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

Optimal. Leaf size=22 \[ e^{x^4+\frac {(5 x+x (-1+\log (x))) \log (x)}{x}} \]

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Rubi [A]  time = 0.08, antiderivative size = 36, normalized size of antiderivative = 1.64, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2288} \begin {gather*} \frac {x^3 e^{x^4+\log ^2(x)} \left (2 x^4+\log (x)\right )}{2 x^3+\frac {\log (x)}{x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {e^{x^4+\log ^2(x)} x^3 \left (2 x^4+\log (x)\right )}{2 x^3+\frac {\log (x)}{x}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 14, normalized size = 0.64 \begin {gather*} e^{x^4+\log ^2(x)} x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(x^4 + Log[x]^2)*x^4

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fricas [A]  time = 1.04, size = 13, normalized size = 0.59 \begin {gather*} e^{\left (x^{4} + \log \relax (x)^{2} + 4 \, \log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(x^4 + log(x)^2 + 4*log(x))

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giac [A]  time = 0.22, size = 13, normalized size = 0.59 \begin {gather*} e^{\left (x^{4} + \log \relax (x)^{2} + 4 \, \log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(x^4 + log(x)^2 + 4*log(x))

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maple [A]  time = 0.02, size = 14, normalized size = 0.64

method	result	size
norman	\({\mathrm e}^{\ln \relax (x )^{2}+4 \ln \relax (x )+x^{4}}\)	\(14\)
risch	\(x^{4} {\mathrm e}^{\ln \relax (x )^{2}+x^{4}}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*ln(x)+4*x^4+4)*exp(ln(x)^2+4*ln(x)+x^4)/x,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.41, size = 13, normalized size = 0.59 \begin {gather*} x^{4} e^{\left (x^{4} + \log \relax (x)^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^4*e^(x^4 + log(x)^2)

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mupad [B]  time = 3.30, size = 13, normalized size = 0.59 \begin {gather*} x^4\,{\mathrm {e}}^{x^4}\,{\mathrm {e}}^{{\ln \relax (x)}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^4*exp(x^4)*exp(log(x)^2)

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sympy [A]  time = 0.28, size = 12, normalized size = 0.55 \begin {gather*} x^{4} e^{x^{4} + \log {\relax (x )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**4*exp(x**4 + log(x)**2)

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