3.20.60 \(\int \frac {4 e^{\log ^2(-3 x^2)} \log (-3 x^2)}{x} \, dx\)

Optimal. Leaf size=10 \[ e^{\log ^2\left (-3 x^2\right )} \]

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Rubi [A]  time = 0.11, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {12, 2276, 2204, 2209} \begin {gather*} e^{\log ^2\left (-3 x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^Log[-3*x^2]^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 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2276

Int[(F_)^(((a_.) + Log[(c_.)*(x_)^(n_.)]^2*(b_.))*(d_.))*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[(e*x)^(m + 1)/(
e*n*(c*x^n)^((m + 1)/n)), Subst[Int[E^(a*d*Log[F] + ((m + 1)*x)/n + b*d*Log[F]*x^2), x], x, Log[c*x^n]], x] /;
FreeQ[{F, a, b, c, d, e, m, n}, x]

Rubi steps

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

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Mathematica [A]  time = 0.04, size = 10, normalized size = 1.00 \begin {gather*} e^{\log ^2\left (-3 x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^Log[-3*x^2]^2

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fricas [A]  time = 0.94, size = 9, normalized size = 0.90 \begin {gather*} e^{\left (\log \left (-3 \, x^{2}\right )^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(log(-3*x^2)^2)

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giac [A]  time = 0.23, size = 9, normalized size = 0.90 \begin {gather*} e^{\left (\log \left (-3 \, x^{2}\right )^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(log(-3*x^2)^2)

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maple [A]  time = 0.02, size = 10, normalized size = 1.00

method	result	size
derivativedivides	\({\mathrm e}^{\ln \left (-3 x^{2}\right )^{2}}\)	\(10\)
default	\({\mathrm e}^{\ln \left (-3 x^{2}\right )^{2}}\)	\(10\)
norman	\({\mathrm e}^{\ln \left (-3 x^{2}\right )^{2}}\)	\(10\)
risch	\({\mathrm e}^{\ln \left (-3 x^{2}\right )^{2}}\)	\(10\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(4*ln(-3*x^2)*exp(ln(-3*x^2)^2)/x,x,method=_RETURNVERBOSE)

[Out]

exp(ln(-3*x^2)^2)

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maxima [A]  time = 0.58, size = 9, normalized size = 0.90 \begin {gather*} e^{\left (\log \left (-3 \, x^{2}\right )^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(log(-3*x^2)^2)

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mupad [B]  time = 1.39, size = 38, normalized size = 3.80 \begin {gather*} 3^{\pi \,2{}\mathrm {i}}\,{\mathrm {e}}^{{\ln \left (x^2\right )}^2}\,{\mathrm {e}}^{{\ln \relax (3)}^2}\,{\mathrm {e}}^{-\pi ^2}\,{\left (x^2\right )}^{2\,\ln \relax (3)+\pi \,2{}\mathrm {i}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3^(pi*2i)*exp(log(x^2)^2)*exp(log(3)^2)*exp(-pi^2)*(x^2)^(pi*2i + 2*log(3))

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sympy [A]  time = 0.26, size = 10, normalized size = 1.00 \begin {gather*} e^{\log {\left (- 3 x^{2} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(log(-3*x**2)**2)

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