3.66.96 \(\int \frac {x+e^{4 x^2 \log (x)+4 e^8 \log (x) \log (3 x)} (4 x^2+(4 e^8+8 x^2) \log (x)+4 e^8 \log (3 x))}{x} \, dx\)

Optimal. Leaf size=21 \[ -4+x+x^{4 x \left (x+\frac {e^8 \log (3 x)}{x}\right )} \]

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Rubi [F]  time = 0.28, 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 {x+e^{4 x^2 \log (x)+4 e^8 \log (x) \log (3 x)} \left (4 x^2+\left (4 e^8+8 x^2\right ) \log (x)+4 e^8 \log (3 x)\right )}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x + E^(4*x^2*Log[x] + 4*E^8*Log[x]*Log[3*x])*(4*x^2 + (4*E^8 + 8*x^2)*Log[x] + 4*E^8*Log[3*x]))/x,x]

[Out]

x + 4*E^8*Log[3]*Defer[Int][x^(-1 + 4*x^2 + 4*E^8*Log[3*x]), x] + 4*Defer[Int][x^(1 + 4*x^2 + 4*E^8*Log[3*x]),
x] + 8*E^8*Defer[Int][x^(-1 + 4*x^2 + 4*E^8*Log[3*x])*Log[x], x] + 8*Defer[Int][x^(1 + 4*x^2 + 4*E^8*Log[3*x]
)*Log[x], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1+4 x^{-1+4 x^2+4 e^8 \log (3 x)} \left (x^2+e^8 \log (3)+2 e^8 \log (x)+2 x^2 \log (x)\right )\right ) \, dx\\ &=x+4 \int x^{-1+4 x^2+4 e^8 \log (3 x)} \left (x^2+e^8 \log (3)+2 e^8 \log (x)+2 x^2 \log (x)\right ) \, dx\\ &=x+4 \int \left (x^{1+4 x^2+4 e^8 \log (3 x)}+e^8 x^{-1+4 x^2+4 e^8 \log (3 x)} \log (3)+2 e^8 x^{-1+4 x^2+4 e^8 \log (3 x)} \log (x)+2 x^{1+4 x^2+4 e^8 \log (3 x)} \log (x)\right ) \, dx\\ &=x+4 \int x^{1+4 x^2+4 e^8 \log (3 x)} \, dx+8 \int x^{1+4 x^2+4 e^8 \log (3 x)} \log (x) \, dx+\left (8 e^8\right ) \int x^{-1+4 x^2+4 e^8 \log (3 x)} \log (x) \, dx+\left (4 e^8 \log (3)\right ) \int x^{-1+4 x^2+4 e^8 \log (3 x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.54, size = 18, normalized size = 0.86 \begin {gather*} x+x^{4 \left (x^2+e^8 \log (3 x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x + E^(4*x^2*Log[x] + 4*E^8*Log[x]*Log[3*x])*(4*x^2 + (4*E^8 + 8*x^2)*Log[x] + 4*E^8*Log[3*x]))/x,x
]

[Out]

x + x^(4*(x^2 + E^8*Log[3*x]))

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fricas [A]  time = 0.61, size = 25, normalized size = 1.19 \begin {gather*} x + e^{\left (4 \, e^{8} \log \relax (x)^{2} + 4 \, {\left (x^{2} + e^{8} \log \relax (3)\right )} \log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*exp(4)^2*log(3*x)+(4*exp(4)^2+8*x^2)*log(x)+4*x^2)*exp(4*exp(4)^2*log(x)*log(3*x)+4*x^2*log(x))+
x)/x,x, algorithm="fricas")

[Out]

x + e^(4*e^8*log(x)^2 + 4*(x^2 + e^8*log(3))*log(x))

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giac [A]  time = 0.60, size = 21, normalized size = 1.00 \begin {gather*} x + e^{\left (4 \, x^{2} \log \relax (x) + 4 \, e^{8} \log \left (3 \, x\right ) \log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*exp(4)^2*log(3*x)+(4*exp(4)^2+8*x^2)*log(x)+4*x^2)*exp(4*exp(4)^2*log(x)*log(3*x)+4*x^2*log(x))+
x)/x,x, algorithm="giac")

[Out]

x + e^(4*x^2*log(x) + 4*e^8*log(3*x)*log(x))

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maple [A]  time = 0.18, size = 22, normalized size = 1.05

method	result	size
risch	\(x +x^{4 \,{\mathrm e}^{8} \left (\ln \relax (x )+\ln \relax (3)\right )} x^{4 x^{2}}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*exp(4)^2*ln(3*x)+(4*exp(4)^2+8*x^2)*ln(x)+4*x^2)*exp(4*exp(4)^2*ln(x)*ln(3*x)+4*x^2*ln(x))+x)/x,x,meth
od=_RETURNVERBOSE)

[Out]

x+x^(4*exp(8)*(ln(x)+ln(3)))*x^(4*x^2)

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maxima [A]  time = 0.58, size = 27, normalized size = 1.29 \begin {gather*} x + e^{\left (4 \, x^{2} \log \relax (x) + 4 \, e^{8} \log \relax (3) \log \relax (x) + 4 \, e^{8} \log \relax (x)^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*exp(4)^2*log(3*x)+(4*exp(4)^2+8*x^2)*log(x)+4*x^2)*exp(4*exp(4)^2*log(x)*log(3*x)+4*x^2*log(x))+
x)/x,x, algorithm="maxima")

[Out]

x + e^(4*x^2*log(x) + 4*e^8*log(3)*log(x) + 4*e^8*log(x)^2)

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mupad [B]  time = 4.30, size = 27, normalized size = 1.29 \begin {gather*} x+x^{4\,{\mathrm {e}}^8\,\ln \relax (3)}\,x^{4\,x^2}\,{\mathrm {e}}^{4\,{\mathrm {e}}^8\,{\ln \relax (x)}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + exp(4*x^2*log(x) + 4*log(3*x)*exp(8)*log(x))*(4*log(3*x)*exp(8) + log(x)*(4*exp(8) + 8*x^2) + 4*x^2))
/x,x)

[Out]

x + x^(4*exp(8)*log(3))*x^(4*x^2)*exp(4*exp(8)*log(x)^2)

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sympy [A]  time = 0.36, size = 26, normalized size = 1.24 \begin {gather*} x + e^{4 x^{2} \log {\relax (x )} + 4 \left (\log {\relax (x )} + \log {\relax (3 )}\right ) e^{8} \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*exp(4)**2*ln(3*x)+(4*exp(4)**2+8*x**2)*ln(x)+4*x**2)*exp(4*exp(4)**2*ln(x)*ln(3*x)+4*x**2*ln(x))
+x)/x,x)

[Out]

x + exp(4*x**2*log(x) + 4*(log(x) + log(3))*exp(8)*log(x))

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