∫ −16e2xx2−32e3xx2+e4x(−80−16x2)+(e4x(−80−48x2)+e3x(−96x2+32x3)+e2x(−48x2+32x3)+(e4x(−80−48x2)+e3x(−96x2+32x3)+e2x(−48x2+32x3)) log(x)) log(1+log(x))____________________________________________________________________________________________________________________________ (x6+4exx6+e4x(25x2+10x4+x6)+e3x(20x4+4x6)+e2x(10x4+6x6)+(x6+4exx6+e4x(25x2+10x4+x6)+e3x(20x4+4x6)+e2x(10x4+6x6)) log(x)) log2(1+log(x)) dx

3.28.23 \(\int \frac {-16 e^{2 x} x^2-32 e^{3 x} x^2+e^{4 x} (-80-16 x^2)+(e^{4 x} (-80-48 x^2)+e^{3 x} (-96 x^2+32 x^3)+e^{2 x} (-48 x^2+32 x^3)+(e^{4 x} (-80-48 x^2)+e^{3 x} (-96 x^2+32 x^3)+e^{2 x} (-48 x^2+32 x^3)) \log (x)) \log (1+\log (x))}{(x^6+4 e^x x^6+e^{4 x} (25 x^2+10 x^4+x^6)+e^{3 x} (20 x^4+4 x^6)+e^{2 x} (10 x^4+6 x^6)+(x^6+4 e^x x^6+e^{4 x} (25 x^2+10 x^4+x^6)+e^{3 x} (20 x^4+4 x^6)+e^{2 x} (10 x^4+6 x^6)) \log (x)) \log ^2(1+\log (x))} \, dx\)

Optimal. Leaf size=27 \[ \frac {16}{x \left (5+\left (x+e^{-x} x\right )^2\right ) \log (1+\log (x))} \]

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Rubi [F] time = 116.48, 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 {-16 e^{2 x} x^2-32 e^{3 x} x^2+e^{4 x} \left (-80-16 x^2\right )+\left (e^{4 x} \left (-80-48 x^2\right )+e^{3 x} \left (-96 x^2+32 x^3\right )+e^{2 x} \left (-48 x^2+32 x^3\right )+\left (e^{4 x} \left (-80-48 x^2\right )+e^{3 x} \left (-96 x^2+32 x^3\right )+e^{2 x} \left (-48 x^2+32 x^3\right )\right ) \log (x)\right ) \log (1+\log (x))}{\left (x^6+4 e^x x^6+e^{4 x} \left (25 x^2+10 x^4+x^6\right )+e^{3 x} \left (20 x^4+4 x^6\right )+e^{2 x} \left (10 x^4+6 x^6\right )+\left (x^6+4 e^x x^6+e^{4 x} \left (25 x^2+10 x^4+x^6\right )+e^{3 x} \left (20 x^4+4 x^6\right )+e^{2 x} \left (10 x^4+6 x^6\right )\right ) \log (x)\right ) \log ^2(1+\log (x))} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

Int[(-16*E^(2*x)*x^2 - 32*E^(3*x)*x^2 + E^(4*x)*(-80 - 16*x^2) + (E^(4*x)*(-80 - 48*x^2) + E^(3*x)*(-96*x^2 +

32*x^3) + E^(2*x)*(-48*x^2 + 32*x^3) + (E^(4*x)*(-80 - 48*x^2) + E^(3*x)*(-96*x^2 + 32*x^3) + E^(2*x)*(-48*x^2

+ 32*x^3))*Log[x])*Log[1 + Log[x]])/((x^6 + 4*E^x*x^6 + E^(4*x)*(25*x^2 + 10*x^4 + x^6) + E^(3*x)*(20*x^4 + 4

*x^6) + E^(2*x)*(10*x^4 + 6*x^6) + (x^6 + 4*E^x*x^6 + E^(4*x)*(25*x^2 + 10*x^4 + x^6) + E^(3*x)*(20*x^4 + 4*x^

6) + E^(2*x)*(10*x^4 + 6*x^6))*Log[x])*Log[1 + Log[x]]^2),x]

[Out]

-16*Defer[Int][E^(2*x)/(x^2*(5*E^(2*x) + x^2 + 2*E^x*x^2 + E^(2*x)*x^2)*(1 + Log[x])*Log[1 + Log[x]]^2), x] -

(16*I)*Sqrt[5]*Defer[Int][E^(2*x)/((I*Sqrt[5] - x)*(5*E^(2*x) + x^2 + 2*E^x*x^2 + E^(2*x)*x^2)^2*Log[1 + Log[x

]]), x] - (32*I)*Sqrt[5]*Defer[Int][E^(3*x)/((I*Sqrt[5] - x)*(5*E^(2*x) + x^2 + 2*E^x*x^2 + E^(2*x)*x^2)^2*Log

[1 + Log[x]]), x] + 32*Defer[Int][(E^(2*x)*x)/((5*E^(2*x) + x^2 + 2*E^x*x^2 + E^(2*x)*x^2)^2*Log[1 + Log[x]]),

x] + 32*Defer[Int][(E^(3*x)*x)/((5*E^(2*x) + x^2 + 2*E^x*x^2 + E^(2*x)*x^2)^2*Log[1 + Log[x]]), x] - (16*I)*S

qrt[5]*Defer[Int][E^(2*x)/((I*Sqrt[5] + x)*(5*E^(2*x) + x^2 + 2*E^x*x^2 + E^(2*x)*x^2)^2*Log[1 + Log[x]]), x]

- (32*I)*Sqrt[5]*Defer[Int][E^(3*x)/((I*Sqrt[5] + x)*(5*E^(2*x) + x^2 + 2*E^x*x^2 + E^(2*x)*x^2)^2*Log[1 + Log

[x]]), x] - ((16*I)*Defer[Int][E^(2*x)/((I*Sqrt[5] - x)*(5*E^(2*x) + x^2 + 2*E^x*x^2 + E^(2*x)*x^2)*(1 + Log[x

])*Log[1 + Log[x]]), x])/Sqrt[5] - 16*Defer[Int][E^(2*x)/(x^2*(5*E^(2*x) + x^2 + 2*E^x*x^2 + E^(2*x)*x^2)*(1 +

Log[x])*Log[1 + Log[x]]), x] - ((16*I)*Defer[Int][E^(2*x)/((I*Sqrt[5] + x)*(5*E^(2*x) + x^2 + 2*E^x*x^2 + E^(

2*x)*x^2)*(1 + Log[x])*Log[1 + Log[x]]), x])/Sqrt[5] - ((16*I)*Defer[Int][(E^(2*x)*Log[x])/((I*Sqrt[5] - x)*(5

*E^(2*x) + x^2 + 2*E^x*x^2 + E^(2*x)*x^2)*(1 + Log[x])*Log[1 + Log[x]]), x])/Sqrt[5] - 16*Defer[Int][(E^(2*x)*

Log[x])/(x^2*(5*E^(2*x) + x^2 + 2*E^x*x^2 + E^(2*x)*x^2)*(1 + Log[x])*Log[1 + Log[x]]), x] - ((16*I)*Defer[Int

][(E^(2*x)*Log[x])/((I*Sqrt[5] + x)*(5*E^(2*x) + x^2 + 2*E^x*x^2 + E^(2*x)*x^2)*(1 + Log[x])*Log[1 + Log[x]]),

x])/Sqrt[5]

Rubi steps

Aborted

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Mathematica [A] time = 0.20, size = 42, normalized size = 1.56 \begin {gather*} \frac {16 e^{2 x}}{x \left (x^2+2 e^x x^2+e^{2 x} \left (5+x^2\right )\right ) \log (1+\log (x))} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(-16*E^(2*x)*x^2 - 32*E^(3*x)*x^2 + E^(4*x)*(-80 - 16*x^2) + (E^(4*x)*(-80 - 48*x^2) + E^(3*x)*(-96*

x^2 + 32*x^3) + E^(2*x)*(-48*x^2 + 32*x^3) + (E^(4*x)*(-80 - 48*x^2) + E^(3*x)*(-96*x^2 + 32*x^3) + E^(2*x)*(-

48*x^2 + 32*x^3))*Log[x])*Log[1 + Log[x]])/((x^6 + 4*E^x*x^6 + E^(4*x)*(25*x^2 + 10*x^4 + x^6) + E^(3*x)*(20*x

^4 + 4*x^6) + E^(2*x)*(10*x^4 + 6*x^6) + (x^6 + 4*E^x*x^6 + E^(4*x)*(25*x^2 + 10*x^4 + x^6) + E^(3*x)*(20*x^4

+ 4*x^6) + E^(2*x)*(10*x^4 + 6*x^6))*Log[x])*Log[1 + Log[x]]^2),x]

[Out]

(16*E^(2*x))/(x*(x^2 + 2*E^x*x^2 + E^(2*x)*(5 + x^2))*Log[1 + Log[x]])

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fricas [A] time = 0.56, size = 38, normalized size = 1.41 \begin {gather*} \frac {16 \, e^{\left (2 \, x\right )}}{{\left (2 \, x^{3} e^{x} + x^{3} + {\left (x^{3} + 5 \, x\right )} e^{\left (2 \, x\right )}\right )} \log \left (\log \relax (x) + 1\right )} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-48*x^2-80)*exp(x)^4+(32*x^3-96*x^2)*exp(x)^3+(32*x^3-48*x^2)*exp(x)^2)*log(x)+(-48*x^2-80)*exp(

x)^4+(32*x^3-96*x^2)*exp(x)^3+(32*x^3-48*x^2)*exp(x)^2)*log(log(x)+1)+(-16*x^2-80)*exp(x)^4-32*x^2*exp(x)^3-16

*exp(x)^2*x^2)/(((x^6+10*x^4+25*x^2)*exp(x)^4+(4*x^6+20*x^4)*exp(x)^3+(6*x^6+10*x^4)*exp(x)^2+4*x^6*exp(x)+x^6

)*log(x)+(x^6+10*x^4+25*x^2)*exp(x)^4+(4*x^6+20*x^4)*exp(x)^3+(6*x^6+10*x^4)*exp(x)^2+4*x^6*exp(x)+x^6)/log(lo

g(x)+1)^2,x, algorithm="fricas")

[Out]

16*e^(2*x)/((2*x^3*e^x + x^3 + (x^3 + 5*x)*e^(2*x))*log(log(x) + 1))

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giac [B] time = 0.40, size = 55, normalized size = 2.04 \begin {gather*} \frac {16 \, e^{\left (2 \, x\right )}}{x^{3} e^{\left (2 \, x\right )} \log \left (\log \relax (x) + 1\right ) + 2 \, x^{3} e^{x} \log \left (\log \relax (x) + 1\right ) + x^{3} \log \left (\log \relax (x) + 1\right ) + 5 \, x e^{\left (2 \, x\right )} \log \left (\log \relax (x) + 1\right )} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-48*x^2-80)*exp(x)^4+(32*x^3-96*x^2)*exp(x)^3+(32*x^3-48*x^2)*exp(x)^2)*log(x)+(-48*x^2-80)*exp(

x)^4+(32*x^3-96*x^2)*exp(x)^3+(32*x^3-48*x^2)*exp(x)^2)*log(log(x)+1)+(-16*x^2-80)*exp(x)^4-32*x^2*exp(x)^3-16

*exp(x)^2*x^2)/(((x^6+10*x^4+25*x^2)*exp(x)^4+(4*x^6+20*x^4)*exp(x)^3+(6*x^6+10*x^4)*exp(x)^2+4*x^6*exp(x)+x^6

)*log(x)+(x^6+10*x^4+25*x^2)*exp(x)^4+(4*x^6+20*x^4)*exp(x)^3+(6*x^6+10*x^4)*exp(x)^2+4*x^6*exp(x)+x^6)/log(lo

g(x)+1)^2,x, algorithm="giac")

[Out]

16*e^(2*x)/(x^3*e^(2*x)*log(log(x) + 1) + 2*x^3*e^x*log(log(x) + 1) + x^3*log(log(x) + 1) + 5*x*e^(2*x)*log(lo

g(x) + 1))

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maple [A] time = 0.06, size = 44, normalized size = 1.63


method	result	size

risch	\(\frac {16 \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x} x^{2}+2 \,{\mathrm e}^{x} x^{2}+x^{2}+5 \,{\mathrm e}^{2 x}\right ) x \ln \left (\ln \relax (x )+1\right )}\)	\(44\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(((((-48*x^2-80)*exp(x)^4+(32*x^3-96*x^2)*exp(x)^3+(32*x^3-48*x^2)*exp(x)^2)*ln(x)+(-48*x^2-80)*exp(x)^4+(3

2*x^3-96*x^2)*exp(x)^3+(32*x^3-48*x^2)*exp(x)^2)*ln(ln(x)+1)+(-16*x^2-80)*exp(x)^4-32*x^2*exp(x)^3-16*exp(x)^2

*x^2)/(((x^6+10*x^4+25*x^2)*exp(x)^4+(4*x^6+20*x^4)*exp(x)^3+(6*x^6+10*x^4)*exp(x)^2+4*x^6*exp(x)+x^6)*ln(x)+(

x^6+10*x^4+25*x^2)*exp(x)^4+(4*x^6+20*x^4)*exp(x)^3+(6*x^6+10*x^4)*exp(x)^2+4*x^6*exp(x)+x^6)/ln(ln(x)+1)^2,x,

method=_RETURNVERBOSE)

[Out]

16*exp(2*x)/(exp(2*x)*x^2+2*exp(x)*x^2+x^2+5*exp(2*x))/x/ln(ln(x)+1)

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maxima [A] time = 0.75, size = 38, normalized size = 1.41 \begin {gather*} \frac {16 \, e^{\left (2 \, x\right )}}{{\left (2 \, x^{3} e^{x} + x^{3} + {\left (x^{3} + 5 \, x\right )} e^{\left (2 \, x\right )}\right )} \log \left (\log \relax (x) + 1\right )} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-48*x^2-80)*exp(x)^4+(32*x^3-96*x^2)*exp(x)^3+(32*x^3-48*x^2)*exp(x)^2)*log(x)+(-48*x^2-80)*exp(

x)^4+(32*x^3-96*x^2)*exp(x)^3+(32*x^3-48*x^2)*exp(x)^2)*log(log(x)+1)+(-16*x^2-80)*exp(x)^4-32*x^2*exp(x)^3-16

*exp(x)^2*x^2)/(((x^6+10*x^4+25*x^2)*exp(x)^4+(4*x^6+20*x^4)*exp(x)^3+(6*x^6+10*x^4)*exp(x)^2+4*x^6*exp(x)+x^6

)*log(x)+(x^6+10*x^4+25*x^2)*exp(x)^4+(4*x^6+20*x^4)*exp(x)^3+(6*x^6+10*x^4)*exp(x)^2+4*x^6*exp(x)+x^6)/log(lo

g(x)+1)^2,x, algorithm="maxima")

[Out]

16*e^(2*x)/((2*x^3*e^x + x^3 + (x^3 + 5*x)*e^(2*x))*log(log(x) + 1))

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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {\ln \left (\ln \relax (x)+1\right )\,\left ({\mathrm {e}}^{4\,x}\,\left (48\,x^2+80\right )+{\mathrm {e}}^{2\,x}\,\left (48\,x^2-32\,x^3\right )+{\mathrm {e}}^{3\,x}\,\left (96\,x^2-32\,x^3\right )+\ln \relax (x)\,\left ({\mathrm {e}}^{4\,x}\,\left (48\,x^2+80\right )+{\mathrm {e}}^{2\,x}\,\left (48\,x^2-32\,x^3\right )+{\mathrm {e}}^{3\,x}\,\left (96\,x^2-32\,x^3\right )\right )\right )+{\mathrm {e}}^{4\,x}\,\left (16\,x^2+80\right )+16\,x^2\,{\mathrm {e}}^{2\,x}+32\,x^2\,{\mathrm {e}}^{3\,x}}{{\ln \left (\ln \relax (x)+1\right )}^2\,\left (4\,x^6\,{\mathrm {e}}^x+{\mathrm {e}}^{4\,x}\,\left (x^6+10\,x^4+25\,x^2\right )+{\mathrm {e}}^{2\,x}\,\left (6\,x^6+10\,x^4\right )+{\mathrm {e}}^{3\,x}\,\left (4\,x^6+20\,x^4\right )+\ln \relax (x)\,\left (4\,x^6\,{\mathrm {e}}^x+{\mathrm {e}}^{4\,x}\,\left (x^6+10\,x^4+25\,x^2\right )+{\mathrm {e}}^{2\,x}\,\left (6\,x^6+10\,x^4\right )+{\mathrm {e}}^{3\,x}\,\left (4\,x^6+20\,x^4\right )+x^6\right )+x^6\right )} \,d x \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(log(x) + 1)*(exp(4*x)*(48*x^2 + 80) + exp(2*x)*(48*x^2 - 32*x^3) + exp(3*x)*(96*x^2 - 32*x^3) + log(

x)*(exp(4*x)*(48*x^2 + 80) + exp(2*x)*(48*x^2 - 32*x^3) + exp(3*x)*(96*x^2 - 32*x^3))) + exp(4*x)*(16*x^2 + 80

) + 16*x^2*exp(2*x) + 32*x^2*exp(3*x))/(log(log(x) + 1)^2*(4*x^6*exp(x) + exp(4*x)*(25*x^2 + 10*x^4 + x^6) + e

xp(2*x)*(10*x^4 + 6*x^6) + exp(3*x)*(20*x^4 + 4*x^6) + log(x)*(4*x^6*exp(x) + exp(4*x)*(25*x^2 + 10*x^4 + x^6)

+ exp(2*x)*(10*x^4 + 6*x^6) + exp(3*x)*(20*x^4 + 4*x^6) + x^6) + x^6)),x)

[Out]

int(-(log(log(x) + 1)*(exp(4*x)*(48*x^2 + 80) + exp(2*x)*(48*x^2 - 32*x^3) + exp(3*x)*(96*x^2 - 32*x^3) + log(

x)*(exp(4*x)*(48*x^2 + 80) + exp(2*x)*(48*x^2 - 32*x^3) + exp(3*x)*(96*x^2 - 32*x^3))) + exp(4*x)*(16*x^2 + 80

) + 16*x^2*exp(2*x) + 32*x^2*exp(3*x))/(log(log(x) + 1)^2*(4*x^6*exp(x) + exp(4*x)*(25*x^2 + 10*x^4 + x^6) + e

xp(2*x)*(10*x^4 + 6*x^6) + exp(3*x)*(20*x^4 + 4*x^6) + log(x)*(4*x^6*exp(x) + exp(4*x)*(25*x^2 + 10*x^4 + x^6)

+ exp(2*x)*(10*x^4 + 6*x^6) + exp(3*x)*(20*x^4 + 4*x^6) + x^6) + x^6)), x)

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sympy [B] time = 0.91, size = 110, normalized size = 4.07 \begin {gather*} \frac {- 32 x e^{x} - 16 x}{x^{4} \log {\left (\log {\relax (x )} + 1 \right )} + 5 x^{2} \log {\left (\log {\relax (x )} + 1 \right )} + \left (2 x^{4} \log {\left (\log {\relax (x )} + 1 \right )} + 10 x^{2} \log {\left (\log {\relax (x )} + 1 \right )}\right ) e^{x} + \left (x^{4} \log {\left (\log {\relax (x )} + 1 \right )} + 10 x^{2} \log {\left (\log {\relax (x )} + 1 \right )} + 25 \log {\left (\log {\relax (x )} + 1 \right )}\right ) e^{2 x}} + \frac {16}{\left (x^{3} + 5 x\right ) \log {\left (\log {\relax (x )} + 1 \right )}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-48*x**2-80)*exp(x)**4+(32*x**3-96*x**2)*exp(x)**3+(32*x**3-48*x**2)*exp(x)**2)*ln(x)+(-48*x**2-

80)*exp(x)**4+(32*x**3-96*x**2)*exp(x)**3+(32*x**3-48*x**2)*exp(x)**2)*ln(ln(x)+1)+(-16*x**2-80)*exp(x)**4-32*

x**2*exp(x)**3-16*exp(x)**2*x**2)/(((x**6+10*x**4+25*x**2)*exp(x)**4+(4*x**6+20*x**4)*exp(x)**3+(6*x**6+10*x**

4)*exp(x)**2+4*x**6*exp(x)+x**6)*ln(x)+(x**6+10*x**4+25*x**2)*exp(x)**4+(4*x**6+20*x**4)*exp(x)**3+(6*x**6+10*

x**4)*exp(x)**2+4*x**6*exp(x)+x**6)/ln(ln(x)+1)**2,x)

[Out]

(-32*x*exp(x) - 16*x)/(x**4*log(log(x) + 1) + 5*x**2*log(log(x) + 1) + (2*x**4*log(log(x) + 1) + 10*x**2*log(l

og(x) + 1))*exp(x) + (x**4*log(log(x) + 1) + 10*x**2*log(log(x) + 1) + 25*log(log(x) + 1))*exp(2*x)) + 16/((x*

*3 + 5*x)*log(log(x) + 1))

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