3.62.68 \(\int \frac {625 e^x+e^x (-625-625 x) \log (x) \log (\log (x))+(625-1250 x) \log (x) \log ^2(\log (x))}{625 e^{2 x} x^2 \log (x)+e^x (-50 x-1250 x^2+1250 x^3) \log (x) \log (\log (x))+(1+50 x+575 x^2-1250 x^3+625 x^4) \log (x) \log ^2(\log (x))} \, dx\)

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

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Rubi [F]  time = 5.96, 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 {625 e^x+e^x (-625-625 x) \log (x) \log (\log (x))+(625-1250 x) \log (x) \log ^2(\log (x))}{625 e^{2 x} x^2 \log (x)+e^x \left (-50 x-1250 x^2+1250 x^3\right ) \log (x) \log (\log (x))+\left (1+50 x+575 x^2-1250 x^3+625 x^4\right ) \log (x) \log ^2(\log (x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(625*E^x + E^x*(-625 - 625*x)*Log[x]*Log[Log[x]] + (625 - 1250*x)*Log[x]*Log[Log[x]]^2)/(625*E^(2*x)*x^2*L
og[x] + E^x*(-50*x - 1250*x^2 + 1250*x^3)*Log[x]*Log[Log[x]] + (1 + 50*x + 575*x^2 - 1250*x^3 + 625*x^4)*Log[x
]*Log[Log[x]]^2),x]

[Out]

625*Defer[Int][Log[Log[x]]/(Log[x]*(25*E^x*x - Log[Log[x]] - 25*x*Log[Log[x]] + 25*x^2*Log[Log[x]])^2), x] + 2
5*Defer[Int][Log[Log[x]]/(x*Log[x]*(25*E^x*x - Log[Log[x]] - 25*x*Log[Log[x]] + 25*x^2*Log[Log[x]])^2), x] - 6
25*Defer[Int][(x*Log[Log[x]])/(Log[x]*(25*E^x*x - Log[Log[x]] - 25*x*Log[Log[x]] + 25*x^2*Log[Log[x]])^2), x]
- 25*Defer[Int][Log[Log[x]]^2/(25*E^x*x - Log[Log[x]] - 25*x*Log[Log[x]] + 25*x^2*Log[Log[x]])^2, x] - 25*Defe
r[Int][Log[Log[x]]^2/(x*(25*E^x*x - Log[Log[x]] - 25*x*Log[Log[x]] + 25*x^2*Log[Log[x]])^2), x] - 1250*Defer[I
nt][(x*Log[Log[x]]^2)/(25*E^x*x - Log[Log[x]] - 25*x*Log[Log[x]] + 25*x^2*Log[Log[x]])^2, x] + 625*Defer[Int][
(x^2*Log[Log[x]]^2)/(25*E^x*x - Log[Log[x]] - 25*x*Log[Log[x]] + 25*x^2*Log[Log[x]])^2, x] + 25*Defer[Int][1/(
x*Log[x]*(25*E^x*x - Log[Log[x]] - 25*x*Log[Log[x]] + 25*x^2*Log[Log[x]])), x] - 25*Defer[Int][Log[Log[x]]/(25
*E^x*x - Log[Log[x]] - 25*x*Log[Log[x]] + 25*x^2*Log[Log[x]]), x] - 25*Defer[Int][Log[Log[x]]/(x*(25*E^x*x - L
og[Log[x]] - 25*x*Log[Log[x]] + 25*x^2*Log[Log[x]])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {625 \left (e^x-\log (x) \log (\log (x)) \left (e^x (1+x)+(-1+2 x) \log (\log (x))\right )\right )}{\log (x) \left (25 e^x x+\left (-1-25 x+25 x^2\right ) \log (\log (x))\right )^2} \, dx\\ &=625 \int \frac {e^x-\log (x) \log (\log (x)) \left (e^x (1+x)+(-1+2 x) \log (\log (x))\right )}{\log (x) \left (25 e^x x+\left (-1-25 x+25 x^2\right ) \log (\log (x))\right )^2} \, dx\\ &=625 \int \left (-\frac {-1+\log (x) \log (\log (x))+x \log (x) \log (\log (x))}{25 x \log (x) \left (25 e^x x-\log (\log (x))-25 x \log (\log (x))+25 x^2 \log (\log (x))\right )}+\frac {\log (\log (x)) \left (1+25 x-25 x^2-\log (x) \log (\log (x))-x \log (x) \log (\log (x))-50 x^2 \log (x) \log (\log (x))+25 x^3 \log (x) \log (\log (x))\right )}{25 x \log (x) \left (25 e^x x-\log (\log (x))-25 x \log (\log (x))+25 x^2 \log (\log (x))\right )^2}\right ) \, dx\\ &=-\left (25 \int \frac {-1+\log (x) \log (\log (x))+x \log (x) \log (\log (x))}{x \log (x) \left (25 e^x x-\log (\log (x))-25 x \log (\log (x))+25 x^2 \log (\log (x))\right )} \, dx\right )+25 \int \frac {\log (\log (x)) \left (1+25 x-25 x^2-\log (x) \log (\log (x))-x \log (x) \log (\log (x))-50 x^2 \log (x) \log (\log (x))+25 x^3 \log (x) \log (\log (x))\right )}{x \log (x) \left (25 e^x x-\log (\log (x))-25 x \log (\log (x))+25 x^2 \log (\log (x))\right )^2} \, dx\\ &=25 \int \frac {\log (\log (x)) \left (1+25 x-25 x^2+\left (-1-x-50 x^2+25 x^3\right ) \log (x) \log (\log (x))\right )}{x \log (x) \left (25 e^x x+\left (-1-25 x+25 x^2\right ) \log (\log (x))\right )^2} \, dx-25 \int \left (-\frac {1}{x \log (x) \left (25 e^x x-\log (\log (x))-25 x \log (\log (x))+25 x^2 \log (\log (x))\right )}+\frac {\log (\log (x))}{25 e^x x-\log (\log (x))-25 x \log (\log (x))+25 x^2 \log (\log (x))}+\frac {\log (\log (x))}{x \left (25 e^x x-\log (\log (x))-25 x \log (\log (x))+25 x^2 \log (\log (x))\right )}\right ) \, dx\\ &=25 \int \frac {1}{x \log (x) \left (25 e^x x-\log (\log (x))-25 x \log (\log (x))+25 x^2 \log (\log (x))\right )} \, dx-25 \int \frac {\log (\log (x))}{25 e^x x-\log (\log (x))-25 x \log (\log (x))+25 x^2 \log (\log (x))} \, dx-25 \int \frac {\log (\log (x))}{x \left (25 e^x x-\log (\log (x))-25 x \log (\log (x))+25 x^2 \log (\log (x))\right )} \, dx+25 \int \left (\frac {25 \log (\log (x))}{\log (x) \left (25 e^x x-\log (\log (x))-25 x \log (\log (x))+25 x^2 \log (\log (x))\right )^2}+\frac {\log (\log (x))}{x \log (x) \left (25 e^x x-\log (\log (x))-25 x \log (\log (x))+25 x^2 \log (\log (x))\right )^2}-\frac {25 x \log (\log (x))}{\log (x) \left (25 e^x x-\log (\log (x))-25 x \log (\log (x))+25 x^2 \log (\log (x))\right )^2}-\frac {\log ^2(\log (x))}{\left (25 e^x x-\log (\log (x))-25 x \log (\log (x))+25 x^2 \log (\log (x))\right )^2}-\frac {\log ^2(\log (x))}{x \left (25 e^x x-\log (\log (x))-25 x \log (\log (x))+25 x^2 \log (\log (x))\right )^2}-\frac {50 x \log ^2(\log (x))}{\left (25 e^x x-\log (\log (x))-25 x \log (\log (x))+25 x^2 \log (\log (x))\right )^2}+\frac {25 x^2 \log ^2(\log (x))}{\left (25 e^x x-\log (\log (x))-25 x \log (\log (x))+25 x^2 \log (\log (x))\right )^2}\right ) \, dx\\ &=25 \int \frac {\log (\log (x))}{x \log (x) \left (25 e^x x-\log (\log (x))-25 x \log (\log (x))+25 x^2 \log (\log (x))\right )^2} \, dx-25 \int \frac {\log ^2(\log (x))}{\left (25 e^x x-\log (\log (x))-25 x \log (\log (x))+25 x^2 \log (\log (x))\right )^2} \, dx-25 \int \frac {\log ^2(\log (x))}{x \left (25 e^x x-\log (\log (x))-25 x \log (\log (x))+25 x^2 \log (\log (x))\right )^2} \, dx+25 \int \frac {1}{x \log (x) \left (25 e^x x-\log (\log (x))-25 x \log (\log (x))+25 x^2 \log (\log (x))\right )} \, dx-25 \int \frac {\log (\log (x))}{25 e^x x-\log (\log (x))-25 x \log (\log (x))+25 x^2 \log (\log (x))} \, dx-25 \int \frac {\log (\log (x))}{x \left (25 e^x x-\log (\log (x))-25 x \log (\log (x))+25 x^2 \log (\log (x))\right )} \, dx+625 \int \frac {\log (\log (x))}{\log (x) \left (25 e^x x-\log (\log (x))-25 x \log (\log (x))+25 x^2 \log (\log (x))\right )^2} \, dx-625 \int \frac {x \log (\log (x))}{\log (x) \left (25 e^x x-\log (\log (x))-25 x \log (\log (x))+25 x^2 \log (\log (x))\right )^2} \, dx+625 \int \frac {x^2 \log ^2(\log (x))}{\left (25 e^x x-\log (\log (x))-25 x \log (\log (x))+25 x^2 \log (\log (x))\right )^2} \, dx-1250 \int \frac {x \log ^2(\log (x))}{\left (25 e^x x-\log (\log (x))-25 x \log (\log (x))+25 x^2 \log (\log (x))\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.94, size = 28, normalized size = 1.12 \begin {gather*} \frac {25 \log (\log (x))}{25 e^x x+\left (-1-25 x+25 x^2\right ) \log (\log (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(625*E^x + E^x*(-625 - 625*x)*Log[x]*Log[Log[x]] + (625 - 1250*x)*Log[x]*Log[Log[x]]^2)/(625*E^(2*x)
*x^2*Log[x] + E^x*(-50*x - 1250*x^2 + 1250*x^3)*Log[x]*Log[Log[x]] + (1 + 50*x + 575*x^2 - 1250*x^3 + 625*x^4)
*Log[x]*Log[Log[x]]^2),x]

[Out]

(25*Log[Log[x]])/(25*E^x*x + (-1 - 25*x + 25*x^2)*Log[Log[x]])

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fricas [A]  time = 0.54, size = 27, normalized size = 1.08 \begin {gather*} \frac {25 \, \log \left (\log \relax (x)\right )}{25 \, x e^{x} + {\left (25 \, x^{2} - 25 \, x - 1\right )} \log \left (\log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-1250*x+625)*log(x)*log(log(x))^2+(-625*x-625)*exp(x)*log(x)*log(log(x))+625*exp(x))/((625*x^4-125
0*x^3+575*x^2+50*x+1)*log(x)*log(log(x))^2+(1250*x^3-1250*x^2-50*x)*exp(x)*log(x)*log(log(x))+625*x^2*exp(x)^2
*log(x)),x, algorithm="fricas")

[Out]

25*log(log(x))/(25*x*e^x + (25*x^2 - 25*x - 1)*log(log(x)))

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giac [A]  time = 0.47, size = 32, normalized size = 1.28 \begin {gather*} \frac {25 \, \log \left (\log \relax (x)\right )}{25 \, x^{2} \log \left (\log \relax (x)\right ) + 25 \, x e^{x} - 25 \, x \log \left (\log \relax (x)\right ) - \log \left (\log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-1250*x+625)*log(x)*log(log(x))^2+(-625*x-625)*exp(x)*log(x)*log(log(x))+625*exp(x))/((625*x^4-125
0*x^3+575*x^2+50*x+1)*log(x)*log(log(x))^2+(1250*x^3-1250*x^2-50*x)*exp(x)*log(x)*log(log(x))+625*x^2*exp(x)^2
*log(x)),x, algorithm="giac")

[Out]

25*log(log(x))/(25*x^2*log(log(x)) + 25*x*e^x - 25*x*log(log(x)) - log(log(x)))

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maple [B]  time = 0.06, size = 60, normalized size = 2.40

method	result	size
risch	\(\frac {25}{25 x^{2}-25 x -1}-\frac {625 x \,{\mathrm e}^{x}}{\left (25 x^{2}-25 x -1\right ) \left (25 x^{2} \ln \left (\ln \relax (x )\right )-25 x \ln \left (\ln \relax (x )\right )+25 \,{\mathrm e}^{x} x -\ln \left (\ln \relax (x )\right )\right )}\)	\(60\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-1250*x+625)*ln(x)*ln(ln(x))^2+(-625*x-625)*exp(x)*ln(x)*ln(ln(x))+625*exp(x))/((625*x^4-1250*x^3+575*x^
2+50*x+1)*ln(x)*ln(ln(x))^2+(1250*x^3-1250*x^2-50*x)*exp(x)*ln(x)*ln(ln(x))+625*x^2*exp(x)^2*ln(x)),x,method=_
RETURNVERBOSE)

[Out]

25/(25*x^2-25*x-1)-625*x*exp(x)/(25*x^2-25*x-1)/(25*x^2*ln(ln(x))-25*x*ln(ln(x))+25*exp(x)*x-ln(ln(x)))

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maxima [A]  time = 0.41, size = 27, normalized size = 1.08 \begin {gather*} \frac {25 \, \log \left (\log \relax (x)\right )}{25 \, x e^{x} + {\left (25 \, x^{2} - 25 \, x - 1\right )} \log \left (\log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-1250*x+625)*log(x)*log(log(x))^2+(-625*x-625)*exp(x)*log(x)*log(log(x))+625*exp(x))/((625*x^4-125
0*x^3+575*x^2+50*x+1)*log(x)*log(log(x))^2+(1250*x^3-1250*x^2-50*x)*exp(x)*log(x)*log(log(x))+625*x^2*exp(x)^2
*log(x)),x, algorithm="maxima")

[Out]

25*log(log(x))/(25*x*e^x + (25*x^2 - 25*x - 1)*log(log(x)))

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mupad [B]  time = 4.42, size = 30, normalized size = 1.20 \begin {gather*} -\frac {25\,\ln \left (\ln \relax (x)\right )}{\ln \left (\ln \relax (x)\right )+25\,x\,\ln \left (\ln \relax (x)\right )-25\,x^2\,\ln \left (\ln \relax (x)\right )-25\,x\,{\mathrm {e}}^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(log(x))^2*log(x)*(1250*x - 625) - 625*exp(x) + log(log(x))*exp(x)*log(x)*(625*x + 625))/(625*x^2*exp
(2*x)*log(x) + log(log(x))^2*log(x)*(50*x + 575*x^2 - 1250*x^3 + 625*x^4 + 1) - log(log(x))*exp(x)*log(x)*(50*
x + 1250*x^2 - 1250*x^3)),x)

[Out]

-(25*log(log(x)))/(log(log(x)) + 25*x*log(log(x)) - 25*x^2*log(log(x)) - 25*x*exp(x))

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sympy [A]  time = 0.40, size = 36, normalized size = 1.44 \begin {gather*} \frac {25 \log {\left (\log {\relax (x )} \right )}}{25 x^{2} \log {\left (\log {\relax (x )} \right )} + 25 x e^{x} - 25 x \log {\left (\log {\relax (x )} \right )} - \log {\left (\log {\relax (x )} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-1250*x+625)*ln(x)*ln(ln(x))**2+(-625*x-625)*exp(x)*ln(x)*ln(ln(x))+625*exp(x))/((625*x**4-1250*x*
*3+575*x**2+50*x+1)*ln(x)*ln(ln(x))**2+(1250*x**3-1250*x**2-50*x)*exp(x)*ln(x)*ln(ln(x))+625*x**2*exp(x)**2*ln
(x)),x)

[Out]

25*log(log(x))/(25*x**2*log(log(x)) + 25*x*exp(x) - 25*x*log(log(x)) - log(log(x)))

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