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

Optimal. Leaf size=22 \[ 1-e^{2+(2+x)^2 \log ^2(3)}+\log \left (x^4\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 35, normalized size of antiderivative = 1.59, number of steps used = 3, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {14, 2236} \begin {gather*} 4 \log (x)-\exp \left (x^2 \log ^2(3)+4 x \log ^2(3)+2 \left (1+2 \log ^2(3)\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-E^(4*x*Log[3]^2 + x^2*Log[3]^2 + 2*(1 + 2*Log[3]^2)) + 4*Log[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 2236

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(e*F^(a + b*x + c*x^2))/(
2*c*Log[F]), x] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 21, normalized size = 0.95 \begin {gather*} -e^{2+(2+x)^2 \log ^2(3)}+4 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.53, size = 23, normalized size = 1.05 \begin {gather*} -e^{\left ({\left (x^{2} + 4 \, x + 4\right )} \log \relax (3)^{2} + 2\right )} + 4 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.15, size = 31, normalized size = 1.41 \begin {gather*} -e^{\left (x^{2} \log \relax (3)^{2} + 4 \, x \log \relax (3)^{2} + 4 \, \log \relax (3)^{2} + 2\right )} + 4 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.05, size = 24, normalized size = 1.09

method	result	size
norman	\(-{\mathrm e}^{\left (x^{2}+4 x +4\right ) \ln \relax (3)^{2}+2}+4 \ln \relax (x )\)	\(24\)
default	\(4 \ln \relax (x )-{\mathrm e}^{x^{2} \ln \relax (3)^{2}+4 x \ln \relax (3)^{2}+4 \ln \relax (3)^{2}+2}\)	\(32\)
risch	\(4 \ln \relax (x )-{\mathrm e}^{x^{2} \ln \relax (3)^{2}+4 x \ln \relax (3)^{2}+4 \ln \relax (3)^{2}+2}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [C]  time = 0.49, size = 129, normalized size = 5.86 \begin {gather*} 2 i \, \sqrt {\pi } \operatorname {erf}\left (i \, x \log \relax (3) + 2 i \, \log \relax (3)\right ) e^{2} \log \relax (3) + {\left (\frac {2 \, \sqrt {\pi } {\left (x \log \relax (3)^{2} + 2 \, \log \relax (3)^{2}\right )} {\left (\operatorname {erf}\left (\sqrt {-\frac {{\left (x \log \relax (3)^{2} + 2 \, \log \relax (3)^{2}\right )}^{2}}{\log \relax (3)^{2}}}\right ) - 1\right )}}{\sqrt {-\frac {{\left (x \log \relax (3)^{2} + 2 \, \log \relax (3)^{2}\right )}^{2}}{\log \relax (3)^{2}}} \log \relax (3)} - \frac {e^{\left (\frac {{\left (x \log \relax (3)^{2} + 2 \, \log \relax (3)^{2}\right )}^{2}}{\log \relax (3)^{2}}\right )}}{\log \relax (3)}\right )} e^{2} \log \relax (3) + 4 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*I*sqrt(pi)*erf(I*x*log(3) + 2*I*log(3))*e^2*log(3) + (2*sqrt(pi)*(x*log(3)^2 + 2*log(3)^2)*(erf(sqrt(-(x*log
(3)^2 + 2*log(3)^2)^2/log(3)^2)) - 1)/(sqrt(-(x*log(3)^2 + 2*log(3)^2)^2/log(3)^2)*log(3)) - e^((x*log(3)^2 +
2*log(3)^2)^2/log(3)^2)/log(3))*e^2*log(3) + 4*log(x)

________________________________________________________________________________________

mupad [B]  time = 0.12, size = 33, normalized size = 1.50 \begin {gather*} 4\,\ln \relax (x)-{\mathrm {e}}^{4\,x\,{\ln \relax (3)}^2}\,{\mathrm {e}}^2\,{\mathrm {e}}^{x^2\,{\ln \relax (3)}^2}\,{\mathrm {e}}^{4\,{\ln \relax (3)}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*log(x) - exp(4*x*log(3)^2)*exp(2)*exp(x^2*log(3)^2)*exp(4*log(3)^2)

________________________________________________________________________________________

sympy [A]  time = 0.14, size = 20, normalized size = 0.91 \begin {gather*} - e^{\left (x^{2} + 4 x + 4\right ) \log {\relax (3 )}^{2} + 2} + 4 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________