3.48.32 \(\int \frac {-3 x^4 \log ^2(4)-6 x^4 \log ^2(4) \log (x)+(-96+18 x^4 \log ^2(4)) \log ^2(x)}{x^7 \log ^2(4) \log ^2(x)} \, dx\)

Optimal. Leaf size=21 \[ \frac {-9+\frac {16}{x^4 \log ^2(4)}+\frac {3}{\log (x)}}{x^2} \]

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Rubi [A]  time = 0.26, antiderivative size = 24, normalized size of antiderivative = 1.14, number of steps used = 10, number of rules used = 6, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {12, 6742, 14, 2306, 2309, 2178} \begin {gather*} \frac {16}{x^6 \log ^2(4)}-\frac {9}{x^2}+\frac {3}{x^2 \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3*x^4*Log[4]^2 - 6*x^4*Log[4]^2*Log[x] + (-96 + 18*x^4*Log[4]^2)*Log[x]^2)/(x^7*Log[4]^2*Log[x]^2),x]

[Out]

-9/x^2 + 16/(x^6*Log[4]^2) + 3/(x^2*Log[x])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 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 2306

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

Rule 2309

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

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-3 x^4 \log ^2(4)-6 x^4 \log ^2(4) \log (x)+\left (-96+18 x^4 \log ^2(4)\right ) \log ^2(x)}{x^7 \log ^2(x)} \, dx}{\log ^2(4)}\\ &=\frac {\int \left (\frac {6 \left (-16+3 x^4 \log ^2(4)\right )}{x^7}-\frac {3 \log ^2(4)}{x^3 \log ^2(x)}-\frac {6 \log ^2(4)}{x^3 \log (x)}\right ) \, dx}{\log ^2(4)}\\ &=-\left (3 \int \frac {1}{x^3 \log ^2(x)} \, dx\right )-6 \int \frac {1}{x^3 \log (x)} \, dx+\frac {6 \int \frac {-16+3 x^4 \log ^2(4)}{x^7} \, dx}{\log ^2(4)}\\ &=\frac {3}{x^2 \log (x)}+6 \int \frac {1}{x^3 \log (x)} \, dx-6 \operatorname {Subst}\left (\int \frac {e^{-2 x}}{x} \, dx,x,\log (x)\right )+\frac {6 \int \left (-\frac {16}{x^7}+\frac {3 \log ^2(4)}{x^3}\right ) \, dx}{\log ^2(4)}\\ &=-\frac {9}{x^2}-6 \text {Ei}(-2 \log (x))+\frac {16}{x^6 \log ^2(4)}+\frac {3}{x^2 \log (x)}+6 \operatorname {Subst}\left (\int \frac {e^{-2 x}}{x} \, dx,x,\log (x)\right )\\ &=-\frac {9}{x^2}+\frac {16}{x^6 \log ^2(4)}+\frac {3}{x^2 \log (x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 27, normalized size = 1.29 \begin {gather*} 3 \left (-\frac {3}{x^2}+\frac {16}{3 x^6 \log ^2(4)}+\frac {1}{x^2 \log (x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3*x^4*Log[4]^2 - 6*x^4*Log[4]^2*Log[x] + (-96 + 18*x^4*Log[4]^2)*Log[x]^2)/(x^7*Log[4]^2*Log[x]^2)
,x]

[Out]

3*(-3/x^2 + 16/(3*x^6*Log[4]^2) + 1/(x^2*Log[x]))

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fricas [A]  time = 0.54, size = 37, normalized size = 1.76 \begin {gather*} \frac {3 \, x^{4} \log \relax (2)^{2} - {\left (9 \, x^{4} \log \relax (2)^{2} - 4\right )} \log \relax (x)}{x^{6} \log \relax (2)^{2} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((72*x^4*log(2)^2-96)*log(x)^2-24*x^4*log(2)^2*log(x)-12*x^4*log(2)^2)/x^7/log(2)^2/log(x)^2,x,
algorithm="fricas")

[Out]

(3*x^4*log(2)^2 - (9*x^4*log(2)^2 - 4)*log(x))/(x^6*log(2)^2*log(x))

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giac [A]  time = 0.15, size = 35, normalized size = 1.67 \begin {gather*} \frac {\frac {3 \, \log \relax (2)^{2}}{x^{2} \log \relax (x)} - \frac {9 \, x^{4} \log \relax (2)^{2} - 4}{x^{6}}}{\log \relax (2)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((72*x^4*log(2)^2-96)*log(x)^2-24*x^4*log(2)^2*log(x)-12*x^4*log(2)^2)/x^7/log(2)^2/log(x)^2,x,
algorithm="giac")

[Out]

(3*log(2)^2/(x^2*log(x)) - (9*x^4*log(2)^2 - 4)/x^6)/log(2)^2

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maple [A]  time = 0.08, size = 31, normalized size = 1.48

method	result	size
risch	\(-\frac {9 x^{4} \ln \relax (2)^{2}-4}{\ln \relax (2)^{2} x^{6}}+\frac {3}{x^{2} \ln \relax (x )}\)	\(31\)
norman	\(\frac {3 x^{4} \ln \relax (2)+\frac {4 \ln \relax (x )}{\ln \relax (2)}-9 x^{4} \ln \relax (2) \ln \relax (x )}{x^{6} \ln \relax (2) \ln \relax (x )}\)	\(38\)
default	\(\frac {-\frac {36 \ln \relax (2)^{2}}{x^{2}}+24 \ln \relax (2)^{2} \expIntegralEi \left (1, 2 \ln \relax (x )\right )-12 \ln \relax (2)^{2} \left (-\frac {1}{x^{2} \ln \relax (x )}+2 \expIntegralEi \left (1, 2 \ln \relax (x )\right )\right )+\frac {16}{x^{6}}}{4 \ln \relax (2)^{2}}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4*((72*x^4*ln(2)^2-96)*ln(x)^2-24*x^4*ln(2)^2*ln(x)-12*x^4*ln(2)^2)/x^7/ln(2)^2/ln(x)^2,x,method=_RETURN
VERBOSE)

[Out]

-1/ln(2)^2*(9*x^4*ln(2)^2-4)/x^6+3/x^2/ln(x)

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maxima [C]  time = 0.38, size = 44, normalized size = 2.10 \begin {gather*} -\frac {6 \, {\rm Ei}\left (-2 \, \log \relax (x)\right ) \log \relax (2)^{2} - 6 \, \Gamma \left (-1, 2 \, \log \relax (x)\right ) \log \relax (2)^{2} + \frac {9 \, \log \relax (2)^{2}}{x^{2}} - \frac {4}{x^{6}}}{\log \relax (2)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((72*x^4*log(2)^2-96)*log(x)^2-24*x^4*log(2)^2*log(x)-12*x^4*log(2)^2)/x^7/log(2)^2/log(x)^2,x,
algorithm="maxima")

[Out]

-(6*Ei(-2*log(x))*log(2)^2 - 6*gamma(-1, 2*log(x))*log(2)^2 + 9*log(2)^2/x^2 - 4/x^6)/log(2)^2

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mupad [B]  time = 3.21, size = 30, normalized size = 1.43 \begin {gather*} \frac {3}{x^2\,\ln \relax (x)}-\frac {9\,x^4\,{\ln \relax (2)}^2-4}{x^6\,{\ln \relax (2)}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*x^4*log(2)^2 - (log(x)^2*(72*x^4*log(2)^2 - 96))/4 + 6*x^4*log(2)^2*log(x))/(x^7*log(2)^2*log(x)^2),x)

[Out]

3/(x^2*log(x)) - (9*x^4*log(2)^2 - 4)/(x^6*log(2)^2)

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sympy [A]  time = 0.17, size = 27, normalized size = 1.29 \begin {gather*} \frac {3}{x^{2} \log {\relax (x )}} + \frac {- 9 x^{4} \log {\relax (2 )}^{2} + 4}{x^{6} \log {\relax (2 )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((72*x**4*ln(2)**2-96)*ln(x)**2-24*x**4*ln(2)**2*ln(x)-12*x**4*ln(2)**2)/x**7/ln(2)**2/ln(x)**2,
x)

[Out]

3/(x**2*log(x)) + (-9*x**4*log(2)**2 + 4)/(x**6*log(2)**2)

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