3.77.72 \(\int \frac {-4 x^5+48 x^5 \log ^2(4)+12 x^5 \log (x)}{320 \log ^6(4)+240 \log ^4(4) \log (x)+60 \log ^2(4) \log ^2(x)+5 \log ^3(x)} \, dx\)

Optimal. Leaf size=18 \[ \frac {2 x^6}{5 \left (4 \log ^2(4)+\log (x)\right )^2} \]

________________________________________________________________________________________

Rubi [C]  time = 0.67, antiderivative size = 227, normalized size of antiderivative = 12.61, number of steps used = 16, number of rules used = 9, integrand size = 56, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {6, 2561, 6688, 12, 2306, 2309, 2178, 2366, 6482} \begin {gather*} \frac {36}{5} e^{-24 \log ^2(4)} \text {Ei}\left (6 \left (\log (x)+4 \log ^2(4)\right )\right )-\frac {72}{5} e^{-24 \log ^2(4)} \left (-3 \log (x)+1-12 \log ^2(4)\right ) \text {Ei}\left (6 \left (\log (x)+4 \log ^2(4)\right )\right )-\frac {432}{5} e^{-24 \log ^2(4)} \left (\log (x)+4 \log ^2(4)\right ) \text {Ei}\left (6 \left (\log (x)+4 \log ^2(4)\right )\right )+\frac {36}{5} e^{-24 \log ^2(4)} \left (6 \log (x)+1+24 \log ^2(4)\right ) \text {Ei}\left (6 \left (\log (x)+4 \log ^2(4)\right )\right )+\frac {72 x^6}{5}-\frac {6 x^6 \left (6 \log (x)+1+24 \log ^2(4)\right )}{5 \left (\log (x)+4 \log ^2(4)\right )}+\frac {12 x^6 \left (-3 \log (x)+1-12 \log ^2(4)\right )}{5 \left (\log (x)+4 \log ^2(4)\right )}+\frac {2 x^6 \left (-3 \log (x)+1-12 \log ^2(4)\right )}{5 \left (\log (x)+4 \log ^2(4)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4*x^5 + 48*x^5*Log[4]^2 + 12*x^5*Log[x])/(320*Log[4]^6 + 240*Log[4]^4*Log[x] + 60*Log[4]^2*Log[x]^2 + 5*
Log[x]^3),x]

[Out]

(72*x^6)/5 + (36*ExpIntegralEi[6*(4*Log[4]^2 + Log[x])])/(5*E^(24*Log[4]^2)) - (72*ExpIntegralEi[6*(4*Log[4]^2
+ Log[x])]*(1 - 12*Log[4]^2 - 3*Log[x]))/(5*E^(24*Log[4]^2)) + (2*x^6*(1 - 12*Log[4]^2 - 3*Log[x]))/(5*(4*Log
[4]^2 + Log[x])^2) + (12*x^6*(1 - 12*Log[4]^2 - 3*Log[x]))/(5*(4*Log[4]^2 + Log[x])) - (432*ExpIntegralEi[6*(4
*Log[4]^2 + Log[x])]*(4*Log[4]^2 + Log[x]))/(5*E^(24*Log[4]^2)) + (36*ExpIntegralEi[6*(4*Log[4]^2 + Log[x])]*(
1 + 24*Log[4]^2 + 6*Log[x]))/(5*E^(24*Log[4]^2)) - (6*x^6*(1 + 24*Log[4]^2 + 6*Log[x]))/(5*(4*Log[4]^2 + Log[x
]))

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, 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 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 2366

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_.)]*(e_.))*((g_.)*(x_))^(m_.), x_Sy
mbol] :> With[{u = IntHide[(g*x)^m*(a + b*Log[c*x^n])^p, x]}, Dist[d + e*Log[f*x^r], u, x] - Dist[e*r, Int[Sim
plifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, r}, x] &&  !(EqQ[p, 1] && EqQ[a, 0] &&
NeQ[d, 0])

Rule 2561

Int[(u_.)*((a_.)*(x_)^(m_.) + Log[(c_.)*(x_)^(n_.)]^(q_.)*(b_.)*(x_)^(r_.))^(p_.), x_Symbol] :> Int[u*x^(p*r)*
(a*x^(m - r) + b*Log[c*x^n]^q)^p, x] /; FreeQ[{a, b, c, m, n, p, q, r}, x] && IntegerQ[p]

Rule 6482

Int[ExpIntegralEi[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[((a + b*x)*ExpIntegralEi[a + b*x])/b, x] - Simp[E^(a
+ b*x)/b, x] /; FreeQ[{a, b}, x]

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} &=\int \frac {x^5 \left (-4+48 \log ^2(4)\right )+12 x^5 \log (x)}{320 \log ^6(4)+240 \log ^4(4) \log (x)+60 \log ^2(4) \log ^2(x)+5 \log ^3(x)} \, dx\\ &=\int \frac {x^5 \left (-4+48 \log ^2(4)+12 \log (x)\right )}{320 \log ^6(4)+240 \log ^4(4) \log (x)+60 \log ^2(4) \log ^2(x)+5 \log ^3(x)} \, dx\\ &=\int \frac {4 x^5 \left (-1+12 \log ^2(4)+3 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )^3} \, dx\\ &=\frac {4}{5} \int \frac {x^5 \left (-1+12 \log ^2(4)+3 \log (x)\right )}{\left (4 \log ^2(4)+\log (x)\right )^3} \, dx\\ &=-\frac {72}{5} e^{-24 \log ^2(4)} \text {Ei}\left (6 \left (4 \log ^2(4)+\log (x)\right )\right ) \left (1-12 \log ^2(4)-3 \log (x)\right )+\frac {2 x^6 \left (1-12 \log ^2(4)-3 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )^2}+\frac {12 x^6 \left (1-12 \log ^2(4)-3 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )}-\frac {12}{5} \int \left (\frac {18 e^{-24 \log ^2(4)} \text {Ei}\left (6 \left (4 \log ^2(4)+\log (x)\right )\right )}{x}-\frac {x^5 \left (1+24 \log ^2(4)+6 \log (x)\right )}{2 \left (4 \log ^2(4)+\log (x)\right )^2}\right ) \, dx\\ &=-\frac {72}{5} e^{-24 \log ^2(4)} \text {Ei}\left (6 \left (4 \log ^2(4)+\log (x)\right )\right ) \left (1-12 \log ^2(4)-3 \log (x)\right )+\frac {2 x^6 \left (1-12 \log ^2(4)-3 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )^2}+\frac {12 x^6 \left (1-12 \log ^2(4)-3 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )}+\frac {6}{5} \int \frac {x^5 \left (1+24 \log ^2(4)+6 \log (x)\right )}{\left (4 \log ^2(4)+\log (x)\right )^2} \, dx-\frac {1}{5} \left (216 e^{-24 \log ^2(4)}\right ) \int \frac {\text {Ei}\left (6 \left (4 \log ^2(4)+\log (x)\right )\right )}{x} \, dx\\ &=-\frac {72}{5} e^{-24 \log ^2(4)} \text {Ei}\left (6 \left (4 \log ^2(4)+\log (x)\right )\right ) \left (1-12 \log ^2(4)-3 \log (x)\right )+\frac {2 x^6 \left (1-12 \log ^2(4)-3 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )^2}+\frac {12 x^6 \left (1-12 \log ^2(4)-3 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )}+\frac {36}{5} e^{-24 \log ^2(4)} \text {Ei}\left (6 \left (4 \log ^2(4)+\log (x)\right )\right ) \left (1+24 \log ^2(4)+6 \log (x)\right )-\frac {6 x^6 \left (1+24 \log ^2(4)+6 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )}-\frac {36}{5} \int \left (\frac {6 e^{-24 \log ^2(4)} \text {Ei}\left (6 \left (4 \log ^2(4)+\log (x)\right )\right )}{x}-\frac {x^5}{4 \log ^2(4)+\log (x)}\right ) \, dx-\frac {1}{5} \left (216 e^{-24 \log ^2(4)}\right ) \operatorname {Subst}\left (\int \text {Ei}\left (6 \left (x+4 \log ^2(4)\right )\right ) \, dx,x,\log (x)\right )\\ &=-\frac {72}{5} e^{-24 \log ^2(4)} \text {Ei}\left (6 \left (4 \log ^2(4)+\log (x)\right )\right ) \left (1-12 \log ^2(4)-3 \log (x)\right )+\frac {2 x^6 \left (1-12 \log ^2(4)-3 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )^2}+\frac {12 x^6 \left (1-12 \log ^2(4)-3 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )}+\frac {36}{5} e^{-24 \log ^2(4)} \text {Ei}\left (6 \left (4 \log ^2(4)+\log (x)\right )\right ) \left (1+24 \log ^2(4)+6 \log (x)\right )-\frac {6 x^6 \left (1+24 \log ^2(4)+6 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )}+\frac {36}{5} \int \frac {x^5}{4 \log ^2(4)+\log (x)} \, dx-\frac {1}{5} \left (36 e^{-24 \log ^2(4)}\right ) \operatorname {Subst}\left (\int \text {Ei}(x) \, dx,x,24 \log ^2(4)+6 \log (x)\right )-\frac {1}{5} \left (216 e^{-24 \log ^2(4)}\right ) \int \frac {\text {Ei}\left (6 \left (4 \log ^2(4)+\log (x)\right )\right )}{x} \, dx\\ &=\frac {36 x^6}{5}-\frac {72}{5} e^{-24 \log ^2(4)} \text {Ei}\left (6 \left (4 \log ^2(4)+\log (x)\right )\right ) \left (1-12 \log ^2(4)-3 \log (x)\right )+\frac {2 x^6 \left (1-12 \log ^2(4)-3 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )^2}+\frac {12 x^6 \left (1-12 \log ^2(4)-3 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )}-\frac {216}{5} e^{-24 \log ^2(4)} \text {Ei}\left (24 \log ^2(4)+6 \log (x)\right ) \left (4 \log ^2(4)+\log (x)\right )+\frac {36}{5} e^{-24 \log ^2(4)} \text {Ei}\left (6 \left (4 \log ^2(4)+\log (x)\right )\right ) \left (1+24 \log ^2(4)+6 \log (x)\right )-\frac {6 x^6 \left (1+24 \log ^2(4)+6 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )}+\frac {36}{5} \operatorname {Subst}\left (\int \frac {e^{6 x}}{x+4 \log ^2(4)} \, dx,x,\log (x)\right )-\frac {1}{5} \left (216 e^{-24 \log ^2(4)}\right ) \operatorname {Subst}\left (\int \text {Ei}\left (6 \left (x+4 \log ^2(4)\right )\right ) \, dx,x,\log (x)\right )\\ &=\frac {36 x^6}{5}+\frac {36}{5} e^{-24 \log ^2(4)} \text {Ei}\left (6 \left (4 \log ^2(4)+\log (x)\right )\right )-\frac {72}{5} e^{-24 \log ^2(4)} \text {Ei}\left (6 \left (4 \log ^2(4)+\log (x)\right )\right ) \left (1-12 \log ^2(4)-3 \log (x)\right )+\frac {2 x^6 \left (1-12 \log ^2(4)-3 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )^2}+\frac {12 x^6 \left (1-12 \log ^2(4)-3 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )}-\frac {216}{5} e^{-24 \log ^2(4)} \text {Ei}\left (24 \log ^2(4)+6 \log (x)\right ) \left (4 \log ^2(4)+\log (x)\right )+\frac {36}{5} e^{-24 \log ^2(4)} \text {Ei}\left (6 \left (4 \log ^2(4)+\log (x)\right )\right ) \left (1+24 \log ^2(4)+6 \log (x)\right )-\frac {6 x^6 \left (1+24 \log ^2(4)+6 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )}-\frac {1}{5} \left (36 e^{-24 \log ^2(4)}\right ) \operatorname {Subst}\left (\int \text {Ei}(x) \, dx,x,24 \log ^2(4)+6 \log (x)\right )\\ &=\frac {72 x^6}{5}+\frac {36}{5} e^{-24 \log ^2(4)} \text {Ei}\left (6 \left (4 \log ^2(4)+\log (x)\right )\right )-\frac {72}{5} e^{-24 \log ^2(4)} \text {Ei}\left (6 \left (4 \log ^2(4)+\log (x)\right )\right ) \left (1-12 \log ^2(4)-3 \log (x)\right )+\frac {2 x^6 \left (1-12 \log ^2(4)-3 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )^2}+\frac {12 x^6 \left (1-12 \log ^2(4)-3 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )}-\frac {432}{5} e^{-24 \log ^2(4)} \text {Ei}\left (24 \log ^2(4)+6 \log (x)\right ) \left (4 \log ^2(4)+\log (x)\right )+\frac {36}{5} e^{-24 \log ^2(4)} \text {Ei}\left (6 \left (4 \log ^2(4)+\log (x)\right )\right ) \left (1+24 \log ^2(4)+6 \log (x)\right )-\frac {6 x^6 \left (1+24 \log ^2(4)+6 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 18, normalized size = 1.00 \begin {gather*} \frac {2 x^6}{5 \left (4 \log ^2(4)+\log (x)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4*x^5 + 48*x^5*Log[4]^2 + 12*x^5*Log[x])/(320*Log[4]^6 + 240*Log[4]^4*Log[x] + 60*Log[4]^2*Log[x]^
2 + 5*Log[x]^3),x]

[Out]

(2*x^6)/(5*(4*Log[4]^2 + Log[x])^2)

________________________________________________________________________________________

fricas [A]  time = 0.95, size = 26, normalized size = 1.44 \begin {gather*} \frac {2 \, x^{6}}{5 \, {\left (256 \, \log \relax (2)^{4} + 32 \, \log \relax (2)^{2} \log \relax (x) + \log \relax (x)^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x^5*log(x)+192*x^5*log(2)^2-4*x^5)/(5*log(x)^3+240*log(2)^2*log(x)^2+3840*log(2)^4*log(x)+20480*
log(2)^6),x, algorithm="fricas")

[Out]

2/5*x^6/(256*log(2)^4 + 32*log(2)^2*log(x) + log(x)^2)

________________________________________________________________________________________

giac [A]  time = 0.13, size = 26, normalized size = 1.44 \begin {gather*} \frac {2 \, x^{6}}{5 \, {\left (256 \, \log \relax (2)^{4} + 32 \, \log \relax (2)^{2} \log \relax (x) + \log \relax (x)^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x^5*log(x)+192*x^5*log(2)^2-4*x^5)/(5*log(x)^3+240*log(2)^2*log(x)^2+3840*log(2)^4*log(x)+20480*
log(2)^6),x, algorithm="giac")

[Out]

2/5*x^6/(256*log(2)^4 + 32*log(2)^2*log(x) + log(x)^2)

________________________________________________________________________________________

maple [A]  time = 0.11, size = 17, normalized size = 0.94

method	result	size
norman	\(\frac {2 x^{6}}{5 \left (\ln \relax (x )+16 \ln \relax (2)^{2}\right )^{2}}\)	\(17\)
risch	\(\frac {2 x^{6}}{5 \left (\ln \relax (x )+16 \ln \relax (2)^{2}\right )^{2}}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((12*x^5*ln(x)+192*x^5*ln(2)^2-4*x^5)/(5*ln(x)^3+240*ln(2)^2*ln(x)^2+3840*ln(2)^4*ln(x)+20480*ln(2)^6),x,me
thod=_RETURNVERBOSE)

[Out]

2/5*x^6/(ln(x)+16*ln(2)^2)^2

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {72}{5} \, {\left (48 \, \log \relax (2)^{2} - 1\right )} \int \frac {x^{5}}{16 \, \log \relax (2)^{2} + \log \relax (x)}\,{d x} - \frac {192 \, e^{\left (-96 \, \log \relax (2)^{2}\right )} E_{3}\left (-96 \, \log \relax (2)^{2} - 6 \, \log \relax (x)\right ) \log \relax (2)^{2}}{5 \, {\left (16 \, \log \relax (2)^{2} + \log \relax (x)\right )}^{2}} + \frac {12 \, {\left ({\left (48 \, \log \relax (2)^{2} - 1\right )} x^{6} \log \relax (x) + 8 \, {\left (96 \, \log \relax (2)^{4} - \log \relax (2)^{2}\right )} x^{6}\right )}}{5 \, {\left (256 \, \log \relax (2)^{4} + 32 \, \log \relax (2)^{2} \log \relax (x) + \log \relax (x)^{2}\right )}} + \frac {4 \, e^{\left (-96 \, \log \relax (2)^{2}\right )} E_{3}\left (-96 \, \log \relax (2)^{2} - 6 \, \log \relax (x)\right )}{5 \, {\left (16 \, \log \relax (2)^{2} + \log \relax (x)\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x^5*log(x)+192*x^5*log(2)^2-4*x^5)/(5*log(x)^3+240*log(2)^2*log(x)^2+3840*log(2)^4*log(x)+20480*
log(2)^6),x, algorithm="maxima")

[Out]

-72/5*(48*log(2)^2 - 1)*integrate(x^5/(16*log(2)^2 + log(x)), x) - 192/5*e^(-96*log(2)^2)*exp_integral_e(3, -9
6*log(2)^2 - 6*log(x))*log(2)^2/(16*log(2)^2 + log(x))^2 + 12/5*((48*log(2)^2 - 1)*x^6*log(x) + 8*(96*log(2)^4
- log(2)^2)*x^6)/(256*log(2)^4 + 32*log(2)^2*log(x) + log(x)^2) + 4/5*e^(-96*log(2)^2)*exp_integral_e(3, -96*
log(2)^2 - 6*log(x))/(16*log(2)^2 + log(x))^2

________________________________________________________________________________________

mupad [B]  time = 5.02, size = 16, normalized size = 0.89 \begin {gather*} \frac {2\,x^6}{5\,{\left (\ln \relax (x)+16\,{\ln \relax (2)}^2\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((192*x^5*log(2)^2 + 12*x^5*log(x) - 4*x^5)/(3840*log(2)^4*log(x) + 5*log(x)^3 + 240*log(2)^2*log(x)^2 + 20
480*log(2)^6),x)

[Out]

(2*x^6)/(5*(log(x) + 16*log(2)^2)^2)

________________________________________________________________________________________

sympy [A]  time = 0.12, size = 27, normalized size = 1.50 \begin {gather*} \frac {2 x^{6}}{5 \log {\relax (x )}^{2} + 160 \log {\relax (2 )}^{2} \log {\relax (x )} + 1280 \log {\relax (2 )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x**5*ln(x)+192*x**5*ln(2)**2-4*x**5)/(5*ln(x)**3+240*ln(2)**2*ln(x)**2+3840*ln(2)**4*ln(x)+20480
*ln(2)**6),x)

[Out]

2*x**6/(5*log(x)**2 + 160*log(2)**2*log(x) + 1280*log(2)**4)

________________________________________________________________________________________