\(\int \frac {(f+g x^2) \log (c (d+e x^2)^p)}{x^4} \, dx\) [321]

Optimal result

Integrand size = 23, antiderivative size = 108 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^4} \, dx=-\frac {2 e f p}{3 d x}-\frac {2 e^{3/2} f p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 d^{3/2}}+\frac {2 \sqrt {e} g p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}}-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{3 x^3}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{x} \]

[Out]

-2/3*e*f*p/d/x-2/3*e^(3/2)*f*p*arctan(x*e^(1/2)/d^(1/2))/d^(3/2)-1/3*f*ln(c*(e*x^2+d)^p)/x^3-g*ln(c*(e*x^2+d)^
p)/x+2*g*p*arctan(x*e^(1/2)/d^(1/2))*e^(1/2)/d^(1/2)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2526, 2505, 331, 211} \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^4} \, dx=-\frac {2 e^{3/2} f p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 d^{3/2}}+\frac {2 \sqrt {e} g p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}}-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{3 x^3}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{x}-\frac {2 e f p}{3 d x} \]

[In]

Int[((f + g*x^2)*Log[c*(d + e*x^2)^p])/x^4,x]

[Out]

(-2*e*f*p)/(3*d*x) - (2*e^(3/2)*f*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(3*d^(3/2)) + (2*Sqrt[e]*g*p*ArcTan[(Sqrt[e]*
x)/Sqrt[d]])/Sqrt[d] - (f*Log[c*(d + e*x^2)^p])/(3*x^3) - (g*Log[c*(d + e*x^2)^p])/x

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 2505

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

Rule 2526

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x^s)^r, x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, n, p, q, r, s}, x] && IGtQ[q, 0] && IntegerQ[m] && IntegerQ[r] && IntegerQ[s]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{x^4}+\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{x^2}\right ) \, dx \\ & = f \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x^4} \, dx+g \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x^2} \, dx \\ & = -\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{3 x^3}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{x}+\frac {1}{3} (2 e f p) \int \frac {1}{x^2 \left (d+e x^2\right )} \, dx+(2 e g p) \int \frac {1}{d+e x^2} \, dx \\ & = -\frac {2 e f p}{3 d x}+\frac {2 \sqrt {e} g p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}}-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{3 x^3}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{x}-\frac {\left (2 e^2 f p\right ) \int \frac {1}{d+e x^2} \, dx}{3 d} \\ & = -\frac {2 e f p}{3 d x}-\frac {2 e^{3/2} f p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 d^{3/2}}+\frac {2 \sqrt {e} g p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}}-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{3 x^3}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{x} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.03 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.89 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^4} \, dx=\frac {2 \sqrt {e} g p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}}-\frac {2 e f p \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-\frac {e x^2}{d}\right )}{3 d x}-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{3 x^3}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{x} \]

[In]

Integrate[((f + g*x^2)*Log[c*(d + e*x^2)^p])/x^4,x]

[Out]

(2*Sqrt[e]*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[d] - (2*e*f*p*Hypergeometric2F1[-1/2, 1, 1/2, -((e*x^2)/d)])/
(3*d*x) - (f*Log[c*(d + e*x^2)^p])/(3*x^3) - (g*Log[c*(d + e*x^2)^p])/x

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.71

method	result	size
parts	\(-\frac {g \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{x}-\frac {f \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{3 x^{3}}-\frac {2 p e \left (\frac {\left (-3 d g +e f \right ) \arctan \left (\frac {x e}{\sqrt {d e}}\right )}{d \sqrt {d e}}+\frac {f}{d x}\right )}{3}\)	\(77\)
risch	\(-\frac {\left (3 g \,x^{2}+f \right ) \ln \left (\left (e \,x^{2}+d \right )^{p}\right )}{3 x^{3}}-\frac {3 i \pi \,d^{2} g \,x^{2} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}-3 i \pi \,d^{2} g \,x^{2} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )-3 i \pi \,d^{2} g \,x^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}+3 i \pi \,d^{2} g \,x^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )+i \pi \,d^{2} f \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}-i \pi \,d^{2} f \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )-i \pi \,d^{2} f {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}+i \pi \,d^{2} f {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )-6 \sqrt {-d e}\, p \ln \left (-e x -\sqrt {-d e}\right ) g d \,x^{3}+2 \sqrt {-d e}\, p \ln \left (-e x -\sqrt {-d e}\right ) e f \,x^{3}+6 \sqrt {-d e}\, p \ln \left (-e x +\sqrt {-d e}\right ) g d \,x^{3}-2 \sqrt {-d e}\, p \ln \left (-e x +\sqrt {-d e}\right ) e f \,x^{3}+6 \ln \left (c \right ) d^{2} g \,x^{2}+4 d e f p \,x^{2}+2 \ln \left (c \right ) d^{2} f}{6 d^{2} x^{3}}\)	\(442\)

[In]

int((g*x^2+f)*ln(c*(e*x^2+d)^p)/x^4,x,method=_RETURNVERBOSE)

[Out]

-g*ln(c*(e*x^2+d)^p)/x-1/3*f*ln(c*(e*x^2+d)^p)/x^3-2/3*p*e*((-3*d*g+e*f)/d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))
+f/d/x)

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.77 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^4} \, dx=\left [-\frac {{\left (e f - 3 \, d g\right )} p x^{3} \sqrt {-\frac {e}{d}} \log \left (\frac {e x^{2} + 2 \, d x \sqrt {-\frac {e}{d}} - d}{e x^{2} + d}\right ) + 2 \, e f p x^{2} + {\left (3 \, d g p x^{2} + d f p\right )} \log \left (e x^{2} + d\right ) + {\left (3 \, d g x^{2} + d f\right )} \log \left (c\right )}{3 \, d x^{3}}, -\frac {2 \, {\left (e f - 3 \, d g\right )} p x^{3} \sqrt {\frac {e}{d}} \arctan \left (x \sqrt {\frac {e}{d}}\right ) + 2 \, e f p x^{2} + {\left (3 \, d g p x^{2} + d f p\right )} \log \left (e x^{2} + d\right ) + {\left (3 \, d g x^{2} + d f\right )} \log \left (c\right )}{3 \, d x^{3}}\right ] \]

[In]

integrate((g*x^2+f)*log(c*(e*x^2+d)^p)/x^4,x, algorithm="fricas")

[Out]

[-1/3*((e*f - 3*d*g)*p*x^3*sqrt(-e/d)*log((e*x^2 + 2*d*x*sqrt(-e/d) - d)/(e*x^2 + d)) + 2*e*f*p*x^2 + (3*d*g*p
*x^2 + d*f*p)*log(e*x^2 + d) + (3*d*g*x^2 + d*f)*log(c))/(d*x^3), -1/3*(2*(e*f - 3*d*g)*p*x^3*sqrt(e/d)*arctan
(x*sqrt(e/d)) + 2*e*f*p*x^2 + (3*d*g*p*x^2 + d*f*p)*log(e*x^2 + d) + (3*d*g*x^2 + d*f)*log(c))/(d*x^3)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 901 vs. \(2 (104) = 208\).

Time = 36.67 (sec) , antiderivative size = 901, normalized size of antiderivative = 8.34 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^4} \, dx=\begin {cases} \left (- \frac {f}{3 x^{3}} - \frac {g}{x}\right ) \log {\left (0^{p} c \right )} & \text {for}\: d = 0 \wedge e = 0 \\\left (- \frac {f}{3 x^{3}} - \frac {g}{x}\right ) \log {\left (c d^{p} \right )} & \text {for}\: e = 0 \\- \frac {2 f p}{9 x^{3}} - \frac {f \log {\left (c \left (e x^{2}\right )^{p} \right )}}{3 x^{3}} - \frac {2 g p}{x} - \frac {g \log {\left (c \left (e x^{2}\right )^{p} \right )}}{x} & \text {for}\: d = 0 \\\left (- \frac {f}{3 x^{3}} - \frac {g}{x}\right ) \log {\left (0^{p} c \right )} & \text {for}\: d = - e x^{2} \\- \frac {d^{2} f \sqrt {- \frac {d}{e}} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{3 d^{2} x^{3} \sqrt {- \frac {d}{e}} + 3 d e x^{5} \sqrt {- \frac {d}{e}}} + \frac {6 d^{2} g p x^{3} \log {\left (x - \sqrt {- \frac {d}{e}} \right )}}{3 d^{2} x^{3} \sqrt {- \frac {d}{e}} + 3 d e x^{5} \sqrt {- \frac {d}{e}}} - \frac {3 d^{2} g x^{3} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{3 d^{2} x^{3} \sqrt {- \frac {d}{e}} + 3 d e x^{5} \sqrt {- \frac {d}{e}}} - \frac {3 d^{2} g x^{2} \sqrt {- \frac {d}{e}} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{3 d^{2} x^{3} \sqrt {- \frac {d}{e}} + 3 d e x^{5} \sqrt {- \frac {d}{e}}} - \frac {2 d f p x^{3} \log {\left (x - \sqrt {- \frac {d}{e}} \right )}}{\frac {3 d^{2} x^{3} \sqrt {- \frac {d}{e}}}{e} + 3 d x^{5} \sqrt {- \frac {d}{e}}} - \frac {2 d f p x^{2} \sqrt {- \frac {d}{e}}}{\frac {3 d^{2} x^{3} \sqrt {- \frac {d}{e}}}{e} + 3 d x^{5} \sqrt {- \frac {d}{e}}} + \frac {d f x^{3} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{\frac {3 d^{2} x^{3} \sqrt {- \frac {d}{e}}}{e} + 3 d x^{5} \sqrt {- \frac {d}{e}}} - \frac {d f x^{2} \sqrt {- \frac {d}{e}} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{\frac {3 d^{2} x^{3} \sqrt {- \frac {d}{e}}}{e} + 3 d x^{5} \sqrt {- \frac {d}{e}}} + \frac {6 d g p x^{5} \log {\left (x - \sqrt {- \frac {d}{e}} \right )}}{\frac {3 d^{2} x^{3} \sqrt {- \frac {d}{e}}}{e} + 3 d x^{5} \sqrt {- \frac {d}{e}}} - \frac {3 d g x^{5} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{\frac {3 d^{2} x^{3} \sqrt {- \frac {d}{e}}}{e} + 3 d x^{5} \sqrt {- \frac {d}{e}}} - \frac {3 d g x^{4} \sqrt {- \frac {d}{e}} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{\frac {3 d^{2} x^{3} \sqrt {- \frac {d}{e}}}{e} + 3 d x^{5} \sqrt {- \frac {d}{e}}} - \frac {2 e f p x^{5} \log {\left (x - \sqrt {- \frac {d}{e}} \right )}}{\frac {3 d^{2} x^{3} \sqrt {- \frac {d}{e}}}{e} + 3 d x^{5} \sqrt {- \frac {d}{e}}} - \frac {2 e f p x^{4} \sqrt {- \frac {d}{e}}}{\frac {3 d^{2} x^{3} \sqrt {- \frac {d}{e}}}{e} + 3 d x^{5} \sqrt {- \frac {d}{e}}} + \frac {e f x^{5} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{\frac {3 d^{2} x^{3} \sqrt {- \frac {d}{e}}}{e} + 3 d x^{5} \sqrt {- \frac {d}{e}}} & \text {otherwise} \end {cases} \]

[In]

integrate((g*x**2+f)*ln(c*(e*x**2+d)**p)/x**4,x)

[Out]

Piecewise(((-f/(3*x**3) - g/x)*log(0**p*c), Eq(d, 0) & Eq(e, 0)), ((-f/(3*x**3) - g/x)*log(c*d**p), Eq(e, 0)),
(-2*f*p/(9*x**3) - f*log(c*(e*x**2)**p)/(3*x**3) - 2*g*p/x - g*log(c*(e*x**2)**p)/x, Eq(d, 0)), ((-f/(3*x**3)
- g/x)*log(0**p*c), Eq(d, -e*x**2)), (-d**2*f*sqrt(-d/e)*log(c*(d + e*x**2)**p)/(3*d**2*x**3*sqrt(-d/e) + 3*d
*e*x**5*sqrt(-d/e)) + 6*d**2*g*p*x**3*log(x - sqrt(-d/e))/(3*d**2*x**3*sqrt(-d/e) + 3*d*e*x**5*sqrt(-d/e)) - 3
*d**2*g*x**3*log(c*(d + e*x**2)**p)/(3*d**2*x**3*sqrt(-d/e) + 3*d*e*x**5*sqrt(-d/e)) - 3*d**2*g*x**2*sqrt(-d/e
)*log(c*(d + e*x**2)**p)/(3*d**2*x**3*sqrt(-d/e) + 3*d*e*x**5*sqrt(-d/e)) - 2*d*f*p*x**3*log(x - sqrt(-d/e))/(
3*d**2*x**3*sqrt(-d/e)/e + 3*d*x**5*sqrt(-d/e)) - 2*d*f*p*x**2*sqrt(-d/e)/(3*d**2*x**3*sqrt(-d/e)/e + 3*d*x**5
*sqrt(-d/e)) + d*f*x**3*log(c*(d + e*x**2)**p)/(3*d**2*x**3*sqrt(-d/e)/e + 3*d*x**5*sqrt(-d/e)) - d*f*x**2*sqr
t(-d/e)*log(c*(d + e*x**2)**p)/(3*d**2*x**3*sqrt(-d/e)/e + 3*d*x**5*sqrt(-d/e)) + 6*d*g*p*x**5*log(x - sqrt(-d
/e))/(3*d**2*x**3*sqrt(-d/e)/e + 3*d*x**5*sqrt(-d/e)) - 3*d*g*x**5*log(c*(d + e*x**2)**p)/(3*d**2*x**3*sqrt(-d
/e)/e + 3*d*x**5*sqrt(-d/e)) - 3*d*g*x**4*sqrt(-d/e)*log(c*(d + e*x**2)**p)/(3*d**2*x**3*sqrt(-d/e)/e + 3*d*x*
*5*sqrt(-d/e)) - 2*e*f*p*x**5*log(x - sqrt(-d/e))/(3*d**2*x**3*sqrt(-d/e)/e + 3*d*x**5*sqrt(-d/e)) - 2*e*f*p*x
**4*sqrt(-d/e)/(3*d**2*x**3*sqrt(-d/e)/e + 3*d*x**5*sqrt(-d/e)) + e*f*x**5*log(c*(d + e*x**2)**p)/(3*d**2*x**3
*sqrt(-d/e)/e + 3*d*x**5*sqrt(-d/e)), True))

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^4} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((g*x^2+f)*log(c*(e*x^2+d)^p)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.81 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^4} \, dx=-\frac {2 \, {\left (e^{2} f p - 3 \, d e g p\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{3 \, \sqrt {d e} d} - \frac {{\left (3 \, g p x^{2} + f p\right )} \log \left (e x^{2} + d\right )}{3 \, x^{3}} - \frac {2 \, e f p x^{2} + 3 \, d g x^{2} \log \left (c\right ) + d f \log \left (c\right )}{3 \, d x^{3}} \]

[In]

integrate((g*x^2+f)*log(c*(e*x^2+d)^p)/x^4,x, algorithm="giac")

[Out]

-2/3*(e^2*f*p - 3*d*e*g*p)*arctan(e*x/sqrt(d*e))/(sqrt(d*e)*d) - 1/3*(3*g*p*x^2 + f*p)*log(e*x^2 + d)/x^3 - 1/
3*(2*e*f*p*x^2 + 3*d*g*x^2*log(c) + d*f*log(c))/(d*x^3)

Mupad [B] (verification not implemented)

Time = 1.71 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.60 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^4} \, dx=\frac {2\,\sqrt {e}\,p\,\mathrm {atan}\left (\frac {\sqrt {e}\,x}{\sqrt {d}}\right )\,\left (3\,d\,g-e\,f\right )}{3\,d^{3/2}}-\frac {2\,e\,f\,p}{3\,d\,x}-\frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,\left (g\,x^2+\frac {f}{3}\right )}{x^3} \]

[In]

int((log(c*(d + e*x^2)^p)*(f + g*x^2))/x^4,x)

[Out]

(2*e^(1/2)*p*atan((e^(1/2)*x)/d^(1/2))*(3*d*g - e*f))/(3*d^(3/2)) - (2*e*f*p)/(3*d*x) - (log(c*(d + e*x^2)^p)*
(f/3 + g*x^2))/x^3