3.274 \(\int \left (\frac{e^{x^2}}{x}+2 e^{x^2} x \log (x)+\frac{-2+\log (x)}{\left (x+\log ^2(x)\right )^2}+\frac{1+\frac{1}{x}+\frac{2 \log (x)}{x}}{x+\log ^2(x)}\right ) \, dx\)

Optimal. Leaf size=28 \[ e^{x^2} \log (x)-\frac{\log (x)}{x+\log ^2(x)}+\log \left (x+\log ^2(x)\right ) \]

[Out]

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

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Rubi [F]  time = 0.44171, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0. \[ \text{Int}\left (\frac{e^{x^2}}{x}+2 e^{x^2} x \log (x)+\frac{-2+\log (x)}{\left (x+\log ^2(x)\right )^2}+\frac{1+\frac{1}{x}+\frac{2 \log (x)}{x}}{x+\log ^2(x)},x\right ) \]

Verification is Not applicable to the result.

[In]  Int[E^x^2/x + 2*E^x^2*x*Log[x] + (-2 + Log[x])/(x + Log[x]^2)^2 + (1 + x^(-1) + (2*Log[x])/x)/(x + Log[x]^2),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ e^{x^{2}} \log{\left (x \right )} + \int \frac{\log{\left (x \right )} - 2}{\left (x + \log{\left (x \right )}^{2}\right )^{2}}\, dx + \int \frac{1 + \frac{2 \log{\left (x \right )}}{x} + \frac{1}{x}}{x + \log{\left (x \right )}^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(x**2)/x+2*exp(x**2)*x*ln(x)+(-2+ln(x))/(x+ln(x)**2)**2+(1+1/x+2*ln(x)/x)/(x+ln(x)**2),x)

[Out]

exp(x**2)*log(x) + Integral((log(x) - 2)/(x + log(x)**2)**2, x) + Integral((1 +
2*log(x)/x + 1/x)/(x + log(x)**2), x)

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Mathematica [A]  time = 9.8761, size = 28, normalized size = 1. \[ e^{x^2} \log (x)-\frac{\log (x)}{x+\log ^2(x)}+\log \left (x+\log ^2(x)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[E^x^2/x + 2*E^x^2*x*Log[x] + (-2 + Log[x])/(x + Log[x]^2)^2 + (1 + x^(-1) + (2*Log[x])/x)/(x + Log[x]^2),x]

[Out]

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

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Maple [A]  time = 0.03, size = 28, normalized size = 1. \[{{\rm e}^{{x}^{2}}}\ln \left ( x \right ) -{\frac{\ln \left ( x \right ) }{x+ \left ( \ln \left ( x \right ) \right ) ^{2}}}+\ln \left ( x+ \left ( \ln \left ( x \right ) \right ) ^{2} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(x^2)/x+2*exp(x^2)*x*ln(x)+(-2+ln(x))/(x+ln(x)^2)^2+(1+1/x+2*ln(x)/x)/(x+ln(x)^2),x)

[Out]

exp(x^2)*ln(x)-ln(x)/(x+ln(x)^2)+ln(x+ln(x)^2)

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Maxima [A]  time = 1.54193, size = 36, normalized size = 1.29 \[ e^{\left (x^{2}\right )} \log \left (x\right ) - \frac{\log \left (x\right )}{\log \left (x\right )^{2} + x} + \log \left (\log \left (x\right )^{2} + x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(2*x*e^(x^2)*log(x) + (2*log(x)/x + 1/x + 1)/(log(x)^2 + x) + e^(x^2)/x + (log(x) - 2)/(log(x)^2 + x)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.225221, size = 59, normalized size = 2.11 \[ \frac{e^{\left (x^{2}\right )} \log \left (x\right )^{3} +{\left (\log \left (x\right )^{2} + x\right )} \log \left (\log \left (x\right )^{2} + x\right ) +{\left (x e^{\left (x^{2}\right )} - 1\right )} \log \left (x\right )}{\log \left (x\right )^{2} + x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(2*x*e^(x^2)*log(x) + (2*log(x)/x + 1/x + 1)/(log(x)^2 + x) + e^(x^2)/x + (log(x) - 2)/(log(x)^2 + x)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.788409, size = 26, normalized size = 0.93 \[ e^{x^{2}} \log{\left (x \right )} + \log{\left (x + \log{\left (x \right )}^{2} \right )} - \frac{\log{\left (x \right )}}{x + \log{\left (x \right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(x**2)/x+2*exp(x**2)*x*ln(x)+(-2+ln(x))/(x+ln(x)**2)**2+(1+1/x+2*ln(x)/x)/(x+ln(x)**2),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int 2 \, x e^{\left (x^{2}\right )} \log \left (x\right ) + \frac{\frac{2 \, \log \left (x\right )}{x} + \frac{1}{x} + 1}{\log \left (x\right )^{2} + x} + \frac{e^{\left (x^{2}\right )}}{x} + \frac{\log \left (x\right ) - 2}{{\left (\log \left (x\right )^{2} + x\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(2*x*e^(x^2)*log(x) + (2*log(x)/x + 1/x + 1)/(log(x)^2 + x) + e^(x^2)/x + (log(x) - 2)/(log(x)^2 + x)^2,x, algorithm="giac")

[Out]

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