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

Optimal. Leaf size=11 \[ \log ^2\left (-3+\frac {\log (4)}{x^2}\right ) \]

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Rubi [A]  time = 0.16, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {12, 1593, 2480, 2475, 2412, 2390, 2301} \begin {gather*} \log ^2\left (\frac {\log (4)}{x^2}-3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[-3 + Log[4]/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 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2412

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

Rule 2475

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(f + g*x^(s/n))^r*(a + b*Log[c*(d + e*x)^p])^q,
x], x, x^n], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && IntegerQ[r] && IntegerQ[s/n] && Intege
rQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0])

Rule 2480

Int[((a_.) + Log[(c_.)*(v_)^(p_.)]*(b_.))^(q_.)*(u_)^(r_.)*((h_.)*(x_))^(m_.), x_Symbol] :> Int[(h*x)^m*Expand
ToSum[u, x]^r*(a + b*Log[c*ExpandToSum[v, x]^p])^q, x] /; FreeQ[{a, b, c, h, m, p, q, r}, x] && BinomialQ[{u,
v}, x] &&  !BinomialMatchQ[{u, v}, x]

Rubi steps

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

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Mathematica [C]  time = 0.20, size = 290, normalized size = 26.36 \begin {gather*} \frac {24 \log (4) \log (64) \log (x) \log (\log (4))+2 \log ^2(64) \log \left (\frac {\log (4)}{3 x^2}\right ) \log \left (-3+\frac {\log (4)}{x^2}\right )+6 \log (4) \log (64) \log \left (-3+\frac {\log (4)}{x^2}\right ) \log \left (\log (4)-x \sqrt {\log (64)}\right )-18 \log ^2(4) \log (\log (16)) \log \left (\log (4)-x \sqrt {\log (64)}\right )-9 \log ^2(4) \log ^2\left (\log (4)-x \sqrt {\log (64)}\right )+6 \log (4) \log (64) \log \left (-3+\frac {\log (4)}{x^2}\right ) \log \left (\log (4)+x \sqrt {\log (64)}\right )-18 \log ^2(4) \log (\log (16)) \log \left (\log (4)+x \sqrt {\log (64)}\right )-9 \log ^2(4) \log ^2\left (\log (4)+x \sqrt {\log (64)}\right )+2 \log ^2(64) \text {Li}_2\left (1-\frac {\log (4)}{3 x^2}\right )+18 \log ^2(4) \text {Li}_2\left (\frac {\log (4)-x \sqrt {\log (64)}}{\log (16)}\right )+18 \log ^2(4) \text {Li}_2\left (\frac {\log (4)+x \sqrt {\log (64)}}{\log (16)}\right )-12 \log (4) \log (64) \text {Li}_2\left (-\frac {x \sqrt {\log (64)}}{\log (4)}\right )-12 \log (4) \log (64) \text {Li}_2\left (\frac {x \sqrt {\log (64)}}{\log (4)}\right )}{\log ^2(64)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(24*Log[4]*Log[64]*Log[x]*Log[Log[4]] + 2*Log[64]^2*Log[Log[4]/(3*x^2)]*Log[-3 + Log[4]/x^2] + 6*Log[4]*Log[64
]*Log[-3 + Log[4]/x^2]*Log[Log[4] - x*Sqrt[Log[64]]] - 18*Log[4]^2*Log[Log[16]]*Log[Log[4] - x*Sqrt[Log[64]]]
- 9*Log[4]^2*Log[Log[4] - x*Sqrt[Log[64]]]^2 + 6*Log[4]*Log[64]*Log[-3 + Log[4]/x^2]*Log[Log[4] + x*Sqrt[Log[6
4]]] - 18*Log[4]^2*Log[Log[16]]*Log[Log[4] + x*Sqrt[Log[64]]] - 9*Log[4]^2*Log[Log[4] + x*Sqrt[Log[64]]]^2 + 2
*Log[64]^2*PolyLog[2, 1 - Log[4]/(3*x^2)] + 18*Log[4]^2*PolyLog[2, (Log[4] - x*Sqrt[Log[64]])/Log[16]] + 18*Lo
g[4]^2*PolyLog[2, (Log[4] + x*Sqrt[Log[64]])/Log[16]] - 12*Log[4]*Log[64]*PolyLog[2, -((x*Sqrt[Log[64]])/Log[4
])] - 12*Log[4]*Log[64]*PolyLog[2, (x*Sqrt[Log[64]])/Log[4]])/Log[64]^2

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fricas [A]  time = 0.60, size = 18, normalized size = 1.64 \begin {gather*} \log \left (-\frac {3 \, x^{2} - 2 \, \log \relax (2)}{x^{2}}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-8*log(2)*log((2*log(2)-3*x^2)/x^2)/(2*x*log(2)-3*x^3),x, algorithm="fricas")

[Out]

log(-(3*x^2 - 2*log(2))/x^2)^2

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giac [B]  time = 0.55, size = 71, normalized size = 6.45 \begin {gather*} {\left (2 \, {\left (\frac {\log \left (3 \, x^{2} - 2 \, \log \relax (2)\right )}{\log \relax (2)} - \frac {2 \, \log \relax (x)}{\log \relax (2)}\right )} \log \left (-3 \, x^{2} + 2 \, \log \relax (2)\right ) - \frac {\log \left (3 \, x^{2} - 2 \, \log \relax (2)\right )^{2}}{\log \relax (2)} + \frac {4 \, \log \relax (x)^{2}}{\log \relax (2)}\right )} \log \relax (2) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-8*log(2)*log((2*log(2)-3*x^2)/x^2)/(2*x*log(2)-3*x^3),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.56, size = 13, normalized size = 1.18

method	result	size
derivativedivides	\(\ln \left (\frac {2 \ln \relax (2)}{x^{2}}-3\right )^{2}\)	\(13\)
default	\(\ln \left (\frac {2 \ln \relax (2)}{x^{2}}-3\right )^{2}\)	\(13\)
risch	\(\ln \left (\frac {2 \ln \relax (2)}{x^{2}}-3\right )^{2}\)	\(13\)
norman	\(\ln \left (\frac {2 \ln \relax (2)-3 x^{2}}{x^{2}}\right )^{2}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-8*ln(2)*ln((2*ln(2)-3*x^2)/x^2)/(2*x*ln(2)-3*x^3),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.60, size = 82, normalized size = 7.45 \begin {gather*} 2 \, {\left (\frac {\log \left (3 \, x^{2} - 2 \, \log \relax (2)\right )}{\log \relax (2)} - \frac {2 \, \log \relax (x)}{\log \relax (2)}\right )} \log \relax (2) \log \left (-\frac {3 \, x^{2} - 2 \, \log \relax (2)}{x^{2}}\right ) - \log \left (3 \, x^{2} - 2 \, \log \relax (2)\right )^{2} + 4 \, \log \left (3 \, x^{2} - 2 \, \log \relax (2)\right ) \log \relax (x) - 4 \, \log \relax (x)^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-8*log(2)*log((2*log(2)-3*x^2)/x^2)/(2*x*log(2)-3*x^3),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.83, size = 17, normalized size = 1.55 \begin {gather*} {\ln \left (\frac {2\,\ln \relax (2)-3\,x^2}{x^2}\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(8*log((2*log(2) - 3*x^2)/x^2)*log(2))/(2*x*log(2) - 3*x^3),x)

[Out]

log((2*log(2) - 3*x^2)/x^2)^2

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sympy [A]  time = 0.15, size = 15, normalized size = 1.36 \begin {gather*} \log {\left (\frac {- 3 x^{2} + 2 \log {\relax (2 )}}{x^{2}} \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-8*ln(2)*ln((2*ln(2)-3*x**2)/x**2)/(2*x*ln(2)-3*x**3),x)

[Out]

log((-3*x**2 + 2*log(2))/x**2)**2

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