3.102.29 \(\int \frac {e^{4 \log ^2(x)} (2+(16-8 x) \log (x)+e^{e^{1+2 x}+x} (1-x-2 e^{1+2 x} x+8 \log (x)))}{4+e^{2 e^{1+2 x}+2 x}+e^{e^{1+2 x}+x} (4-2 x)-4 x+x^2} \, dx\)

Optimal. Leaf size=28 \[ \frac {e^{4 \log ^2(x)} x}{2+e^{e^{1+2 x}+x}-x} \]

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Rubi [F]  time = 26.42, 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^{4 \log ^2(x)} \left (2+(16-8 x) \log (x)+e^{e^{1+2 x}+x} \left (1-x-2 e^{1+2 x} x+8 \log (x)\right )\right )}{4+e^{2 e^{1+2 x}+2 x}+e^{e^{1+2 x}+x} (4-2 x)-4 x+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(4*Log[x]^2)*(2 + (16 - 8*x)*Log[x] + E^(E^(1 + 2*x) + x)*(1 - x - 2*E^(1 + 2*x)*x + 8*Log[x])))/(4 + E
^(2*E^(1 + 2*x) + 2*x) + E^(E^(1 + 2*x) + x)*(4 - 2*x) - 4*x + x^2),x]

[Out]

Defer[Int][E^(4*Log[x]^2)/(2 + E^(E^(1 + 2*x) + x) - x), x] + 8*Defer[Int][E^(1 - 2*E^(1 + 2*x) + 4*Log[x]^2)*
x, x] - 2*Defer[Int][E^(1 - E^(1 + 2*x) + x + 4*Log[x]^2)*x, x] + 3*Defer[Int][(E^(4*Log[x]^2)*x)/(2 + E^(E^(1
+ 2*x) + x) - x)^2, x] + 16*Defer[Int][(E^(1 - 2*(E^(1 + 2*x) - 2*Log[x]^2))*x)/(2 + E^(E^(1 + 2*x) + x) - x)
^2, x] - Defer[Int][(E^(4*Log[x]^2)*x)/(2 + E^(E^(1 + 2*x) + x) - x), x] - 24*Defer[Int][(E^(1 - 2*(E^(1 + 2*x
) - 2*Log[x]^2))*x)/(2 + E^(E^(1 + 2*x) + x) - x), x] - 4*Defer[Int][E^(1 - 2*E^(1 + 2*x) + 4*Log[x]^2)*x^2, x
] - Defer[Int][(E^(4*Log[x]^2)*x^2)/(2 + E^(E^(1 + 2*x) + x) - x)^2, x] - 24*Defer[Int][(E^(1 - 2*(E^(1 + 2*x)
- 2*Log[x]^2))*x^2)/(2 + E^(E^(1 + 2*x) + x) - x)^2, x] + 24*Defer[Int][(E^(1 - 2*(E^(1 + 2*x) - 2*Log[x]^2))
*x^2)/(2 + E^(E^(1 + 2*x) + x) - x), x] + 12*Defer[Int][(E^(1 - 2*(E^(1 + 2*x) - 2*Log[x]^2))*x^3)/(2 + E^(E^(
1 + 2*x) + x) - x)^2, x] - 6*Defer[Int][(E^(1 - 2*(E^(1 + 2*x) - 2*Log[x]^2))*x^3)/(2 + E^(E^(1 + 2*x) + x) -
x), x] - 2*Defer[Int][(E^(1 - 2*(E^(1 + 2*x) - 2*Log[x]^2))*x^4)/(2 + E^(E^(1 + 2*x) + x) - x)^2, x] + 8*Defer
[Int][(E^(4*Log[x]^2)*Log[x])/(2 + E^(E^(1 + 2*x) + x) - x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{4 \log ^2(x)} \left (2+(16-8 x) \log (x)+e^{e^{1+2 x}+x} \left (1-x-2 e^{1+2 x} x+8 \log (x)\right )\right )}{\left (2+e^{e^{1+2 x}+x}-x\right )^2} \, dx\\ &=\int \left (-2 e^{1-e^{1+2 x}+x+4 \log ^2(x)} x-4 e^{1-2 e^{1+2 x}+4 \log ^2(x)} (-2+x) x-\frac {e^{-2 e^{1+2 x}+4 \log ^2(x)} x \left (-16 e-3 e^{2 e^{1+2 x}}+24 e x+e^{2 e^{1+2 x}} x-12 e x^2+2 e x^3\right )}{\left (2+e^{e^{1+2 x}+x}-x\right )^2}-\frac {e^{-2 e^{1+2 x}+4 \log ^2(x)} \left (-e^{2 e^{1+2 x}}+24 e x+e^{2 e^{1+2 x}} x-24 e x^2+6 e x^3-8 e^{2 e^{1+2 x}} \log (x)\right )}{2+e^{e^{1+2 x}+x}-x}\right ) \, dx\\ &=-\left (2 \int e^{1-e^{1+2 x}+x+4 \log ^2(x)} x \, dx\right )-4 \int e^{1-2 e^{1+2 x}+4 \log ^2(x)} (-2+x) x \, dx-\int \frac {e^{-2 e^{1+2 x}+4 \log ^2(x)} x \left (-16 e-3 e^{2 e^{1+2 x}}+24 e x+e^{2 e^{1+2 x}} x-12 e x^2+2 e x^3\right )}{\left (2+e^{e^{1+2 x}+x}-x\right )^2} \, dx-\int \frac {e^{-2 e^{1+2 x}+4 \log ^2(x)} \left (-e^{2 e^{1+2 x}}+24 e x+e^{2 e^{1+2 x}} x-24 e x^2+6 e x^3-8 e^{2 e^{1+2 x}} \log (x)\right )}{2+e^{e^{1+2 x}+x}-x} \, dx\\ &=-\left (2 \int e^{1-e^{1+2 x}+x+4 \log ^2(x)} x \, dx\right )-4 \int \left (-2 e^{1-2 e^{1+2 x}+4 \log ^2(x)} x+e^{1-2 e^{1+2 x}+4 \log ^2(x)} x^2\right ) \, dx-\int \frac {e^{-2 \left (e^{1+2 x}-2 \log ^2(x)\right )} \left (e^{2 e^{1+2 x}} (-3+x)+2 e (-2+x)^3\right ) x}{\left (2+e^{e^{1+2 x}+x}-x\right )^2} \, dx-\int \frac {e^{-2 \left (e^{1+2 x}-2 \log ^2(x)\right )} \left (-e^{2 e^{1+2 x}}+24 e x+e^{2 e^{1+2 x}} x-24 e x^2+6 e x^3-8 e^{2 e^{1+2 x}} \log (x)\right )}{2+e^{e^{1+2 x}+x}-x} \, dx\\ &=-\left (2 \int e^{1-e^{1+2 x}+x+4 \log ^2(x)} x \, dx\right )-4 \int e^{1-2 e^{1+2 x}+4 \log ^2(x)} x^2 \, dx+8 \int e^{1-2 e^{1+2 x}+4 \log ^2(x)} x \, dx-\int \left (-\frac {16 e^{1-2 \left (e^{1+2 x}-2 \log ^2(x)\right )} x}{\left (2+e^{e^{1+2 x}+x}-x\right )^2}-\frac {3 \exp \left (2 e^{1+2 x}-2 \left (e^{1+2 x}-2 \log ^2(x)\right )\right ) x}{\left (2+e^{e^{1+2 x}+x}-x\right )^2}+\frac {24 e^{1-2 \left (e^{1+2 x}-2 \log ^2(x)\right )} x^2}{\left (2+e^{e^{1+2 x}+x}-x\right )^2}+\frac {\exp \left (2 e^{1+2 x}-2 \left (e^{1+2 x}-2 \log ^2(x)\right )\right ) x^2}{\left (2+e^{e^{1+2 x}+x}-x\right )^2}-\frac {12 e^{1-2 \left (e^{1+2 x}-2 \log ^2(x)\right )} x^3}{\left (2+e^{e^{1+2 x}+x}-x\right )^2}+\frac {2 e^{1-2 \left (e^{1+2 x}-2 \log ^2(x)\right )} x^4}{\left (2+e^{e^{1+2 x}+x}-x\right )^2}\right ) \, dx-\int \frac {e^{-2 \left (e^{1+2 x}-2 \log ^2(x)\right )} \left (e^{2 e^{1+2 x}} (-1+x)+6 e (-2+x)^2 x-8 e^{2 e^{1+2 x}} \log (x)\right )}{2+e^{e^{1+2 x}+x}-x} \, dx\\ &=-\left (2 \int e^{1-e^{1+2 x}+x+4 \log ^2(x)} x \, dx\right )-2 \int \frac {e^{1-2 \left (e^{1+2 x}-2 \log ^2(x)\right )} x^4}{\left (2+e^{e^{1+2 x}+x}-x\right )^2} \, dx+3 \int \frac {\exp \left (2 e^{1+2 x}-2 \left (e^{1+2 x}-2 \log ^2(x)\right )\right ) x}{\left (2+e^{e^{1+2 x}+x}-x\right )^2} \, dx-4 \int e^{1-2 e^{1+2 x}+4 \log ^2(x)} x^2 \, dx+8 \int e^{1-2 e^{1+2 x}+4 \log ^2(x)} x \, dx+12 \int \frac {e^{1-2 \left (e^{1+2 x}-2 \log ^2(x)\right )} x^3}{\left (2+e^{e^{1+2 x}+x}-x\right )^2} \, dx+16 \int \frac {e^{1-2 \left (e^{1+2 x}-2 \log ^2(x)\right )} x}{\left (2+e^{e^{1+2 x}+x}-x\right )^2} \, dx-24 \int \frac {e^{1-2 \left (e^{1+2 x}-2 \log ^2(x)\right )} x^2}{\left (2+e^{e^{1+2 x}+x}-x\right )^2} \, dx-\int \frac {\exp \left (2 e^{1+2 x}-2 \left (e^{1+2 x}-2 \log ^2(x)\right )\right ) x^2}{\left (2+e^{e^{1+2 x}+x}-x\right )^2} \, dx-\int \left (-\frac {\exp \left (2 e^{1+2 x}-2 \left (e^{1+2 x}-2 \log ^2(x)\right )\right )}{2+e^{e^{1+2 x}+x}-x}+\frac {24 e^{1-2 \left (e^{1+2 x}-2 \log ^2(x)\right )} x}{2+e^{e^{1+2 x}+x}-x}+\frac {\exp \left (2 e^{1+2 x}-2 \left (e^{1+2 x}-2 \log ^2(x)\right )\right ) x}{2+e^{e^{1+2 x}+x}-x}-\frac {24 e^{1-2 \left (e^{1+2 x}-2 \log ^2(x)\right )} x^2}{2+e^{e^{1+2 x}+x}-x}+\frac {6 e^{1-2 \left (e^{1+2 x}-2 \log ^2(x)\right )} x^3}{2+e^{e^{1+2 x}+x}-x}-\frac {8 \exp \left (2 e^{1+2 x}-2 \left (e^{1+2 x}-2 \log ^2(x)\right )\right ) \log (x)}{2+e^{e^{1+2 x}+x}-x}\right ) \, dx\\ &=-\left (2 \int e^{1-e^{1+2 x}+x+4 \log ^2(x)} x \, dx\right )-2 \int \frac {e^{1-2 \left (e^{1+2 x}-2 \log ^2(x)\right )} x^4}{\left (2+e^{e^{1+2 x}+x}-x\right )^2} \, dx+3 \int \frac {e^{4 \log ^2(x)} x}{\left (2+e^{e^{1+2 x}+x}-x\right )^2} \, dx-4 \int e^{1-2 e^{1+2 x}+4 \log ^2(x)} x^2 \, dx-6 \int \frac {e^{1-2 \left (e^{1+2 x}-2 \log ^2(x)\right )} x^3}{2+e^{e^{1+2 x}+x}-x} \, dx+8 \int e^{1-2 e^{1+2 x}+4 \log ^2(x)} x \, dx+8 \int \frac {\exp \left (2 e^{1+2 x}-2 \left (e^{1+2 x}-2 \log ^2(x)\right )\right ) \log (x)}{2+e^{e^{1+2 x}+x}-x} \, dx+12 \int \frac {e^{1-2 \left (e^{1+2 x}-2 \log ^2(x)\right )} x^3}{\left (2+e^{e^{1+2 x}+x}-x\right )^2} \, dx+16 \int \frac {e^{1-2 \left (e^{1+2 x}-2 \log ^2(x)\right )} x}{\left (2+e^{e^{1+2 x}+x}-x\right )^2} \, dx-24 \int \frac {e^{1-2 \left (e^{1+2 x}-2 \log ^2(x)\right )} x}{2+e^{e^{1+2 x}+x}-x} \, dx-24 \int \frac {e^{1-2 \left (e^{1+2 x}-2 \log ^2(x)\right )} x^2}{\left (2+e^{e^{1+2 x}+x}-x\right )^2} \, dx+24 \int \frac {e^{1-2 \left (e^{1+2 x}-2 \log ^2(x)\right )} x^2}{2+e^{e^{1+2 x}+x}-x} \, dx+\int \frac {\exp \left (2 e^{1+2 x}-2 \left (e^{1+2 x}-2 \log ^2(x)\right )\right )}{2+e^{e^{1+2 x}+x}-x} \, dx-\int \frac {\exp \left (2 e^{1+2 x}-2 \left (e^{1+2 x}-2 \log ^2(x)\right )\right ) x}{2+e^{e^{1+2 x}+x}-x} \, dx-\int \frac {e^{4 \log ^2(x)} x^2}{\left (2+e^{e^{1+2 x}+x}-x\right )^2} \, dx\\ &=-\left (2 \int e^{1-e^{1+2 x}+x+4 \log ^2(x)} x \, dx\right )-2 \int \frac {e^{1-2 \left (e^{1+2 x}-2 \log ^2(x)\right )} x^4}{\left (2+e^{e^{1+2 x}+x}-x\right )^2} \, dx+3 \int \frac {e^{4 \log ^2(x)} x}{\left (2+e^{e^{1+2 x}+x}-x\right )^2} \, dx-4 \int e^{1-2 e^{1+2 x}+4 \log ^2(x)} x^2 \, dx-6 \int \frac {e^{1-2 \left (e^{1+2 x}-2 \log ^2(x)\right )} x^3}{2+e^{e^{1+2 x}+x}-x} \, dx+8 \int e^{1-2 e^{1+2 x}+4 \log ^2(x)} x \, dx+8 \int \frac {e^{4 \log ^2(x)} \log (x)}{2+e^{e^{1+2 x}+x}-x} \, dx+12 \int \frac {e^{1-2 \left (e^{1+2 x}-2 \log ^2(x)\right )} x^3}{\left (2+e^{e^{1+2 x}+x}-x\right )^2} \, dx+16 \int \frac {e^{1-2 \left (e^{1+2 x}-2 \log ^2(x)\right )} x}{\left (2+e^{e^{1+2 x}+x}-x\right )^2} \, dx-24 \int \frac {e^{1-2 \left (e^{1+2 x}-2 \log ^2(x)\right )} x}{2+e^{e^{1+2 x}+x}-x} \, dx-24 \int \frac {e^{1-2 \left (e^{1+2 x}-2 \log ^2(x)\right )} x^2}{\left (2+e^{e^{1+2 x}+x}-x\right )^2} \, dx+24 \int \frac {e^{1-2 \left (e^{1+2 x}-2 \log ^2(x)\right )} x^2}{2+e^{e^{1+2 x}+x}-x} \, dx+\int \frac {e^{4 \log ^2(x)}}{2+e^{e^{1+2 x}+x}-x} \, dx-\int \frac {e^{4 \log ^2(x)} x}{2+e^{e^{1+2 x}+x}-x} \, dx-\int \frac {e^{4 \log ^2(x)} x^2}{\left (2+e^{e^{1+2 x}+x}-x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 28, normalized size = 1.00 \begin {gather*} \frac {e^{4 \log ^2(x)} x}{2+e^{e^{1+2 x}+x}-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(4*Log[x]^2)*(2 + (16 - 8*x)*Log[x] + E^(E^(1 + 2*x) + x)*(1 - x - 2*E^(1 + 2*x)*x + 8*Log[x])))/
(4 + E^(2*E^(1 + 2*x) + 2*x) + E^(E^(1 + 2*x) + x)*(4 - 2*x) - 4*x + x^2),x]

[Out]

(E^(4*Log[x]^2)*x)/(2 + E^(E^(1 + 2*x) + x) - x)

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fricas [A]  time = 0.59, size = 26, normalized size = 0.93 \begin {gather*} -\frac {x e^{\left (4 \, \log \relax (x)^{2}\right )}}{x - e^{\left (x + e^{\left (2 \, x + 1\right )}\right )} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*log(x)-2*x*exp(2*x+1)-x+1)*exp(exp(2*x+1)+x)+(-8*x+16)*log(x)+2)*exp(log(x)^2)^4/(exp(exp(2*x+1)
+x)^2+(4-2*x)*exp(exp(2*x+1)+x)+x^2-4*x+4),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.86, size = 146, normalized size = 5.21 \begin {gather*} -\frac {x^{2} e^{\left (4 \, \log \relax (x)^{2}\right )} + 2 \, x^{2} e^{\left (4 \, \log \relax (x)^{2} + 2 \, x + 1\right )} - 3 \, x e^{\left (4 \, \log \relax (x)^{2}\right )} - 4 \, x e^{\left (4 \, \log \relax (x)^{2} + 2 \, x + 1\right )}}{2 \, x^{2} e^{\left (2 \, x + 1\right )} + x^{2} - 2 \, x e^{\left (3 \, x + e^{\left (2 \, x + 1\right )} + 1\right )} - 8 \, x e^{\left (2 \, x + 1\right )} - x e^{\left (x + e^{\left (2 \, x + 1\right )}\right )} - 5 \, x + 4 \, e^{\left (3 \, x + e^{\left (2 \, x + 1\right )} + 1\right )} + 8 \, e^{\left (2 \, x + 1\right )} + 3 \, e^{\left (x + e^{\left (2 \, x + 1\right )}\right )} + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*log(x)-2*x*exp(2*x+1)-x+1)*exp(exp(2*x+1)+x)+(-8*x+16)*log(x)+2)*exp(log(x)^2)^4/(exp(exp(2*x+1)
+x)^2+(4-2*x)*exp(exp(2*x+1)+x)+x^2-4*x+4),x, algorithm="giac")

[Out]

-(x^2*e^(4*log(x)^2) + 2*x^2*e^(4*log(x)^2 + 2*x + 1) - 3*x*e^(4*log(x)^2) - 4*x*e^(4*log(x)^2 + 2*x + 1))/(2*
x^2*e^(2*x + 1) + x^2 - 2*x*e^(3*x + e^(2*x + 1) + 1) - 8*x*e^(2*x + 1) - x*e^(x + e^(2*x + 1)) - 5*x + 4*e^(3
*x + e^(2*x + 1) + 1) + 8*e^(2*x + 1) + 3*e^(x + e^(2*x + 1)) + 6)

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maple [A]  time = 0.04, size = 27, normalized size = 0.96

method	result	size
risch	\(-\frac {x \,{\mathrm e}^{4 \ln \relax (x )^{2}}}{x -{\mathrm e}^{{\mathrm e}^{2 x +1}+x}-2}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*ln(x)-2*x*exp(2*x+1)-x+1)*exp(exp(2*x+1)+x)+(-8*x+16)*ln(x)+2)*exp(ln(x)^2)^4/(exp(exp(2*x+1)+x)^2+(4-
2*x)*exp(exp(2*x+1)+x)+x^2-4*x+4),x,method=_RETURNVERBOSE)

[Out]

-x*exp(4*ln(x)^2)/(x-exp(exp(2*x+1)+x)-2)

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maxima [A]  time = 0.41, size = 26, normalized size = 0.93 \begin {gather*} -\frac {x e^{\left (4 \, \log \relax (x)^{2}\right )}}{x - e^{\left (x + e^{\left (2 \, x + 1\right )}\right )} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*log(x)-2*x*exp(2*x+1)-x+1)*exp(exp(2*x+1)+x)+(-8*x+16)*log(x)+2)*exp(log(x)^2)^4/(exp(exp(2*x+1)
+x)^2+(4-2*x)*exp(exp(2*x+1)+x)+x^2-4*x+4),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 8.17, size = 91, normalized size = 3.25 \begin {gather*} -\frac {x\,\left (3\,{\mathrm {e}}^{4\,{\ln \relax (x)}^2}+4\,{\mathrm {e}}^{4\,{\ln \relax (x)}^2+2\,x+1}-2\,x\,{\mathrm {e}}^{4\,{\ln \relax (x)}^2+2\,x+1}-x\,{\mathrm {e}}^{4\,{\ln \relax (x)}^2}\right )}{\left ({\mathrm {e}}^{x+{\mathrm {e}}^{2\,x}\,\mathrm {e}}-x+2\right )\,\left (x-4\,{\mathrm {e}}^{2\,x+1}+2\,x\,{\mathrm {e}}^{2\,x+1}-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(4*log(x)^2)*(log(x)*(8*x - 16) + exp(x + exp(2*x + 1))*(x - 8*log(x) + 2*x*exp(2*x + 1) - 1) - 2))/(
exp(2*x + 2*exp(2*x + 1)) - 4*x - exp(x + exp(2*x + 1))*(2*x - 4) + x^2 + 4),x)

[Out]

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

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sympy [A]  time = 0.48, size = 22, normalized size = 0.79 \begin {gather*} \frac {x e^{4 \log {\relax (x )}^{2}}}{- x + e^{x + e^{2 x + 1}} + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*ln(x)-2*x*exp(2*x+1)-x+1)*exp(exp(2*x+1)+x)+(-8*x+16)*ln(x)+2)*exp(ln(x)**2)**4/(exp(exp(2*x+1)+
x)**2+(4-2*x)*exp(exp(2*x+1)+x)+x**2-4*x+4),x)

[Out]

x*exp(4*log(x)**2)/(-x + exp(x + exp(2*x + 1)) + 2)

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