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

Optimal. Leaf size=10 \[ e^{4+x} \log \left (x^2\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {14, 2178, 2194, 2554, 12} \begin {gather*} e^{x+4} \log \left (x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(4 + x)*Log[x^2]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2554

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[(w*D[u, x]
)/u, x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 10, normalized size = 1.00 \begin {gather*} e^{4+x} \log \left (x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(4 + x)*Log[x^2]

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fricas [A]  time = 0.69, size = 9, normalized size = 0.90 \begin {gather*} e^{\left (x + 4\right )} \log \left (x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(x + 4)*log(x^2)

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giac [A]  time = 0.16, size = 9, normalized size = 0.90 \begin {gather*} e^{\left (x + 4\right )} \log \left (x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(x + 4)*log(x^2)

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maple [A]  time = 0.10, size = 10, normalized size = 1.00

method	result	size
norman	\(\ln \left (x^{2}\right ) {\mathrm e}^{4} {\mathrm e}^{x}\)	\(10\)
default	\({\mathrm e}^{4} \left (\ln \left (x^{2}\right )-2 \ln \relax (x )\right ) {\mathrm e}^{x}+2 \,{\mathrm e}^{4} {\mathrm e}^{x} \ln \relax (x )\)	\(24\)
risch	\(2 \,{\mathrm e}^{4+x} \ln \relax (x )-\frac {i \left (\mathrm {csgn}\left (i x \right )^{2}-2 \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )+\mathrm {csgn}\left (i x^{2}\right )^{2}\right ) \mathrm {csgn}\left (i x^{2}\right ) \pi \,{\mathrm e}^{4+x}}{2}\)	\(56\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.39, size = 9, normalized size = 0.90 \begin {gather*} e^{\left (x + 4\right )} \log \left (x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(x + 4)*log(x^2)

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mupad [B]  time = 6.81, size = 9, normalized size = 0.90 \begin {gather*} \ln \left (x^2\right )\,{\mathrm {e}}^4\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.36, size = 10, normalized size = 1.00 \begin {gather*} e^{4} e^{x} \log {\left (x^{2} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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