3.89.17 \(\int \frac {e^{\frac {\log (x) \log (x^2)}{5 x+x^3+e^{x^2} \log (x^2)}} ((10 x+2 x^3) \log (x)+(5 x+x^3+(-5 x-3 x^3) \log (x)) \log (x^2)+(e^{x^2}-2 e^{x^2} x^2 \log (x)) \log ^2(x^2))}{25 x^3+10 x^5+x^7+e^{x^2} (10 x^2+2 x^4) \log (x^2)+e^{2 x^2} x \log ^2(x^2)} \, dx\)

Optimal. Leaf size=23 \[ x^{\frac {1}{e^{x^2}+\frac {x \left (5+x^2\right )}{\log \left (x^2\right )}}} \]

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Rubi [F]  time = 40.02, 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^{\frac {\log (x) \log \left (x^2\right )}{5 x+x^3+e^{x^2} \log \left (x^2\right )}} \left (\left (10 x+2 x^3\right ) \log (x)+\left (5 x+x^3+\left (-5 x-3 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (e^{x^2}-2 e^{x^2} x^2 \log (x)\right ) \log ^2\left (x^2\right )\right )}{25 x^3+10 x^5+x^7+e^{x^2} \left (10 x^2+2 x^4\right ) \log \left (x^2\right )+e^{2 x^2} x \log ^2\left (x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((Log[x]*Log[x^2])/(5*x + x^3 + E^x^2*Log[x^2]))*((10*x + 2*x^3)*Log[x] + (5*x + x^3 + (-5*x - 3*x^3)*L
og[x])*Log[x^2] + (E^x^2 - 2*E^x^2*x^2*Log[x])*Log[x^2]^2))/(25*x^3 + 10*x^5 + x^7 + E^x^2*(10*x^2 + 2*x^4)*Lo
g[x^2] + E^(2*x^2)*x*Log[x^2]^2),x]

[Out]

10*Defer[Int][(E^((Log[x]*Log[x^2])/(5*x + x^3 + E^x^2*Log[x^2]))*Log[x])/(5*x + x^3 + E^x^2*Log[x^2])^2, x] +
2*Defer[Int][(E^((Log[x]*Log[x^2])/(5*x + x^3 + E^x^2*Log[x^2]))*x^2*Log[x])/(5*x + x^3 + E^x^2*Log[x^2])^2,
x] - 5*Defer[Int][(E^((Log[x]*Log[x^2])/(5*x + x^3 + E^x^2*Log[x^2]))*Log[x]*Log[x^2])/(5*x + x^3 + E^x^2*Log[
x^2])^2, x] + 7*Defer[Int][(E^((Log[x]*Log[x^2])/(5*x + x^3 + E^x^2*Log[x^2]))*x^2*Log[x]*Log[x^2])/(5*x + x^3
+ E^x^2*Log[x^2])^2, x] + 2*Defer[Int][(E^((Log[x]*Log[x^2])/(5*x + x^3 + E^x^2*Log[x^2]))*x^4*Log[x]*Log[x^2
])/(5*x + x^3 + E^x^2*Log[x^2])^2, x] + Defer[Int][(E^((Log[x]*Log[x^2])/(5*x + x^3 + E^x^2*Log[x^2]))*Log[x^2
])/(x*(5*x + x^3 + E^x^2*Log[x^2])), x] - 2*Defer[Int][(E^((Log[x]*Log[x^2])/(5*x + x^3 + E^x^2*Log[x^2]))*x*L
og[x]*Log[x^2])/(5*x + x^3 + E^x^2*Log[x^2]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {\log (x) \log \left (x^2\right )}{5 x+x^3+e^{x^2} \log \left (x^2\right )}} \left (\left (10 x+2 x^3\right ) \log (x)+\left (5 x+x^3+\left (-5 x-3 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (e^{x^2}-2 e^{x^2} x^2 \log (x)\right ) \log ^2\left (x^2\right )\right )}{x \left (5 x+x^3+e^{x^2} \log \left (x^2\right )\right )^2} \, dx\\ &=\int \left (-\frac {e^{\frac {\log (x) \log \left (x^2\right )}{5 x+x^3+e^{x^2} \log \left (x^2\right )}} \left (-1+2 x^2 \log (x)\right ) \log \left (x^2\right )}{x \left (5 x+x^3+e^{x^2} \log \left (x^2\right )\right )}+\frac {e^{\frac {\log (x) \log \left (x^2\right )}{5 x+x^3+e^{x^2} \log \left (x^2\right )}} \log (x) \left (10+2 x^2-5 \log \left (x^2\right )+7 x^2 \log \left (x^2\right )+2 x^4 \log \left (x^2\right )\right )}{\left (5 x+x^3+e^{x^2} \log \left (x^2\right )\right )^2}\right ) \, dx\\ &=-\int \frac {e^{\frac {\log (x) \log \left (x^2\right )}{5 x+x^3+e^{x^2} \log \left (x^2\right )}} \left (-1+2 x^2 \log (x)\right ) \log \left (x^2\right )}{x \left (5 x+x^3+e^{x^2} \log \left (x^2\right )\right )} \, dx+\int \frac {e^{\frac {\log (x) \log \left (x^2\right )}{5 x+x^3+e^{x^2} \log \left (x^2\right )}} \log (x) \left (10+2 x^2-5 \log \left (x^2\right )+7 x^2 \log \left (x^2\right )+2 x^4 \log \left (x^2\right )\right )}{\left (5 x+x^3+e^{x^2} \log \left (x^2\right )\right )^2} \, dx\\ &=\int \left (\frac {10 e^{\frac {\log (x) \log \left (x^2\right )}{5 x+x^3+e^{x^2} \log \left (x^2\right )}} \log (x)}{\left (5 x+x^3+e^{x^2} \log \left (x^2\right )\right )^2}+\frac {2 e^{\frac {\log (x) \log \left (x^2\right )}{5 x+x^3+e^{x^2} \log \left (x^2\right )}} x^2 \log (x)}{\left (5 x+x^3+e^{x^2} \log \left (x^2\right )\right )^2}-\frac {5 e^{\frac {\log (x) \log \left (x^2\right )}{5 x+x^3+e^{x^2} \log \left (x^2\right )}} \log (x) \log \left (x^2\right )}{\left (5 x+x^3+e^{x^2} \log \left (x^2\right )\right )^2}+\frac {7 e^{\frac {\log (x) \log \left (x^2\right )}{5 x+x^3+e^{x^2} \log \left (x^2\right )}} x^2 \log (x) \log \left (x^2\right )}{\left (5 x+x^3+e^{x^2} \log \left (x^2\right )\right )^2}+\frac {2 e^{\frac {\log (x) \log \left (x^2\right )}{5 x+x^3+e^{x^2} \log \left (x^2\right )}} x^4 \log (x) \log \left (x^2\right )}{\left (5 x+x^3+e^{x^2} \log \left (x^2\right )\right )^2}\right ) \, dx-\int \left (-\frac {e^{\frac {\log (x) \log \left (x^2\right )}{5 x+x^3+e^{x^2} \log \left (x^2\right )}} \log \left (x^2\right )}{x \left (5 x+x^3+e^{x^2} \log \left (x^2\right )\right )}+\frac {2 e^{\frac {\log (x) \log \left (x^2\right )}{5 x+x^3+e^{x^2} \log \left (x^2\right )}} x \log (x) \log \left (x^2\right )}{5 x+x^3+e^{x^2} \log \left (x^2\right )}\right ) \, dx\\ &=2 \int \frac {e^{\frac {\log (x) \log \left (x^2\right )}{5 x+x^3+e^{x^2} \log \left (x^2\right )}} x^2 \log (x)}{\left (5 x+x^3+e^{x^2} \log \left (x^2\right )\right )^2} \, dx+2 \int \frac {e^{\frac {\log (x) \log \left (x^2\right )}{5 x+x^3+e^{x^2} \log \left (x^2\right )}} x^4 \log (x) \log \left (x^2\right )}{\left (5 x+x^3+e^{x^2} \log \left (x^2\right )\right )^2} \, dx-2 \int \frac {e^{\frac {\log (x) \log \left (x^2\right )}{5 x+x^3+e^{x^2} \log \left (x^2\right )}} x \log (x) \log \left (x^2\right )}{5 x+x^3+e^{x^2} \log \left (x^2\right )} \, dx-5 \int \frac {e^{\frac {\log (x) \log \left (x^2\right )}{5 x+x^3+e^{x^2} \log \left (x^2\right )}} \log (x) \log \left (x^2\right )}{\left (5 x+x^3+e^{x^2} \log \left (x^2\right )\right )^2} \, dx+7 \int \frac {e^{\frac {\log (x) \log \left (x^2\right )}{5 x+x^3+e^{x^2} \log \left (x^2\right )}} x^2 \log (x) \log \left (x^2\right )}{\left (5 x+x^3+e^{x^2} \log \left (x^2\right )\right )^2} \, dx+10 \int \frac {e^{\frac {\log (x) \log \left (x^2\right )}{5 x+x^3+e^{x^2} \log \left (x^2\right )}} \log (x)}{\left (5 x+x^3+e^{x^2} \log \left (x^2\right )\right )^2} \, dx+\int \frac {e^{\frac {\log (x) \log \left (x^2\right )}{5 x+x^3+e^{x^2} \log \left (x^2\right )}} \log \left (x^2\right )}{x \left (5 x+x^3+e^{x^2} \log \left (x^2\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.47, size = 123, normalized size = 5.35 \begin {gather*} e^{-\frac {1}{2} e^{-2 x^2} x \left (5+x^2\right )} x^{\frac {e^{-2 x^2} \left (x^2 \left (5+x^2\right )^2+2 e^{x^2} \left (5 x+x^3+e^{x^2} \log (x)\right ) \log \left (x^2\right )+e^{2 x^2} \log ^2\left (x^2\right )\right )}{2 \log (x) \left (5 x+x^3+e^{x^2} \log \left (x^2\right )\right )}} \left (x^2\right )^{-\frac {e^{-x^2}}{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((Log[x]*Log[x^2])/(5*x + x^3 + E^x^2*Log[x^2]))*((10*x + 2*x^3)*Log[x] + (5*x + x^3 + (-5*x - 3*
x^3)*Log[x])*Log[x^2] + (E^x^2 - 2*E^x^2*x^2*Log[x])*Log[x^2]^2))/(25*x^3 + 10*x^5 + x^7 + E^x^2*(10*x^2 + 2*x
^4)*Log[x^2] + E^(2*x^2)*x*Log[x^2]^2),x]

[Out]

x^((x^2*(5 + x^2)^2 + 2*E^x^2*(5*x + x^3 + E^x^2*Log[x])*Log[x^2] + E^(2*x^2)*Log[x^2]^2)/(2*E^(2*x^2)*Log[x]*
(5*x + x^3 + E^x^2*Log[x^2])))/(E^((x*(5 + x^2))/(2*E^(2*x^2)))*(x^2)^(1/(2*E^x^2)))

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fricas [A]  time = 0.47, size = 24, normalized size = 1.04 \begin {gather*} e^{\left (\frac {2 \, \log \relax (x)^{2}}{x^{3} + 2 \, e^{\left (x^{2}\right )} \log \relax (x) + 5 \, x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2*exp(x^2)*log(x)+exp(x^2))*log(x^2)^2+((-3*x^3-5*x)*log(x)+x^3+5*x)*log(x^2)+(2*x^3+10*x)*lo
g(x))*exp(log(x)*log(x^2)/(exp(x^2)*log(x^2)+x^3+5*x))/(x*exp(x^2)^2*log(x^2)^2+(2*x^4+10*x^2)*exp(x^2)*log(x^
2)+x^7+10*x^5+25*x^3),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left ({\left (2 \, x^{2} e^{\left (x^{2}\right )} \log \relax (x) - e^{\left (x^{2}\right )}\right )} \log \left (x^{2}\right )^{2} - {\left (x^{3} - {\left (3 \, x^{3} + 5 \, x\right )} \log \relax (x) + 5 \, x\right )} \log \left (x^{2}\right ) - 2 \, {\left (x^{3} + 5 \, x\right )} \log \relax (x)\right )} e^{\left (\frac {\log \left (x^{2}\right ) \log \relax (x)}{x^{3} + e^{\left (x^{2}\right )} \log \left (x^{2}\right ) + 5 \, x}\right )}}{x^{7} + 10 \, x^{5} + x e^{\left (2 \, x^{2}\right )} \log \left (x^{2}\right )^{2} + 25 \, x^{3} + 2 \, {\left (x^{4} + 5 \, x^{2}\right )} e^{\left (x^{2}\right )} \log \left (x^{2}\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2*exp(x^2)*log(x)+exp(x^2))*log(x^2)^2+((-3*x^3-5*x)*log(x)+x^3+5*x)*log(x^2)+(2*x^3+10*x)*lo
g(x))*exp(log(x)*log(x^2)/(exp(x^2)*log(x^2)+x^3+5*x))/(x*exp(x^2)^2*log(x^2)^2+(2*x^4+10*x^2)*exp(x^2)*log(x^
2)+x^7+10*x^5+25*x^3),x, algorithm="giac")

[Out]

integrate(-((2*x^2*e^(x^2)*log(x) - e^(x^2))*log(x^2)^2 - (x^3 - (3*x^3 + 5*x)*log(x) + 5*x)*log(x^2) - 2*(x^3
+ 5*x)*log(x))*e^(log(x^2)*log(x)/(x^3 + e^(x^2)*log(x^2) + 5*x))/(x^7 + 10*x^5 + x*e^(2*x^2)*log(x^2)^2 + 25
*x^3 + 2*(x^4 + 5*x^2)*e^(x^2)*log(x^2)), x)

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maple [F]  time = 0.20, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (-2 x^{2} {\mathrm e}^{x^{2}} \ln \relax (x )+{\mathrm e}^{x^{2}}\right ) \ln \left (x^{2}\right )^{2}+\left (\left (-3 x^{3}-5 x \right ) \ln \relax (x )+x^{3}+5 x \right ) \ln \left (x^{2}\right )+\left (2 x^{3}+10 x \right ) \ln \relax (x )\right ) {\mathrm e}^{\frac {\ln \relax (x ) \ln \left (x^{2}\right )}{{\mathrm e}^{x^{2}} \ln \left (x^{2}\right )+x^{3}+5 x}}}{x \,{\mathrm e}^{2 x^{2}} \ln \left (x^{2}\right )^{2}+\left (2 x^{4}+10 x^{2}\right ) {\mathrm e}^{x^{2}} \ln \left (x^{2}\right )+x^{7}+10 x^{5}+25 x^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^2*exp(x^2)*ln(x)+exp(x^2))*ln(x^2)^2+((-3*x^3-5*x)*ln(x)+x^3+5*x)*ln(x^2)+(2*x^3+10*x)*ln(x))*exp(l
n(x)*ln(x^2)/(exp(x^2)*ln(x^2)+x^3+5*x))/(x*exp(x^2)^2*ln(x^2)^2+(2*x^4+10*x^2)*exp(x^2)*ln(x^2)+x^7+10*x^5+25
*x^3),x)

[Out]

int(((-2*x^2*exp(x^2)*ln(x)+exp(x^2))*ln(x^2)^2+((-3*x^3-5*x)*ln(x)+x^3+5*x)*ln(x^2)+(2*x^3+10*x)*ln(x))*exp(l
n(x)*ln(x^2)/(exp(x^2)*ln(x^2)+x^3+5*x))/(x*exp(x^2)^2*ln(x^2)^2+(2*x^4+10*x^2)*exp(x^2)*ln(x^2)+x^7+10*x^5+25
*x^3),x)

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maxima [A]  time = 0.58, size = 24, normalized size = 1.04 \begin {gather*} e^{\left (\frac {2 \, \log \relax (x)^{2}}{x^{3} + 2 \, e^{\left (x^{2}\right )} \log \relax (x) + 5 \, x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2*exp(x^2)*log(x)+exp(x^2))*log(x^2)^2+((-3*x^3-5*x)*log(x)+x^3+5*x)*log(x^2)+(2*x^3+10*x)*lo
g(x))*exp(log(x)*log(x^2)/(exp(x^2)*log(x^2)+x^3+5*x))/(x*exp(x^2)^2*log(x^2)^2+(2*x^4+10*x^2)*exp(x^2)*log(x^
2)+x^7+10*x^5+25*x^3),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 6.24, size = 26, normalized size = 1.13 \begin {gather*} {\mathrm {e}}^{\frac {\ln \left (x^2\right )\,\ln \relax (x)}{5\,x+x^3+\ln \left (x^2\right )\,{\mathrm {e}}^{x^2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((log(x^2)*log(x))/(5*x + x^3 + log(x^2)*exp(x^2)))*(log(x^2)*(5*x - log(x)*(5*x + 3*x^3) + x^3) + log
(x)*(10*x + 2*x^3) + log(x^2)^2*(exp(x^2) - 2*x^2*exp(x^2)*log(x))))/(25*x^3 + 10*x^5 + x^7 + x*log(x^2)^2*exp
(2*x^2) + log(x^2)*exp(x^2)*(10*x^2 + 2*x^4)),x)

[Out]

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

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sympy [A]  time = 1.76, size = 24, normalized size = 1.04 \begin {gather*} e^{\frac {2 \log {\relax (x )}^{2}}{x^{3} + 5 x + 2 e^{x^{2}} \log {\relax (x )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**2*exp(x**2)*ln(x)+exp(x**2))*ln(x**2)**2+((-3*x**3-5*x)*ln(x)+x**3+5*x)*ln(x**2)+(2*x**3+10*
x)*ln(x))*exp(ln(x)*ln(x**2)/(exp(x**2)*ln(x**2)+x**3+5*x))/(x*exp(x**2)**2*ln(x**2)**2+(2*x**4+10*x**2)*exp(x
**2)*ln(x**2)+x**7+10*x**5+25*x**3),x)

[Out]

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

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