3.69.54 \(\int \frac {6+2 \log (4)+(3+3 x+(1+x) \log (4)) \log (x^2)}{x \log (x^2)} \, dx\)

Optimal. Leaf size=15 \[ (3+\log (4)) \left (x+\log \left (9 x \log \left (x^2\right )\right )\right ) \]

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Rubi [A]  time = 0.18, antiderivative size = 24, normalized size of antiderivative = 1.60, number of steps used = 8, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6688, 12, 6742, 43, 2302, 29} \begin {gather*} (3+\log (4)) \log \left (\log \left (x^2\right )\right )+x (3+\log (4))+(3+\log (4)) \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(6 + 2*Log[4] + (3 + 3*x + (1 + x)*Log[4])*Log[x^2])/(x*Log[x^2]),x]

[Out]

x*(3 + Log[4]) + (3 + Log[4])*Log[x] + (3 + Log[4])*Log[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 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

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

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Mathematica [B]  time = 0.02, size = 32, normalized size = 2.13 \begin {gather*} 3 x+x \log (4)+3 \log (x)+\log (4) \log (x)+3 \log \left (\log \left (x^2\right )\right )+\log (4) \log \left (\log \left (x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(6 + 2*Log[4] + (3 + 3*x + (1 + x)*Log[4])*Log[x^2])/(x*Log[x^2]),x]

[Out]

3*x + x*Log[4] + 3*Log[x] + Log[4]*Log[x] + 3*Log[Log[x^2]] + Log[4]*Log[Log[x^2]]

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fricas [A]  time = 0.65, size = 33, normalized size = 2.20 \begin {gather*} 2 \, x \log \relax (2) + \frac {1}{2} \, {\left (2 \, \log \relax (2) + 3\right )} \log \left (x^{2}\right ) + {\left (2 \, \log \relax (2) + 3\right )} \log \left (\log \left (x^{2}\right )\right ) + 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*log(2)*(x+1)+3*x+3)*log(x^2)+4*log(2)+6)/x/log(x^2),x, algorithm="fricas")

[Out]

2*x*log(2) + 1/2*(2*log(2) + 3)*log(x^2) + (2*log(2) + 3)*log(log(x^2)) + 3*x

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giac [A]  time = 0.27, size = 30, normalized size = 2.00 \begin {gather*} x {\left (2 \, \log \relax (2) + 3\right )} + {\left (2 \, \log \relax (2) + 3\right )} \log \relax (x) + {\left (2 \, \log \relax (2) + 3\right )} \log \left (\log \left (x^{2}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*log(2)*(x+1)+3*x+3)*log(x^2)+4*log(2)+6)/x/log(x^2),x, algorithm="giac")

[Out]

x*(2*log(2) + 3) + (2*log(2) + 3)*log(x) + (2*log(2) + 3)*log(log(x^2))

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maple [A]  time = 0.03, size = 31, normalized size = 2.07

method	result	size
norman	\(x \left (2 \ln \relax (2)+3\right )+\left (\frac {3}{2}+\ln \relax (2)\right ) \ln \left (x^{2}\right )+\left (2 \ln \relax (2)+3\right ) \ln \left (\ln \left (x^{2}\right )\right )\)	\(31\)
default	\(2 \ln \relax (2) \ln \relax (x )+2 \ln \relax (2) \ln \left (\ln \left (x^{2}\right )\right )+2 x \ln \relax (2)+3 \ln \left (\ln \left (x^{2}\right )\right )+3 x +3 \ln \relax (x )\)	\(36\)
risch	\(2 \ln \relax (2) \ln \relax (x )+2 \ln \relax (2) \ln \left (\ln \left (x^{2}\right )\right )+2 x \ln \relax (2)+3 \ln \left (\ln \left (x^{2}\right )\right )+3 x +3 \ln \relax (x )\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*ln(2)*(x+1)+3*x+3)*ln(x^2)+4*ln(2)+6)/x/ln(x^2),x,method=_RETURNVERBOSE)

[Out]

x*(2*ln(2)+3)+(3/2+ln(2))*ln(x^2)+(2*ln(2)+3)*ln(ln(x^2))

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maxima [B]  time = 0.36, size = 35, normalized size = 2.33 \begin {gather*} 2 \, x \log \relax (2) + 2 \, \log \relax (2) \log \relax (x) + 2 \, \log \relax (2) \log \left (\log \left (x^{2}\right )\right ) + 3 \, x + 3 \, \log \relax (x) + 3 \, \log \left (\log \left (x^{2}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*log(2)*(x+1)+3*x+3)*log(x^2)+4*log(2)+6)/x/log(x^2),x, algorithm="maxima")

[Out]

2*x*log(2) + 2*log(2)*log(x) + 2*log(2)*log(log(x^2)) + 3*x + 3*log(x) + 3*log(log(x^2))

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mupad [B]  time = 4.00, size = 36, normalized size = 2.40 \begin {gather*} \ln \left (\ln \left (x^2\right )\right )\,\left (\ln \relax (4)+3\right )+\frac {x^3\,\left (\ln \relax (4)+3\right )+x^2\,\ln \left (x^2\right )\,\left (\ln \relax (2)+\frac {3}{2}\right )}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*log(2) + log(x^2)*(3*x + 2*log(2)*(x + 1) + 3) + 6)/(x*log(x^2)),x)

[Out]

log(log(x^2))*(log(4) + 3) + (x^3*(log(4) + 3) + x^2*log(x^2)*(log(2) + 3/2))/x^2

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sympy [A]  time = 0.18, size = 31, normalized size = 2.07 \begin {gather*} x \left (2 \log {\relax (2 )} + 3\right ) + \left (2 \log {\relax (2 )} + 3\right ) \log {\relax (x )} + \left (2 \log {\relax (2 )} + 3\right ) \log {\left (\log {\left (x^{2} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*ln(2)*(x+1)+3*x+3)*ln(x**2)+4*ln(2)+6)/x/ln(x**2),x)

[Out]

x*(2*log(2) + 3) + (2*log(2) + 3)*log(x) + (2*log(2) + 3)*log(log(x**2))

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