))

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mupad [B]  time = 3.44, size = 68, normalized size = 2.52 \begin {gather*} {\mathrm {e}}^{\frac {8\,x\,{\mathrm {e}}^{4\,{\mathrm {e}}^{{\mathrm {e}}^x}}\,{\mathrm {e}}^2}{x^2+6\,x+9}}\,{\mathrm {e}}^{\frac {4\,x^2\,{\mathrm {e}}^{4\,{\mathrm {e}}^{{\mathrm {e}}^x}}\,{\mathrm {e}}^2}{x^2+6\,x+9}}\,{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^{4\,{\mathrm {e}}^{{\mathrm {e}}^x}}\,{\mathrm {e}}^2}{x^2+6\,x+9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*log((2*x + 2)/(x + 3)) + 4*exp(exp(x)) + 2)*exp(exp(2*log((2*x + 2)/(x + 3)) + 4*exp(exp(x)) + 2))*
(exp(exp(x))*exp(x)*(16*x + 4*x^2 + 12) + 4))/(4*x + x^2 + 3),x)

[Out]

exp((8*x*exp(4*exp(exp(x)))*exp(2))/(6*x + x^2 + 9))*exp((4*x^2*exp(4*exp(exp(x)))*exp(2))/(6*x + x^2 + 9))*ex
p((4*exp(4*exp(exp(x)))*exp(2))/(6*x + x^2 + 9))

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sympy [A]  time = 2.16, size = 22, normalized size = 0.81 \begin {gather*} e^{\frac {\left (2 x + 2\right )^{2} e^{4 e^{e^{x}} + 2}}{\left (x + 3\right )^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**2+16*x+12)*exp(x)*exp(exp(x))+4)*exp(2*exp(exp(x))+ln((2*x+2)/(3+x))+1)**2*exp(exp(2*exp(exp(
x))+ln((2*x+2)/(3+x))+1)**2)/(x**2+4*x+3),x)

[Out]

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

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