3.36.26 \(\int \frac {7 e^3-7 e^{x^2}+e^{x^2} (7-14 x^2) \log (x)}{5 e^6+10 e^3 x+5 x^2+e^{x^2} (10 e^3+10 x) \log (x)+5 e^{2 x^2} \log ^2(x)} \, dx\)

Optimal. Leaf size=20 \[ \frac {7 x}{5 \left (e^3+x+e^{x^2} \log (x)\right )} \]

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Rubi [F]  time = 1.49, 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 {7 e^3-7 e^{x^2}+e^{x^2} \left (7-14 x^2\right ) \log (x)}{5 e^6+10 e^3 x+5 x^2+e^{x^2} \left (10 e^3+10 x\right ) \log (x)+5 e^{2 x^2} \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(7*E^3 - 7*E^x^2 + E^x^2*(7 - 14*x^2)*Log[x])/(5*E^6 + 10*E^3*x + 5*x^2 + E^x^2*(10*E^3 + 10*x)*Log[x] + 5
*E^(2*x^2)*Log[x]^2),x]

[Out]

(-7*Defer[Int][x/(E^3 + x + E^x^2*Log[x])^2, x])/5 + (14*E^3*Defer[Int][x^2/(E^3 + x + E^x^2*Log[x])^2, x])/5
+ (14*Defer[Int][x^3/(E^3 + x + E^x^2*Log[x])^2, x])/5 + (7*E^3*Defer[Int][1/(Log[x]*(E^3 + x + E^x^2*Log[x])^
2), x])/5 + (7*Defer[Int][x/(Log[x]*(E^3 + x + E^x^2*Log[x])^2), x])/5 + (7*Defer[Int][(E^3 + x + E^x^2*Log[x]
)^(-1), x])/5 - (14*Defer[Int][x^2/(E^3 + x + E^x^2*Log[x]), x])/5 - (7*Defer[Int][1/(Log[x]*(E^3 + x + E^x^2*
Log[x])), x])/5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {7 \left (e^3-e^{x^2}-e^{x^2} \left (-1+2 x^2\right ) \log (x)\right )}{5 \left (e^3+x+e^{x^2} \log (x)\right )^2} \, dx\\ &=\frac {7}{5} \int \frac {e^3-e^{x^2}-e^{x^2} \left (-1+2 x^2\right ) \log (x)}{\left (e^3+x+e^{x^2} \log (x)\right )^2} \, dx\\ &=\frac {7}{5} \int \left (-\frac {1-\log (x)+2 x^2 \log (x)}{\log (x) \left (e^3+x+e^{x^2} \log (x)\right )}+\frac {e^3+x-x \log (x)+2 e^3 x^2 \log (x)+2 x^3 \log (x)}{\log (x) \left (e^3+x+e^{x^2} \log (x)\right )^2}\right ) \, dx\\ &=-\left (\frac {7}{5} \int \frac {1-\log (x)+2 x^2 \log (x)}{\log (x) \left (e^3+x+e^{x^2} \log (x)\right )} \, dx\right )+\frac {7}{5} \int \frac {e^3+x-x \log (x)+2 e^3 x^2 \log (x)+2 x^3 \log (x)}{\log (x) \left (e^3+x+e^{x^2} \log (x)\right )^2} \, dx\\ &=\frac {7}{5} \int \left (-\frac {x}{\left (e^3+x+e^{x^2} \log (x)\right )^2}+\frac {2 e^3 x^2}{\left (e^3+x+e^{x^2} \log (x)\right )^2}+\frac {2 x^3}{\left (e^3+x+e^{x^2} \log (x)\right )^2}+\frac {e^3}{\log (x) \left (e^3+x+e^{x^2} \log (x)\right )^2}+\frac {x}{\log (x) \left (e^3+x+e^{x^2} \log (x)\right )^2}\right ) \, dx-\frac {7}{5} \int \left (-\frac {1}{e^3+x+e^{x^2} \log (x)}+\frac {2 x^2}{e^3+x+e^{x^2} \log (x)}+\frac {1}{\log (x) \left (e^3+x+e^{x^2} \log (x)\right )}\right ) \, dx\\ &=-\left (\frac {7}{5} \int \frac {x}{\left (e^3+x+e^{x^2} \log (x)\right )^2} \, dx\right )+\frac {7}{5} \int \frac {x}{\log (x) \left (e^3+x+e^{x^2} \log (x)\right )^2} \, dx+\frac {7}{5} \int \frac {1}{e^3+x+e^{x^2} \log (x)} \, dx-\frac {7}{5} \int \frac {1}{\log (x) \left (e^3+x+e^{x^2} \log (x)\right )} \, dx+\frac {14}{5} \int \frac {x^3}{\left (e^3+x+e^{x^2} \log (x)\right )^2} \, dx-\frac {14}{5} \int \frac {x^2}{e^3+x+e^{x^2} \log (x)} \, dx+\frac {1}{5} \left (7 e^3\right ) \int \frac {1}{\log (x) \left (e^3+x+e^{x^2} \log (x)\right )^2} \, dx+\frac {1}{5} \left (14 e^3\right ) \int \frac {x^2}{\left (e^3+x+e^{x^2} \log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.34, size = 20, normalized size = 1.00 \begin {gather*} \frac {7 x}{5 \left (e^3+x+e^{x^2} \log (x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(7*E^3 - 7*E^x^2 + E^x^2*(7 - 14*x^2)*Log[x])/(5*E^6 + 10*E^3*x + 5*x^2 + E^x^2*(10*E^3 + 10*x)*Log[
x] + 5*E^(2*x^2)*Log[x]^2),x]

[Out]

(7*x)/(5*(E^3 + x + E^x^2*Log[x]))

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fricas [A]  time = 0.66, size = 16, normalized size = 0.80 \begin {gather*} \frac {7 \, x}{5 \, {\left (e^{\left (x^{2}\right )} \log \relax (x) + x + e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-14*x^2+7)*exp(x^2)*log(x)-7*exp(x^2)+7*exp(3))/(5*exp(x^2)^2*log(x)^2+(10*exp(3)+10*x)*exp(x^2)*l
og(x)+5*exp(3)^2+10*x*exp(3)+5*x^2),x, algorithm="fricas")

[Out]

7/5*x/(e^(x^2)*log(x) + x + e^3)

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giac [A]  time = 0.18, size = 16, normalized size = 0.80 \begin {gather*} \frac {7 \, x}{5 \, {\left (e^{\left (x^{2}\right )} \log \relax (x) + x + e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-14*x^2+7)*exp(x^2)*log(x)-7*exp(x^2)+7*exp(3))/(5*exp(x^2)^2*log(x)^2+(10*exp(3)+10*x)*exp(x^2)*l
og(x)+5*exp(3)^2+10*x*exp(3)+5*x^2),x, algorithm="giac")

[Out]

7/5*x/(e^(x^2)*log(x) + x + e^3)

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maple [A]  time = 0.04, size = 17, normalized size = 0.85

method	result	size
risch	\(\frac {7 x}{5 \left ({\mathrm e}^{3}+{\mathrm e}^{x^{2}} \ln \relax (x )+x \right )}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-14*x^2+7)*exp(x^2)*ln(x)-7*exp(x^2)+7*exp(3))/(5*exp(x^2)^2*ln(x)^2+(10*exp(3)+10*x)*exp(x^2)*ln(x)+5*e
xp(3)^2+10*x*exp(3)+5*x^2),x,method=_RETURNVERBOSE)

[Out]

7/5*x/(exp(3)+exp(x^2)*ln(x)+x)

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maxima [A]  time = 0.74, size = 16, normalized size = 0.80 \begin {gather*} \frac {7 \, x}{5 \, {\left (e^{\left (x^{2}\right )} \log \relax (x) + x + e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-14*x^2+7)*exp(x^2)*log(x)-7*exp(x^2)+7*exp(3))/(5*exp(x^2)^2*log(x)^2+(10*exp(3)+10*x)*exp(x^2)*l
og(x)+5*exp(3)^2+10*x*exp(3)+5*x^2),x, algorithm="maxima")

[Out]

7/5*x/(e^(x^2)*log(x) + x + e^3)

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mupad [B]  time = 2.57, size = 110, normalized size = 5.50 \begin {gather*} \frac {7\,\left (2\,x^5\,{\ln \relax (x)}^2+2\,{\mathrm {e}}^3\,x^4\,{\ln \relax (x)}^2-x^3\,{\ln \relax (x)}^2+x^3\,\ln \relax (x)+{\mathrm {e}}^3\,x^2\,\ln \relax (x)\right )}{5\,\left ({\mathrm {e}}^{x^2}+\frac {x+{\mathrm {e}}^3}{\ln \relax (x)}\right )\,\left (2\,x^4\,{\ln \relax (x)}^3+2\,{\mathrm {e}}^3\,x^3\,{\ln \relax (x)}^3-x^2\,{\ln \relax (x)}^3+x^2\,{\ln \relax (x)}^2+{\mathrm {e}}^3\,x\,{\ln \relax (x)}^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(7*exp(x^2) - 7*exp(3) + exp(x^2)*log(x)*(14*x^2 - 7))/(5*exp(6) + 10*x*exp(3) + 5*exp(2*x^2)*log(x)^2 +
5*x^2 + exp(x^2)*log(x)*(10*x + 10*exp(3))),x)

[Out]

(7*(x^3*log(x) - x^3*log(x)^2 + 2*x^5*log(x)^2 + x^2*exp(3)*log(x) + 2*x^4*exp(3)*log(x)^2))/(5*(exp(x^2) + (x
+ exp(3))/log(x))*(x^2*log(x)^2 - x^2*log(x)^3 + 2*x^4*log(x)^3 + x*exp(3)*log(x)^2 + 2*x^3*exp(3)*log(x)^3))

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sympy [A]  time = 0.30, size = 20, normalized size = 1.00 \begin {gather*} \frac {7 x}{5 x + 5 e^{x^{2}} \log {\relax (x )} + 5 e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-14*x**2+7)*exp(x**2)*ln(x)-7*exp(x**2)+7*exp(3))/(5*exp(x**2)**2*ln(x)**2+(10*exp(3)+10*x)*exp(x*
*2)*ln(x)+5*exp(3)**2+10*x*exp(3)+5*x**2),x)

[Out]

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

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