Integrand size = 56, antiderivative size = 18 \[ \int \frac {-4 x^5+48 x^5 \log ^2(4)+12 x^5 \log (x)}{320 \log ^6(4)+240 \log ^4(4) \log (x)+60 \log ^2(4) \log ^2(x)+5 \log ^3(x)} \, dx=\frac {2 x^6}{5 \left (4 \log ^2(4)+\log (x)\right )^2} \] Output:
2/5*x^6/(ln(x)+16*ln(2)^2)^2
Time = 0.07 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {-4 x^5+48 x^5 \log ^2(4)+12 x^5 \log (x)}{320 \log ^6(4)+240 \log ^4(4) \log (x)+60 \log ^2(4) \log ^2(x)+5 \log ^3(x)} \, dx=\frac {2 x^6}{5 \left (4 \log ^2(4)+\log (x)\right )^2} \] Input:
Integrate[(-4*x^5 + 48*x^5*Log[4]^2 + 12*x^5*Log[x])/(320*Log[4]^6 + 240*L og[4]^4*Log[x] + 60*Log[4]^2*Log[x]^2 + 5*Log[x]^3),x]
Output:
(2*x^6)/(5*(4*Log[4]^2 + Log[x])^2)
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.99 (sec) , antiderivative size = 222, normalized size of antiderivative = 12.33, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6, 3041, 7239, 27, 25, 2813, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {-4 x^5+48 x^5 \log ^2(4)+12 x^5 \log (x)}{240 \log ^4(4) \log (x)+5 \log ^3(x)+60 \log ^2(4) \log ^2(x)+320 \log ^6(4)} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {x^5 \left (48 \log ^2(4)-4\right )+12 x^5 \log (x)}{240 \log ^4(4) \log (x)+5 \log ^3(x)+60 \log ^2(4) \log ^2(x)+320 \log ^6(4)}dx\) |
\(\Big \downarrow \) 3041 |
\(\displaystyle \int \frac {x^5 \left (12 \log (x)-4+48 \log ^2(4)\right )}{240 \log ^4(4) \log (x)+5 \log ^3(x)+60 \log ^2(4) \log ^2(x)+320 \log ^6(4)}dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {4 x^5 \left (3 \log (x)-1+12 \log ^2(4)\right )}{5 \left (\log (x)+4 \log ^2(4)\right )^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {4}{5} \int -\frac {x^5 \left (-3 \log (x)-12 \log ^2(4)+1\right )}{\left (\log (x)+4 \log ^2(4)\right )^3}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {4}{5} \int \frac {x^5 \left (-3 \log (x)-12 \log ^2(4)+1\right )}{\left (\log (x)+4 \log ^2(4)\right )^3}dx\) |
\(\Big \downarrow \) 2813 |
\(\displaystyle -\frac {4}{5} \left (3 \int \left (\frac {18 e^{-24 \log ^2(4)} \operatorname {ExpIntegralEi}\left (6 \left (\log (x)+4 \log ^2(4)\right )\right )}{x}-\frac {x^5 \left (6 \log (x)+24 \log ^2(4)+1\right )}{2 \left (\log (x)+4 \log ^2(4)\right )^2}\right )dx+18 e^{-24 \log ^2(4)} \left (-3 \log (x)+1-12 \log ^2(4)\right ) \operatorname {ExpIntegralEi}\left (6 \left (\log (x)+4 \log ^2(4)\right )\right )-\frac {3 x^6 \left (-3 \log (x)+1-12 \log ^2(4)\right )}{\log (x)+4 \log ^2(4)}-\frac {x^6 \left (-3 \log (x)+1-12 \log ^2(4)\right )}{2 \left (\log (x)+4 \log ^2(4)\right )^2}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {4}{5} \left (3 \left (-3 e^{-24 \log ^2(4)} \operatorname {ExpIntegralEi}\left (6 \left (\log (x)+4 \log ^2(4)\right )\right )+36 e^{-24 \log ^2(4)} \left (\log (x)+4 \log ^2(4)\right ) \operatorname {ExpIntegralEi}\left (6 \left (\log (x)+4 \log ^2(4)\right )\right )-3 e^{-24 \log ^2(4)} \left (6 \log (x)+1+24 \log ^2(4)\right ) \operatorname {ExpIntegralEi}\left (6 \left (\log (x)+4 \log ^2(4)\right )\right )-6 x^6+\frac {x^6 \left (6 \log (x)+1+24 \log ^2(4)\right )}{2 \left (\log (x)+4 \log ^2(4)\right )}\right )+18 e^{-24 \log ^2(4)} \left (-3 \log (x)+1-12 \log ^2(4)\right ) \operatorname {ExpIntegralEi}\left (6 \left (\log (x)+4 \log ^2(4)\right )\right )-\frac {3 x^6 \left (-3 \log (x)+1-12 \log ^2(4)\right )}{\log (x)+4 \log ^2(4)}-\frac {x^6 \left (-3 \log (x)+1-12 \log ^2(4)\right )}{2 \left (\log (x)+4 \log ^2(4)\right )^2}\right )\) |
Input:
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]
Output:
(-4*((18*ExpIntegralEi[6*(4*Log[4]^2 + Log[x])]*(1 - 12*Log[4]^2 - 3*Log[x ]))/E^(24*Log[4]^2) - (x^6*(1 - 12*Log[4]^2 - 3*Log[x]))/(2*(4*Log[4]^2 + Log[x])^2) - (3*x^6*(1 - 12*Log[4]^2 - 3*Log[x]))/(4*Log[4]^2 + Log[x]) + 3*(-6*x^6 - (3*ExpIntegralEi[6*(4*Log[4]^2 + Log[x])])/E^(24*Log[4]^2) + ( 36*ExpIntegralEi[6*(4*Log[4]^2 + Log[x])]*(4*Log[4]^2 + Log[x]))/E^(24*Log [4]^2) - (3*ExpIntegralEi[6*(4*Log[4]^2 + Log[x])]*(1 + 24*Log[4]^2 + 6*Lo g[x]))/E^(24*Log[4]^2) + (x^6*(1 + 24*Log[4]^2 + 6*Log[x]))/(2*(4*Log[4]^2 + Log[x])))))/5
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_ .)]*(e_.))*((g_.)*(x_))^(m_.), x_Symbol] :> With[{u = IntHide[(g*x)^m*(a + b*Log[c*x^n])^p, x]}, Simp[(d + e*Log[f*x^r]) u, x] - Simp[e*r Int[Simp lifyIntegrand[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])
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]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94
method | result | size |
norman | \(\frac {2 x^{6}}{5 \left (\ln \left (x \right )+16 \ln \left (2\right )^{2}\right )^{2}}\) | \(17\) |
risch | \(\frac {2 x^{6}}{5 \left (\ln \left (x \right )+16 \ln \left (2\right )^{2}\right )^{2}}\) | \(17\) |
default | \(\frac {2 x^{6}}{5 \left (256 \ln \left (2\right )^{4}+32 \ln \left (x \right ) \ln \left (2\right )^{2}+\ln \left (x \right )^{2}\right )}\) | \(27\) |
parallelrisch | \(\frac {2 x^{6}}{5 \left (256 \ln \left (2\right )^{4}+32 \ln \left (x \right ) \ln \left (2\right )^{2}+\ln \left (x \right )^{2}\right )}\) | \(27\) |
Input:
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+38 40*ln(2)^4*ln(x)+20480*ln(2)^6),x,method=_RETURNVERBOSE)
Output:
2/5*x^6/(ln(x)+16*ln(2)^2)^2
Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.44 \[ \int \frac {-4 x^5+48 x^5 \log ^2(4)+12 x^5 \log (x)}{320 \log ^6(4)+240 \log ^4(4) \log (x)+60 \log ^2(4) \log ^2(x)+5 \log ^3(x)} \, dx=\frac {2 \, x^{6}}{5 \, {\left (256 \, \log \left (2\right )^{4} + 32 \, \log \left (2\right )^{2} \log \left (x\right ) + \log \left (x\right )^{2}\right )}} \] Input:
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")
Output:
2/5*x^6/(256*log(2)^4 + 32*log(2)^2*log(x) + log(x)^2)
Time = 0.06 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.50 \[ \int \frac {-4 x^5+48 x^5 \log ^2(4)+12 x^5 \log (x)}{320 \log ^6(4)+240 \log ^4(4) \log (x)+60 \log ^2(4) \log ^2(x)+5 \log ^3(x)} \, dx=\frac {2 x^{6}}{5 \log {\left (x \right )}^{2} + 160 \log {\left (2 \right )}^{2} \log {\left (x \right )} + 1280 \log {\left (2 \right )}^{4}} \] Input:
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)
Output:
2*x**6/(5*log(x)**2 + 160*log(2)**2*log(x) + 1280*log(2)**4)
\[ \int \frac {-4 x^5+48 x^5 \log ^2(4)+12 x^5 \log (x)}{320 \log ^6(4)+240 \log ^4(4) \log (x)+60 \log ^2(4) \log ^2(x)+5 \log ^3(x)} \, dx=\int { \frac {4 \, {\left (48 \, x^{5} \log \left (2\right )^{2} + 3 \, x^{5} \log \left (x\right ) - x^{5}\right )}}{5 \, {\left (4096 \, \log \left (2\right )^{6} + 768 \, \log \left (2\right )^{4} \log \left (x\right ) + 48 \, \log \left (2\right )^{2} \log \left (x\right )^{2} + \log \left (x\right )^{3}\right )}} \,d x } \] Input:
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")
Output:
-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, -96*log(2)^2 - 6*log(x))*log(2)^2/(16*lo g(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^(-9 6*log(2)^2)*exp_integral_e(3, -96*log(2)^2 - 6*log(x))/(16*log(2)^2 + log( x))^2
Time = 0.12 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.44 \[ \int \frac {-4 x^5+48 x^5 \log ^2(4)+12 x^5 \log (x)}{320 \log ^6(4)+240 \log ^4(4) \log (x)+60 \log ^2(4) \log ^2(x)+5 \log ^3(x)} \, dx=\frac {2 \, x^{6}}{5 \, {\left (256 \, \log \left (2\right )^{4} + 32 \, \log \left (2\right )^{2} \log \left (x\right ) + \log \left (x\right )^{2}\right )}} \] Input:
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")
Output:
2/5*x^6/(256*log(2)^4 + 32*log(2)^2*log(x) + log(x)^2)
Time = 3.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {-4 x^5+48 x^5 \log ^2(4)+12 x^5 \log (x)}{320 \log ^6(4)+240 \log ^4(4) \log (x)+60 \log ^2(4) \log ^2(x)+5 \log ^3(x)} \, dx=\frac {2\,x^6}{5\,{\left (\ln \left (x\right )+16\,{\ln \left (2\right )}^2\right )}^2} \] Input:
int((192*x^5*log(2)^2 + 12*x^5*log(x) - 4*x^5)/(3840*log(2)^4*log(x) + 5*l og(x)^3 + 240*log(2)^2*log(x)^2 + 20480*log(2)^6),x)
Output:
(2*x^6)/(5*(log(x) + 16*log(2)^2)^2)
Time = 0.15 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.56 \[ \int \frac {-4 x^5+48 x^5 \log ^2(4)+12 x^5 \log (x)}{320 \log ^6(4)+240 \log ^4(4) \log (x)+60 \log ^2(4) \log ^2(x)+5 \log ^3(x)} \, dx=\frac {2 x^{6}}{5 \mathrm {log}\left (x \right )^{2}+160 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (2\right )^{2}+1280 \mathrm {log}\left (2\right )^{4}} \] Input:
int((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)
Output:
(2*x**6)/(5*(log(x)**2 + 32*log(x)*log(2)**2 + 256*log(2)**4))