3.13.33 \(\int \frac {4 x+(18-6 x) \log (\frac {x^4}{\log ^2(3)})+2 x \log (\frac {x^4}{\log ^2(3)}) \log (\log (\frac {x^4}{\log ^2(3)}))}{3 \log (\frac {x^4}{\log ^2(3)})} \, dx\)

Optimal. Leaf size=23 \[ x-x \left (-5+x-\frac {1}{3} x \log \left (\log \left (\frac {x^4}{\log ^2(3)}\right )\right )\right ) \]

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Rubi [A]  time = 0.16, antiderivative size = 26, normalized size of antiderivative = 1.13, number of steps used = 8, number of rules used = 5, integrand size = 56, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.089, Rules used = {12, 6688, 2310, 2178, 2522} \begin {gather*} -x^2+\frac {1}{3} x^2 \log \left (\log \left (\frac {x^4}{\log ^2(3)}\right )\right )+6 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4*x + (18 - 6*x)*Log[x^4/Log[3]^2] + 2*x*Log[x^4/Log[3]^2]*Log[Log[x^4/Log[3]^2]])/(3*Log[x^4/Log[3]^2]),
x]

[Out]

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

Rule 12

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

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 2310

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

Rule 2522

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

Rule 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \frac {4 x+(18-6 x) \log \left (\frac {x^4}{\log ^2(3)}\right )+2 x \log \left (\frac {x^4}{\log ^2(3)}\right ) \log \left (\log \left (\frac {x^4}{\log ^2(3)}\right )\right )}{\log \left (\frac {x^4}{\log ^2(3)}\right )} \, dx\\ &=\frac {1}{3} \int \left (18-6 x+\frac {4 x}{\log \left (\frac {x^4}{\log ^2(3)}\right )}+2 x \log \left (\log \left (\frac {x^4}{\log ^2(3)}\right )\right )\right ) \, dx\\ &=6 x-x^2+\frac {2}{3} \int x \log \left (\log \left (\frac {x^4}{\log ^2(3)}\right )\right ) \, dx+\frac {4}{3} \int \frac {x}{\log \left (\frac {x^4}{\log ^2(3)}\right )} \, dx\\ &=6 x-x^2+\frac {1}{3} x^2 \log \left (\log \left (\frac {x^4}{\log ^2(3)}\right )\right )-\frac {4}{3} \int \frac {x}{\log \left (\frac {x^4}{\log ^2(3)}\right )} \, dx+\frac {\left (x^2 \log (3)\right ) \operatorname {Subst}\left (\int \frac {e^{x/2}}{x} \, dx,x,\log \left (\frac {x^4}{\log ^2(3)}\right )\right )}{3 \sqrt {x^4}}\\ &=6 x-x^2+\frac {x^2 \text {Ei}\left (\frac {1}{2} \log \left (\frac {x^4}{\log ^2(3)}\right )\right ) \log (3)}{3 \sqrt {x^4}}+\frac {1}{3} x^2 \log \left (\log \left (\frac {x^4}{\log ^2(3)}\right )\right )-\frac {\left (x^2 \log (3)\right ) \operatorname {Subst}\left (\int \frac {e^{x/2}}{x} \, dx,x,\log \left (\frac {x^4}{\log ^2(3)}\right )\right )}{3 \sqrt {x^4}}\\ &=6 x-x^2+\frac {1}{3} x^2 \log \left (\log \left (\frac {x^4}{\log ^2(3)}\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 26, normalized size = 1.13 \begin {gather*} 6 x-x^2+\frac {1}{3} x^2 \log \left (\log \left (\frac {x^4}{\log ^2(3)}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*x + (18 - 6*x)*Log[x^4/Log[3]^2] + 2*x*Log[x^4/Log[3]^2]*Log[Log[x^4/Log[3]^2]])/(3*Log[x^4/Log[3
]^2]),x]

[Out]

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

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fricas [A]  time = 1.01, size = 24, normalized size = 1.04 \begin {gather*} \frac {1}{3} \, x^{2} \log \left (\log \left (\frac {x^{4}}{\log \relax (3)^{2}}\right )\right ) - x^{2} + 6 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(2*x*log(x^4/log(3)^2)*log(log(x^4/log(3)^2))+(-6*x+18)*log(x^4/log(3)^2)+4*x)/log(x^4/log(3)^2)
,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.52, size = 25, normalized size = 1.09 \begin {gather*} \frac {1}{3} \, x^{2} \log \left (\log \left (x^{4}\right ) - 2 \, \log \left (\log \relax (3)\right )\right ) - x^{2} + 6 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(2*x*log(x^4/log(3)^2)*log(log(x^4/log(3)^2))+(-6*x+18)*log(x^4/log(3)^2)+4*x)/log(x^4/log(3)^2)
,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 25, normalized size = 1.09

method	result	size
norman	\(6 x -x^{2}+\frac {x^{2} \ln \left (\ln \left (\frac {x^{4}}{\ln \relax (3)^{2}}\right )\right )}{3}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(2*x*ln(x^4/ln(3)^2)*ln(ln(x^4/ln(3)^2))+(-6*x+18)*ln(x^4/ln(3)^2)+4*x)/ln(x^4/ln(3)^2),x,method=_RETU
RNVERBOSE)

[Out]

6*x-x^2+1/3*x^2*ln(ln(x^4/ln(3)^2))

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maxima [A]  time = 0.51, size = 32, normalized size = 1.39 \begin {gather*} \frac {1}{3} \, x^{2} \log \relax (2) + \frac {1}{3} \, x^{2} \log \left (2 \, \log \relax (x) - \log \left (\log \relax (3)\right )\right ) - x^{2} + 6 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(2*x*log(x^4/log(3)^2)*log(log(x^4/log(3)^2))+(-6*x+18)*log(x^4/log(3)^2)+4*x)/log(x^4/log(3)^2)
,x, algorithm="maxima")

[Out]

1/3*x^2*log(2) + 1/3*x^2*log(2*log(x) - log(log(3))) - x^2 + 6*x

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mupad [B]  time = 1.06, size = 25, normalized size = 1.09 \begin {gather*} 6\,x+\frac {x^2\,\ln \left (\ln \left (x^4\right )-2\,\ln \left (\ln \relax (3)\right )\right )}{3}-x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x)/3 - (log(x^4/log(3)^2)*(6*x - 18))/3 + (2*x*log(x^4/log(3)^2)*log(log(x^4/log(3)^2)))/3)/log(x^4/lo
g(3)^2),x)

[Out]

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

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sympy [A]  time = 0.33, size = 22, normalized size = 0.96 \begin {gather*} \frac {x^{2} \log {\left (\log {\left (\frac {x^{4}}{\log {\relax (3 )}^{2}} \right )} \right )}}{3} - x^{2} + 6 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(2*x*ln(x**4/ln(3)**2)*ln(ln(x**4/ln(3)**2))+(-6*x+18)*ln(x**4/ln(3)**2)+4*x)/ln(x**4/ln(3)**2),
x)

[Out]

x**2*log(log(x**4/log(3)**2))/3 - x**2 + 6*x

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