3.101.82 \(\int \frac {e^{\frac {1}{25} (25 x^2+10 x \log (\log (x^2))+\log ^2(\log (x^2)))} (60 x+20 x^2+(-25 x+150 x^2+50 x^3) \log (x^2)+(12+4 x+(30 x+10 x^2) \log (x^2)) \log (\log (x^2)))}{(225 x+150 x^2+25 x^3) \log (x^2)} \, dx\)

Optimal. Leaf size=21 \[ \frac {e^{\left (x+\frac {1}{5} \log \left (\log \left (x^2\right )\right )\right )^2}}{3+x} \]

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Rubi [F]  time = 5.39, 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 (\frac {1}{25} \left (25 x^2+10 x \log \left (\log \left (x^2\right )\right )+\log ^2\left (\log \left (x^2\right )\right )\right )\right ) \left (60 x+20 x^2+\left (-25 x+150 x^2+50 x^3\right ) \log \left (x^2\right )+\left (12+4 x+\left (30 x+10 x^2\right ) \log \left (x^2\right )\right ) \log \left (\log \left (x^2\right )\right )\right )}{\left (225 x+150 x^2+25 x^3\right ) \log \left (x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((25*x^2 + 10*x*Log[Log[x^2]] + Log[Log[x^2]]^2)/25)*(60*x + 20*x^2 + (-25*x + 150*x^2 + 50*x^3)*Log[x^
2] + (12 + 4*x + (30*x + 10*x^2)*Log[x^2])*Log[Log[x^2]]))/((225*x + 150*x^2 + 25*x^3)*Log[x^2]),x]

[Out]

2*Defer[Int][E^((5*x + Log[Log[x^2]])^2/25), x] - Defer[Int][E^((5*x + Log[Log[x^2]])^2/25)/(3 + x)^2, x] - 6*
Defer[Int][E^((5*x + Log[Log[x^2]])^2/25)/(3 + x), x] + (4*Defer[Int][E^((5*x + Log[Log[x^2]])^2/25)/((3 + x)*
Log[x^2]), x])/5 + (2*Defer[Int][(E^((5*x + Log[Log[x^2]])^2/25)*Log[Log[x^2]])/(3 + x), x])/5 + (4*Defer[Int]
[(E^((5*x + Log[Log[x^2]])^2/25)*Log[Log[x^2]])/(x*Log[x^2]), x])/75 - (4*Defer[Int][(E^((5*x + Log[Log[x^2]])
^2/25)*Log[Log[x^2]])/((3 + x)*Log[x^2]), x])/75

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {1}{25} \left (25 x^2+10 x \log \left (\log \left (x^2\right )\right )+\log ^2\left (\log \left (x^2\right )\right )\right )\right ) \left (60 x+20 x^2+\left (-25 x+150 x^2+50 x^3\right ) \log \left (x^2\right )+\left (12+4 x+\left (30 x+10 x^2\right ) \log \left (x^2\right )\right ) \log \left (\log \left (x^2\right )\right )\right )}{x \left (225+150 x+25 x^2\right ) \log \left (x^2\right )} \, dx\\ &=\int \frac {\exp \left (\frac {1}{25} \left (25 x^2+10 x \log \left (\log \left (x^2\right )\right )+\log ^2\left (\log \left (x^2\right )\right )\right )\right ) \left (60 x+20 x^2+\left (-25 x+150 x^2+50 x^3\right ) \log \left (x^2\right )+\left (12+4 x+\left (30 x+10 x^2\right ) \log \left (x^2\right )\right ) \log \left (\log \left (x^2\right )\right )\right )}{25 x (3+x)^2 \log \left (x^2\right )} \, dx\\ &=\frac {1}{25} \int \frac {\exp \left (\frac {1}{25} \left (25 x^2+10 x \log \left (\log \left (x^2\right )\right )+\log ^2\left (\log \left (x^2\right )\right )\right )\right ) \left (60 x+20 x^2+\left (-25 x+150 x^2+50 x^3\right ) \log \left (x^2\right )+\left (12+4 x+\left (30 x+10 x^2\right ) \log \left (x^2\right )\right ) \log \left (\log \left (x^2\right )\right )\right )}{x (3+x)^2 \log \left (x^2\right )} \, dx\\ &=\frac {1}{25} \int \frac {e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2} \left (60 x+20 x^2+\left (-25 x+150 x^2+50 x^3\right ) \log \left (x^2\right )+\left (12+4 x+\left (30 x+10 x^2\right ) \log \left (x^2\right )\right ) \log \left (\log \left (x^2\right )\right )\right )}{x (3+x)^2 \log \left (x^2\right )} \, dx\\ &=\frac {1}{25} \int \left (\frac {5 e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2} \left (12+4 x-5 \log \left (x^2\right )+30 x \log \left (x^2\right )+10 x^2 \log \left (x^2\right )\right )}{(3+x)^2 \log \left (x^2\right )}+\frac {2 e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2} \left (2+5 x \log \left (x^2\right )\right ) \log \left (\log \left (x^2\right )\right )}{x (3+x) \log \left (x^2\right )}\right ) \, dx\\ &=\frac {2}{25} \int \frac {e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2} \left (2+5 x \log \left (x^2\right )\right ) \log \left (\log \left (x^2\right )\right )}{x (3+x) \log \left (x^2\right )} \, dx+\frac {1}{5} \int \frac {e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2} \left (12+4 x-5 \log \left (x^2\right )+30 x \log \left (x^2\right )+10 x^2 \log \left (x^2\right )\right )}{(3+x)^2 \log \left (x^2\right )} \, dx\\ &=\frac {2}{25} \int \left (\frac {e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2} \left (2+5 x \log \left (x^2\right )\right ) \log \left (\log \left (x^2\right )\right )}{3 x \log \left (x^2\right )}-\frac {e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2} \left (2+5 x \log \left (x^2\right )\right ) \log \left (\log \left (x^2\right )\right )}{3 (3+x) \log \left (x^2\right )}\right ) \, dx+\frac {1}{5} \int \left (\frac {5 e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2} \left (-1+6 x+2 x^2\right )}{(3+x)^2}+\frac {4 e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2}}{(3+x) \log \left (x^2\right )}\right ) \, dx\\ &=\frac {2}{75} \int \frac {e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2} \left (2+5 x \log \left (x^2\right )\right ) \log \left (\log \left (x^2\right )\right )}{x \log \left (x^2\right )} \, dx-\frac {2}{75} \int \frac {e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2} \left (2+5 x \log \left (x^2\right )\right ) \log \left (\log \left (x^2\right )\right )}{(3+x) \log \left (x^2\right )} \, dx+\frac {4}{5} \int \frac {e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2}}{(3+x) \log \left (x^2\right )} \, dx+\int \frac {e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2} \left (-1+6 x+2 x^2\right )}{(3+x)^2} \, dx\\ &=\frac {2}{75} \int \left (5 e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2} \log \left (\log \left (x^2\right )\right )+\frac {2 e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2} \log \left (\log \left (x^2\right )\right )}{x \log \left (x^2\right )}\right ) \, dx-\frac {2}{75} \int \left (\frac {5 e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2} x \log \left (\log \left (x^2\right )\right )}{3+x}+\frac {2 e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2} \log \left (\log \left (x^2\right )\right )}{(3+x) \log \left (x^2\right )}\right ) \, dx+\frac {4}{5} \int \frac {e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2}}{(3+x) \log \left (x^2\right )} \, dx+\int \left (2 e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2}-\frac {e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2}}{(3+x)^2}-\frac {6 e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2}}{3+x}\right ) \, dx\\ &=\frac {4}{75} \int \frac {e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2} \log \left (\log \left (x^2\right )\right )}{x \log \left (x^2\right )} \, dx-\frac {4}{75} \int \frac {e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2} \log \left (\log \left (x^2\right )\right )}{(3+x) \log \left (x^2\right )} \, dx+\frac {2}{15} \int e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2} \log \left (\log \left (x^2\right )\right ) \, dx-\frac {2}{15} \int \frac {e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2} x \log \left (\log \left (x^2\right )\right )}{3+x} \, dx+\frac {4}{5} \int \frac {e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2}}{(3+x) \log \left (x^2\right )} \, dx+2 \int e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2} \, dx-6 \int \frac {e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2}}{3+x} \, dx-\int \frac {e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2}}{(3+x)^2} \, dx\\ &=\frac {4}{75} \int \frac {e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2} \log \left (\log \left (x^2\right )\right )}{x \log \left (x^2\right )} \, dx-\frac {4}{75} \int \frac {e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2} \log \left (\log \left (x^2\right )\right )}{(3+x) \log \left (x^2\right )} \, dx+\frac {2}{15} \int e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2} \log \left (\log \left (x^2\right )\right ) \, dx-\frac {2}{15} \int \left (e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2} \log \left (\log \left (x^2\right )\right )-\frac {3 e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2} \log \left (\log \left (x^2\right )\right )}{3+x}\right ) \, dx+\frac {4}{5} \int \frac {e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2}}{(3+x) \log \left (x^2\right )} \, dx+2 \int e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2} \, dx-6 \int \frac {e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2}}{3+x} \, dx-\int \frac {e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2}}{(3+x)^2} \, dx\\ &=\frac {4}{75} \int \frac {e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2} \log \left (\log \left (x^2\right )\right )}{x \log \left (x^2\right )} \, dx-\frac {4}{75} \int \frac {e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2} \log \left (\log \left (x^2\right )\right )}{(3+x) \log \left (x^2\right )} \, dx+\frac {2}{5} \int \frac {e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2} \log \left (\log \left (x^2\right )\right )}{3+x} \, dx+\frac {4}{5} \int \frac {e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2}}{(3+x) \log \left (x^2\right )} \, dx+2 \int e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2} \, dx-6 \int \frac {e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2}}{3+x} \, dx-\int \frac {e^{\frac {1}{25} \left (5 x+\log \left (\log \left (x^2\right )\right )\right )^2}}{(3+x)^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 33, normalized size = 1.57 \begin {gather*} \frac {e^{x^2+\frac {1}{25} \log ^2\left (\log \left (x^2\right )\right )} \log ^{\frac {2 x}{5}}\left (x^2\right )}{3+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((25*x^2 + 10*x*Log[Log[x^2]] + Log[Log[x^2]]^2)/25)*(60*x + 20*x^2 + (-25*x + 150*x^2 + 50*x^3)*
Log[x^2] + (12 + 4*x + (30*x + 10*x^2)*Log[x^2])*Log[Log[x^2]]))/((225*x + 150*x^2 + 25*x^3)*Log[x^2]),x]

[Out]

(E^(x^2 + Log[Log[x^2]]^2/25)*Log[x^2]^((2*x)/5))/(3 + x)

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fricas [A]  time = 0.63, size = 28, normalized size = 1.33 \begin {gather*} \frac {e^{\left (x^{2} + \frac {2}{5} \, x \log \left (\log \left (x^{2}\right )\right ) + \frac {1}{25} \, \log \left (\log \left (x^{2}\right )\right )^{2}\right )}}{x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((10*x^2+30*x)*log(x^2)+4*x+12)*log(log(x^2))+(50*x^3+150*x^2-25*x)*log(x^2)+20*x^2+60*x)*exp(1/25*
log(log(x^2))^2+2/5*x*log(log(x^2))+x^2)/(25*x^3+150*x^2+225*x)/log(x^2),x, algorithm="fricas")

[Out]

e^(x^2 + 2/5*x*log(log(x^2)) + 1/25*log(log(x^2))^2)/(x + 3)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {undef} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((10*x^2+30*x)*log(x^2)+4*x+12)*log(log(x^2))+(50*x^3+150*x^2-25*x)*log(x^2)+20*x^2+60*x)*exp(1/25*
log(log(x^2))^2+2/5*x*log(log(x^2))+x^2)/(25*x^3+150*x^2+225*x)/log(x^2),x, algorithm="giac")

[Out]

undef

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maple [F]  time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (\left (10 x^{2}+30 x \right ) \ln \left (x^{2}\right )+4 x +12\right ) \ln \left (\ln \left (x^{2}\right )\right )+\left (50 x^{3}+150 x^{2}-25 x \right ) \ln \left (x^{2}\right )+20 x^{2}+60 x \right ) {\mathrm e}^{\frac {\ln \left (\ln \left (x^{2}\right )\right )^{2}}{25}+\frac {2 x \ln \left (\ln \left (x^{2}\right )\right )}{5}+x^{2}}}{\left (25 x^{3}+150 x^{2}+225 x \right ) \ln \left (x^{2}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((10*x^2+30*x)*ln(x^2)+4*x+12)*ln(ln(x^2))+(50*x^3+150*x^2-25*x)*ln(x^2)+20*x^2+60*x)*exp(1/25*ln(ln(x^2)
)^2+2/5*x*ln(ln(x^2))+x^2)/(25*x^3+150*x^2+225*x)/ln(x^2),x)

[Out]

int((((10*x^2+30*x)*ln(x^2)+4*x+12)*ln(ln(x^2))+(50*x^3+150*x^2-25*x)*ln(x^2)+20*x^2+60*x)*exp(1/25*ln(ln(x^2)
)^2+2/5*x*ln(ln(x^2))+x^2)/(25*x^3+150*x^2+225*x)/ln(x^2),x)

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maxima [B]  time = 0.57, size = 42, normalized size = 2.00 \begin {gather*} \frac {e^{\left (x^{2} + \frac {2}{5} \, x \log \relax (2) + \frac {1}{25} \, \log \relax (2)^{2} + \frac {2}{5} \, x \log \left (\log \relax (x)\right ) + \frac {2}{25} \, \log \relax (2) \log \left (\log \relax (x)\right ) + \frac {1}{25} \, \log \left (\log \relax (x)\right )^{2}\right )}}{x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((10*x^2+30*x)*log(x^2)+4*x+12)*log(log(x^2))+(50*x^3+150*x^2-25*x)*log(x^2)+20*x^2+60*x)*exp(1/25*
log(log(x^2))^2+2/5*x*log(log(x^2))+x^2)/(25*x^3+150*x^2+225*x)/log(x^2),x, algorithm="maxima")

[Out]

e^(x^2 + 2/5*x*log(2) + 1/25*log(2)^2 + 2/5*x*log(log(x)) + 2/25*log(2)*log(log(x)) + 1/25*log(log(x))^2)/(x +
3)

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mupad [B]  time = 8.68, size = 28, normalized size = 1.33 \begin {gather*} \frac {{\ln \left (x^2\right )}^{\frac {2\,x}{5}}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{\frac {{\ln \left (\ln \left (x^2\right )\right )}^2}{25}}}{x+3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(log(log(x^2))^2/25 + (2*x*log(log(x^2)))/5 + x^2)*(60*x + log(x^2)*(150*x^2 - 25*x + 50*x^3) + log(lo
g(x^2))*(4*x + log(x^2)*(30*x + 10*x^2) + 12) + 20*x^2))/(log(x^2)*(225*x + 150*x^2 + 25*x^3)),x)

[Out]

(log(x^2)^((2*x)/5)*exp(x^2)*exp(log(log(x^2))^2/25))/(x + 3)

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sympy [A]  time = 0.51, size = 29, normalized size = 1.38 \begin {gather*} \frac {e^{x^{2} + \frac {2 x \log {\left (\log {\left (x^{2} \right )} \right )}}{5} + \frac {\log {\left (\log {\left (x^{2} \right )} \right )}^{2}}{25}}}{x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((10*x**2+30*x)*ln(x**2)+4*x+12)*ln(ln(x**2))+(50*x**3+150*x**2-25*x)*ln(x**2)+20*x**2+60*x)*exp(1/
25*ln(ln(x**2))**2+2/5*x*ln(ln(x**2))+x**2)/(25*x**3+150*x**2+225*x)/ln(x**2),x)

[Out]

exp(x**2 + 2*x*log(log(x**2))/5 + log(log(x**2))**2/25)/(x + 3)

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