3.58.39 \(\int \frac {e^x (-8+e^{20} (-32+6 x)) \log (\log (-\frac {e^{20} x^4}{-1+e^{20} (-4+x)}))+e^x (1-x+e^{20} (4-5 x+x^2)) \log (-\frac {e^{20} x^4}{-1+e^{20} (-4+x)}) \log ^2(\log (-\frac {e^{20} x^4}{-1+e^{20} (-4+x)}))}{(-x^2+e^{20} (-4 x^2+x^3)) \log (-\frac {e^{20} x^4}{-1+e^{20} (-4+x)})} \, dx\)

Optimal. Leaf size=25 \[ \frac {e^x \log ^2\left (\log \left (\frac {x^4}{4+\frac {1}{e^{20}}-x}\right )\right )}{x} \]

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Rubi [F]  time = 11.33, 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^x \left (-8+e^{20} (-32+6 x)\right ) \log \left (\log \left (-\frac {e^{20} x^4}{-1+e^{20} (-4+x)}\right )\right )+e^x \left (1-x+e^{20} \left (4-5 x+x^2\right )\right ) \log \left (-\frac {e^{20} x^4}{-1+e^{20} (-4+x)}\right ) \log ^2\left (\log \left (-\frac {e^{20} x^4}{-1+e^{20} (-4+x)}\right )\right )}{\left (-x^2+e^{20} \left (-4 x^2+x^3\right )\right ) \log \left (-\frac {e^{20} x^4}{-1+e^{20} (-4+x)}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^x*(-8 + E^20*(-32 + 6*x))*Log[Log[-((E^20*x^4)/(-1 + E^20*(-4 + x)))]] + E^x*(1 - x + E^20*(4 - 5*x + x
^2))*Log[-((E^20*x^4)/(-1 + E^20*(-4 + x)))]*Log[Log[-((E^20*x^4)/(-1 + E^20*(-4 + x)))]]^2)/((-x^2 + E^20*(-4
*x^2 + x^3))*Log[-((E^20*x^4)/(-1 + E^20*(-4 + x)))]),x]

[Out]

8*Defer[Int][(E^x*Log[Log[-((E^20*x^4)/(-1 - 4*E^20 + E^20*x))]])/(x^2*Log[-((E^20*x^4)/(-1 - 4*E^20 + E^20*x)
)]), x] + (2*Defer[Int][(E^(20 + x)*Log[Log[-((E^20*x^4)/(-1 - 4*E^20 + E^20*x))]])/(x*Log[-((E^20*x^4)/(-1 -
4*E^20 + E^20*x))]), x])/(1 + 4*E^20) + (2*Defer[Int][(E^(40 + x)*Log[Log[-((E^20*x^4)/(-1 - 4*E^20 + E^20*x))
]])/((1 + 4*E^20 - E^20*x)*Log[-((E^20*x^4)/(-1 - 4*E^20 + E^20*x))]), x])/(1 + 4*E^20) - Defer[Int][(E^x*Log[
Log[-((E^20*x^4)/(-1 - 4*E^20 + E^20*x))]]^2)/x^2, x] + ((1 + 5*E^20)*Defer[Int][(E^x*Log[Log[-((E^20*x^4)/(-1
- 4*E^20 + E^20*x))]]^2)/x, x])/(1 + 4*E^20) - Defer[Int][(E^(20 + x)*Log[Log[-((E^20*x^4)/(-1 - 4*E^20 + E^2
0*x))]]^2)/x, x]/(1 + 4*E^20) + ((1 + 5*E^20)*Defer[Int][(E^(20 + x)*Log[Log[-((E^20*x^4)/(-1 - 4*E^20 + E^20*
x))]]^2)/(1 + 4*E^20 - E^20*x), x])/(1 + 4*E^20) + Defer[Int][(E^(20 + x)*Log[Log[-((E^20*x^4)/(-1 - 4*E^20 +
E^20*x))]]^2)/(-1 - 4*E^20 + E^20*x), x] + Defer[Int][(E^(40 + x)*Log[Log[-((E^20*x^4)/(-1 - 4*E^20 + E^20*x))
]]^2)/(-1 - 4*E^20 + E^20*x), x]/(1 + 4*E^20)

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-e^x \left (-8+e^{20} (-32+6 x)\right ) \log \left (\log \left (-\frac {e^{20} x^4}{-1+e^{20} (-4+x)}\right )\right )-e^x \left (1-x+e^{20} \left (4-5 x+x^2\right )\right ) \log \left (-\frac {e^{20} x^4}{-1+e^{20} (-4+x)}\right ) \log ^2\left (\log \left (-\frac {e^{20} x^4}{-1+e^{20} (-4+x)}\right )\right )}{x^2 \left (1+4 e^{20}-e^{20} x\right ) \log \left (-\frac {e^{20} x^4}{-1-4 e^{20}+e^{20} x}\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [F]  time = 0.24, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^x \left (-8+e^{20} (-32+6 x)\right ) \log \left (\log \left (-\frac {e^{20} x^4}{-1+e^{20} (-4+x)}\right )\right )+e^x \left (1-x+e^{20} \left (4-5 x+x^2\right )\right ) \log \left (-\frac {e^{20} x^4}{-1+e^{20} (-4+x)}\right ) \log ^2\left (\log \left (-\frac {e^{20} x^4}{-1+e^{20} (-4+x)}\right )\right )}{\left (-x^2+e^{20} \left (-4 x^2+x^3\right )\right ) \log \left (-\frac {e^{20} x^4}{-1+e^{20} (-4+x)}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(E^x*(-8 + E^20*(-32 + 6*x))*Log[Log[-((E^20*x^4)/(-1 + E^20*(-4 + x)))]] + E^x*(1 - x + E^20*(4 - 5
*x + x^2))*Log[-((E^20*x^4)/(-1 + E^20*(-4 + x)))]*Log[Log[-((E^20*x^4)/(-1 + E^20*(-4 + x)))]]^2)/((-x^2 + E^
20*(-4*x^2 + x^3))*Log[-((E^20*x^4)/(-1 + E^20*(-4 + x)))]),x]

[Out]

Integrate[(E^x*(-8 + E^20*(-32 + 6*x))*Log[Log[-((E^20*x^4)/(-1 + E^20*(-4 + x)))]] + E^x*(1 - x + E^20*(4 - 5
*x + x^2))*Log[-((E^20*x^4)/(-1 + E^20*(-4 + x)))]*Log[Log[-((E^20*x^4)/(-1 + E^20*(-4 + x)))]]^2)/((-x^2 + E^
20*(-4*x^2 + x^3))*Log[-((E^20*x^4)/(-1 + E^20*(-4 + x)))]), x]

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fricas [A]  time = 0.88, size = 27, normalized size = 1.08 \begin {gather*} \frac {e^{x} \log \left (\log \left (-\frac {x^{4} e^{20}}{{\left (x - 4\right )} e^{20} - 1}\right )\right )^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2-5*x+4)*exp(10)^2-x+1)*exp(x)*log(-x^4*exp(10)^2/((x-4)*exp(10)^2-1))*log(log(-x^4*exp(10)^2/(
(x-4)*exp(10)^2-1)))^2+((6*x-32)*exp(10)^2-8)*exp(x)*log(log(-x^4*exp(10)^2/((x-4)*exp(10)^2-1))))/((x^3-4*x^2
)*exp(10)^2-x^2)/log(-x^4*exp(10)^2/((x-4)*exp(10)^2-1)),x, algorithm="fricas")

[Out]

e^x*log(log(-x^4*e^20/((x - 4)*e^20 - 1)))^2/x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left ({\left (x^{2} - 5 \, x + 4\right )} e^{20} - x + 1\right )} e^{x} \log \left (-\frac {x^{4} e^{20}}{{\left (x - 4\right )} e^{20} - 1}\right ) \log \left (\log \left (-\frac {x^{4} e^{20}}{{\left (x - 4\right )} e^{20} - 1}\right )\right )^{2} + 2 \, {\left ({\left (3 \, x - 16\right )} e^{20} - 4\right )} e^{x} \log \left (\log \left (-\frac {x^{4} e^{20}}{{\left (x - 4\right )} e^{20} - 1}\right )\right )}{{\left (x^{2} - {\left (x^{3} - 4 \, x^{2}\right )} e^{20}\right )} \log \left (-\frac {x^{4} e^{20}}{{\left (x - 4\right )} e^{20} - 1}\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2-5*x+4)*exp(10)^2-x+1)*exp(x)*log(-x^4*exp(10)^2/((x-4)*exp(10)^2-1))*log(log(-x^4*exp(10)^2/(
(x-4)*exp(10)^2-1)))^2+((6*x-32)*exp(10)^2-8)*exp(x)*log(log(-x^4*exp(10)^2/((x-4)*exp(10)^2-1))))/((x^3-4*x^2
)*exp(10)^2-x^2)/log(-x^4*exp(10)^2/((x-4)*exp(10)^2-1)),x, algorithm="giac")

[Out]

integrate(-(((x^2 - 5*x + 4)*e^20 - x + 1)*e^x*log(-x^4*e^20/((x - 4)*e^20 - 1))*log(log(-x^4*e^20/((x - 4)*e^
20 - 1)))^2 + 2*((3*x - 16)*e^20 - 4)*e^x*log(log(-x^4*e^20/((x - 4)*e^20 - 1))))/((x^2 - (x^3 - 4*x^2)*e^20)*
log(-x^4*e^20/((x - 4)*e^20 - 1))), x)

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maple [C]  time = 1.32, size = 269, normalized size = 10.76

method	result	size
risch	\(\frac {{\mathrm e}^{x} \ln \left (20+i \pi +4 \ln \relax (x )-\ln \left (\left (x -4\right ) {\mathrm e}^{20}-1\right )-\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}-\frac {i \pi \,\mathrm {csgn}\left (i x^{3}\right ) \left (-\mathrm {csgn}\left (i x^{3}\right )+\mathrm {csgn}\left (i x^{2}\right )\right ) \left (-\mathrm {csgn}\left (i x^{3}\right )+\mathrm {csgn}\left (i x \right )\right )}{2}-\frac {i \pi \,\mathrm {csgn}\left (i x^{4}\right ) \left (-\mathrm {csgn}\left (i x^{4}\right )+\mathrm {csgn}\left (i x^{3}\right )\right ) \left (-\mathrm {csgn}\left (i x^{4}\right )+\mathrm {csgn}\left (i x \right )\right )}{2}-\frac {i \pi \,\mathrm {csgn}\left (\frac {i x^{4}}{\left (x -4\right ) {\mathrm e}^{20}-1}\right ) \left (-\mathrm {csgn}\left (\frac {i x^{4}}{\left (x -4\right ) {\mathrm e}^{20}-1}\right )+\mathrm {csgn}\left (i x^{4}\right )\right ) \left (-\mathrm {csgn}\left (\frac {i x^{4}}{\left (x -4\right ) {\mathrm e}^{20}-1}\right )+\mathrm {csgn}\left (\frac {i}{\left (x -4\right ) {\mathrm e}^{20}-1}\right )\right )}{2}+i \pi \mathrm {csgn}\left (\frac {i x^{4}}{\left (x -4\right ) {\mathrm e}^{20}-1}\right )^{2} \left (\mathrm {csgn}\left (\frac {i x^{4}}{\left (x -4\right ) {\mathrm e}^{20}-1}\right )-1\right )\right )^{2}}{x}\)	\(269\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^2-5*x+4)*exp(10)^2-x+1)*exp(x)*ln(-x^4*exp(10)^2/((x-4)*exp(10)^2-1))*ln(ln(-x^4*exp(10)^2/((x-4)*exp
(10)^2-1)))^2+((6*x-32)*exp(10)^2-8)*exp(x)*ln(ln(-x^4*exp(10)^2/((x-4)*exp(10)^2-1))))/((x^3-4*x^2)*exp(10)^2
-x^2)/ln(-x^4*exp(10)^2/((x-4)*exp(10)^2-1)),x,method=_RETURNVERBOSE)

[Out]

1/x*exp(x)*ln(20+I*Pi+4*ln(x)-ln((x-4)*exp(20)-1)-1/2*I*Pi*csgn(I*x^2)*(-csgn(I*x^2)+csgn(I*x))^2-1/2*I*Pi*csg
n(I*x^3)*(-csgn(I*x^3)+csgn(I*x^2))*(-csgn(I*x^3)+csgn(I*x))-1/2*I*Pi*csgn(I*x^4)*(-csgn(I*x^4)+csgn(I*x^3))*(
-csgn(I*x^4)+csgn(I*x))-1/2*I*Pi*csgn(I*x^4/((x-4)*exp(20)-1))*(-csgn(I*x^4/((x-4)*exp(20)-1))+csgn(I*x^4))*(-
csgn(I*x^4/((x-4)*exp(20)-1))+csgn(I/((x-4)*exp(20)-1)))+I*Pi*csgn(I*x^4/((x-4)*exp(20)-1))^2*(csgn(I*x^4/((x-
4)*exp(20)-1))-1))^2

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maxima [A]  time = 0.49, size = 29, normalized size = 1.16 \begin {gather*} \frac {e^{x} \log \left (-\log \left (-x e^{20} + 4 \, e^{20} + 1\right ) + 4 \, \log \relax (x) + 20\right )^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2-5*x+4)*exp(10)^2-x+1)*exp(x)*log(-x^4*exp(10)^2/((x-4)*exp(10)^2-1))*log(log(-x^4*exp(10)^2/(
(x-4)*exp(10)^2-1)))^2+((6*x-32)*exp(10)^2-8)*exp(x)*log(log(-x^4*exp(10)^2/((x-4)*exp(10)^2-1))))/((x^3-4*x^2
)*exp(10)^2-x^2)/log(-x^4*exp(10)^2/((x-4)*exp(10)^2-1)),x, algorithm="maxima")

[Out]

e^x*log(-log(-x*e^20 + 4*e^20 + 1) + 4*log(x) + 20)^2/x

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {\ln \left (-\frac {x^4\,{\mathrm {e}}^{20}}{{\mathrm {e}}^{20}\,\left (x-4\right )-1}\right )\,{\mathrm {e}}^x\,\left ({\mathrm {e}}^{20}\,\left (x^2-5\,x+4\right )-x+1\right )\,{\ln \left (\ln \left (-\frac {x^4\,{\mathrm {e}}^{20}}{{\mathrm {e}}^{20}\,\left (x-4\right )-1}\right )\right )}^2+{\mathrm {e}}^x\,\left ({\mathrm {e}}^{20}\,\left (6\,x-32\right )-8\right )\,\ln \left (\ln \left (-\frac {x^4\,{\mathrm {e}}^{20}}{{\mathrm {e}}^{20}\,\left (x-4\right )-1}\right )\right )}{\ln \left (-\frac {x^4\,{\mathrm {e}}^{20}}{{\mathrm {e}}^{20}\,\left (x-4\right )-1}\right )\,\left ({\mathrm {e}}^{20}\,\left (4\,x^2-x^3\right )+x^2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x)*log(log(-(x^4*exp(20))/(exp(20)*(x - 4) - 1)))*(exp(20)*(6*x - 32) - 8) + log(-(x^4*exp(20))/(exp
(20)*(x - 4) - 1))*exp(x)*log(log(-(x^4*exp(20))/(exp(20)*(x - 4) - 1)))^2*(exp(20)*(x^2 - 5*x + 4) - x + 1))/
(log(-(x^4*exp(20))/(exp(20)*(x - 4) - 1))*(exp(20)*(4*x^2 - x^3) + x^2)),x)

[Out]

int(-(exp(x)*log(log(-(x^4*exp(20))/(exp(20)*(x - 4) - 1)))*(exp(20)*(6*x - 32) - 8) + log(-(x^4*exp(20))/(exp
(20)*(x - 4) - 1))*exp(x)*log(log(-(x^4*exp(20))/(exp(20)*(x - 4) - 1)))^2*(exp(20)*(x^2 - 5*x + 4) - x + 1))/
(log(-(x^4*exp(20))/(exp(20)*(x - 4) - 1))*(exp(20)*(4*x^2 - x^3) + x^2)), x)

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sympy [A]  time = 1.01, size = 26, normalized size = 1.04 \begin {gather*} \frac {e^{x} \log {\left (\log {\left (- \frac {x^{4} e^{20}}{\left (x - 4\right ) e^{20} - 1} \right )} \right )}^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**2-5*x+4)*exp(10)**2-x+1)*exp(x)*ln(-x**4*exp(10)**2/((x-4)*exp(10)**2-1))*ln(ln(-x**4*exp(10)*
*2/((x-4)*exp(10)**2-1)))**2+((6*x-32)*exp(10)**2-8)*exp(x)*ln(ln(-x**4*exp(10)**2/((x-4)*exp(10)**2-1))))/((x
**3-4*x**2)*exp(10)**2-x**2)/ln(-x**4*exp(10)**2/((x-4)*exp(10)**2-1)),x)

[Out]

exp(x)*log(log(-x**4*exp(20)/((x - 4)*exp(20) - 1)))**2/x

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