Integrand size = 86, antiderivative size = 21 \[ \int \frac {-2 x^3 \log ^2(25) \log ^2(x)+\left (x^3 \log ^2(25) \log (x)+2 x^3 \log ^2(25) \log ^2(x)\right ) \log (4 x)+128 x \log (25) \log (x) \log ^2(4 x)+(-64 x \log (25)-128 x \log (25) \log (x)) \log ^3(4 x)}{128 \log ^5(4 x)} \, dx=\left (4-\frac {x^2 \log (25) \log (x)}{16 \log ^2(4 x)}\right )^2 \]
Time = 0.29 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.62 \[ \int \frac {-2 x^3 \log ^2(25) \log ^2(x)+\left (x^3 \log ^2(25) \log (x)+2 x^3 \log ^2(25) \log ^2(x)\right ) \log (4 x)+128 x \log (25) \log (x) \log ^2(4 x)+(-64 x \log (25)-128 x \log (25) \log (x)) \log ^3(4 x)}{128 \log ^5(4 x)} \, dx=\frac {x^2 \log (25) \log (x) \left (x^2 \log (25) \log (x)-128 \log ^2(4 x)\right )}{256 \log ^4(4 x)} \]
Integrate[(-2*x^3*Log[25]^2*Log[x]^2 + (x^3*Log[25]^2*Log[x] + 2*x^3*Log[2 5]^2*Log[x]^2)*Log[4*x] + 128*x*Log[25]*Log[x]*Log[4*x]^2 + (-64*x*Log[25] - 128*x*Log[25]*Log[x])*Log[4*x]^3)/(128*Log[4*x]^5),x]
Time = 0.48 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.070, Rules used = {27, 25, 7239, 27, 7263, 17}
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.
| \(\displaystyle \int \frac {-2 x^3 \log ^2(25) \log ^2(x)+\left (2 x^3 \log ^2(25) \log ^2(x)+x^3 \log ^2(25) \log (x)\right ) \log (4 x)+(-128 x \log (25) \log (x)-64 x \log (25)) \log ^3(4 x)+128 x \log (25) \log (x) \log ^2(4 x)}{128 \log ^5(4 x)} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{128} \int -\frac {2 \log ^2(25) \log ^2(x) x^3-128 \log (25) \log (x) \log ^2(4 x) x+64 (2 \log (25) \log (x) x+\log (25) x) \log ^3(4 x)-\left (2 \log ^2(25) \log ^2(x) x^3+\log ^2(25) \log (x) x^3\right ) \log (4 x)}{\log ^5(4 x)}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{128} \int \frac {2 \log ^2(25) \log ^2(x) x^3-128 \log (25) \log (x) \log ^2(4 x) x+64 (2 \log (25) \log (x) x+\log (25) x) \log ^3(4 x)-\left (2 \log ^2(25) \log ^2(x) x^3+\log ^2(25) \log (x) x^3\right ) \log (4 x)}{\log ^5(4 x)}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {1}{128} \int \frac {x \log (25) (2 \log (x) (\log (4 x)-1)+\log (4 x)) \left (64 \log ^2(4 x)-x^2 \log (25) \log (x)\right )}{\log ^5(4 x)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{128} \log (25) \int \frac {x (2 \log (x) (1-\log (4 x))-\log (4 x)) \left (x^2 \log (25) \log (x)-64 \log ^2(4 x)\right )}{\log ^5(4 x)}dx\) |
| \(\Big \downarrow \) 7263 |
| \(\displaystyle \frac {1}{128} \int \left (\frac {x^2 \log (25) \log (x)}{\log ^2(4 x)}-64\right )d\frac {x^2 \log (25) \log (x)}{\log ^2(4 x)}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle \frac {1}{256} \left (64-\frac {x^2 \log (25) \log (x)}{\log ^2(4 x)}\right )^2\) |
Int[(-2*x^3*Log[25]^2*Log[x]^2 + (x^3*Log[25]^2*Log[x] + 2*x^3*Log[25]^2*L og[x]^2)*Log[4*x] + 128*x*Log[25]*Log[x]*Log[4*x]^2 + (-64*x*Log[25] - 128 *x*Log[25]*Log[x])*Log[4*x]^3)/(128*Log[4*x]^5),x]
3.16.67.3.1 Defintions of rubi rules used
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Int[(u_)*(v_)^(r_.)*((a_.)*(v_)^(p_.) + (b_.)*(w_)^(q_.))^(m_.), x_Symbol] :> With[{c = Simplify[u/(p*w*D[v, x] - q*v*D[w, x])]}, Simp[(-c)*q Subst[ Int[(a + b*x^q)^m, x], x, v^(m*p + r + 1)*w], x] /; FreeQ[c, x]] /; FreeQ[{ a, b, m, p, q, r}, x] && EqQ[p + q*(m*p + r + 1), 0] && IntegerQ[q] && Inte gerQ[m]
Time = 0.72 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.81
| method | result | size |
| parallelrisch | \(\frac {2 \ln \left (5\right )^{2} x^{4} \ln \left (x \right )^{2}-128 \ln \left (x \right ) \ln \left (5\right ) \ln \left (4 x \right )^{2} x^{2}}{128 \ln \left (4 x \right )^{4}}\) | \(38\) |
| default | \(-\frac {\ln \left (5\right ) \ln \left (x \right ) x^{2}}{\left (\ln \left (x \right )+2 \ln \left (2\right )\right )^{2}}+\frac {\ln \left (5\right )^{2} \ln \left (x \right )^{2} x^{4}}{64 \left (\ln \left (x \right )+2 \ln \left (2\right )\right )^{4}}\) | \(42\) |
| risch | \(-\frac {64 x^{2} \ln \left (5\right ) \ln \left (x \right )^{3}-\ln \left (5\right )^{2} x^{4} \ln \left (x \right )^{2}+256 \ln \left (2\right )^{2} \ln \left (5\right ) x^{2} \ln \left (x \right )+256 x^{2} \ln \left (5\right ) \ln \left (x \right )^{2} \ln \left (2\right )}{4 \left (2 \ln \left (x \right )+4 \ln \left (2\right )\right )^{4}}\) | \(65\) |
| parts | \(\frac {\ln \left (5\right )^{2} \left (-\frac {4 \ln \left (2\right ) x^{4}}{3 \left (\ln \left (x \right )+2 \ln \left (2\right )\right )^{3}}+\frac {x^{4}}{\ln \left (x \right )+2 \ln \left (2\right )}+\frac {x^{4}}{2 \left (\ln \left (x \right )+2 \ln \left (2\right )\right )^{2}}+\frac {\operatorname {Ei}_{1}\left (-4 \ln \left (x \right )-8 \ln \left (2\right )\right )}{64}-\frac {2 \ln \left (2\right ) x^{4}}{3 \left (\ln \left (x \right )+2 \ln \left (2\right )\right )^{2}}-\frac {8 \ln \left (2\right ) x^{4}}{3 \left (\ln \left (x \right )+2 \ln \left (2\right )\right )}-\frac {\ln \left (2\right ) \operatorname {Ei}_{1}\left (-4 \ln \left (x \right )-8 \ln \left (2\right )\right )}{24}+\frac {\ln \left (2\right )^{2} x^{4}}{\left (\ln \left (x \right )+2 \ln \left (2\right )\right )^{4}}\right )}{16}+\frac {\ln \left (5\right )^{2} \left (-\frac {x^{4}}{2 \left (\ln \left (x \right )+2 \ln \left (2\right )\right )^{2}}-\frac {2 x^{4}}{\ln \left (x \right )+2 \ln \left (2\right )}-\frac {\operatorname {Ei}_{1}\left (-4 \ln \left (x \right )-8 \ln \left (2\right )\right )}{32}-2 \ln \left (2\right ) \left (-\frac {x^{4}}{3 \left (\ln \left (x \right )+2 \ln \left (2\right )\right )^{3}}-\frac {2 x^{4}}{3 \left (\ln \left (x \right )+2 \ln \left (2\right )\right )^{2}}-\frac {8 x^{4}}{3 \left (\ln \left (x \right )+2 \ln \left (2\right )\right )}-\frac {\operatorname {Ei}_{1}\left (-4 \ln \left (x \right )-8 \ln \left (2\right )\right )}{24}\right )\right )}{32}-2 \ln \left (5\right ) \left (\frac {\operatorname {Ei}_{1}\left (-2 \ln \left (x \right )-4 \ln \left (2\right )\right )}{16}-\frac {\ln \left (2\right ) x^{2}}{\left (\ln \left (x \right )+2 \ln \left (2\right )\right )^{2}}+\frac {x^{2}}{\ln \left (x \right )+2 \ln \left (2\right )}\right )-\ln \left (5\right ) \left (-\frac {x^{2}}{\ln \left (4 x \right )}-\frac {\operatorname {Ei}_{1}\left (-2 \ln \left (4 x \right )\right )}{8}\right )\) | \(310\) |
int(1/128*((-256*x*ln(5)*ln(x)-128*x*ln(5))*ln(4*x)^3+256*x*ln(5)*ln(x)*ln (4*x)^2+(8*x^3*ln(5)^2*ln(x)^2+4*x^3*ln(5)^2*ln(x))*ln(4*x)-8*x^3*ln(5)^2* ln(x)^2)/ln(4*x)^5,x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (20) = 40\).
Time = 0.24 (sec) , antiderivative size = 90, normalized size of antiderivative = 4.29 \[ \int \frac {-2 x^3 \log ^2(25) \log ^2(x)+\left (x^3 \log ^2(25) \log (x)+2 x^3 \log ^2(25) \log ^2(x)\right ) \log (4 x)+128 x \log (25) \log (x) \log ^2(4 x)+(-64 x \log (25)-128 x \log (25) \log (x)) \log ^3(4 x)}{128 \log ^5(4 x)} \, dx=-\frac {256 \, x^{2} \log \left (5\right ) \log \left (2\right )^{2} \log \left (x\right ) + 64 \, x^{2} \log \left (5\right ) \log \left (x\right )^{3} - {\left (x^{4} \log \left (5\right )^{2} - 256 \, x^{2} \log \left (5\right ) \log \left (2\right )\right )} \log \left (x\right )^{2}}{64 \, {\left (16 \, \log \left (2\right )^{4} + 32 \, \log \left (2\right )^{3} \log \left (x\right ) + 24 \, \log \left (2\right )^{2} \log \left (x\right )^{2} + 8 \, \log \left (2\right ) \log \left (x\right )^{3} + \log \left (x\right )^{4}\right )}} \]
integrate(1/128*((-256*x*log(5)*log(x)-128*x*log(5))*log(4*x)^3+256*x*log( 5)*log(x)*log(4*x)^2+(8*x^3*log(5)^2*log(x)^2+4*x^3*log(5)^2*log(x))*log(4 *x)-8*x^3*log(5)^2*log(x)^2)/log(4*x)^5,x, algorithm=\
-1/64*(256*x^2*log(5)*log(2)^2*log(x) + 64*x^2*log(5)*log(x)^3 - (x^4*log( 5)^2 - 256*x^2*log(5)*log(2))*log(x)^2)/(16*log(2)^4 + 32*log(2)^3*log(x) + 24*log(2)^2*log(x)^2 + 8*log(2)*log(x)^3 + log(x)^4)
Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (20) = 40\).
Time = 0.15 (sec) , antiderivative size = 100, normalized size of antiderivative = 4.76 \[ \int \frac {-2 x^3 \log ^2(25) \log ^2(x)+\left (x^3 \log ^2(25) \log (x)+2 x^3 \log ^2(25) \log ^2(x)\right ) \log (4 x)+128 x \log (25) \log (x) \log ^2(4 x)+(-64 x \log (25)-128 x \log (25) \log (x)) \log ^3(4 x)}{128 \log ^5(4 x)} \, dx=\frac {- 64 x^{2} \log {\left (5 \right )} \log {\left (x \right )}^{3} - 256 x^{2} \log {\left (2 \right )}^{2} \log {\left (5 \right )} \log {\left (x \right )} + \left (x^{4} \log {\left (5 \right )}^{2} - 256 x^{2} \log {\left (2 \right )} \log {\left (5 \right )}\right ) \log {\left (x \right )}^{2}}{64 \log {\left (x \right )}^{4} + 512 \log {\left (2 \right )} \log {\left (x \right )}^{3} + 1536 \log {\left (2 \right )}^{2} \log {\left (x \right )}^{2} + 2048 \log {\left (2 \right )}^{3} \log {\left (x \right )} + 1024 \log {\left (2 \right )}^{4}} \]
integrate(1/128*((-256*x*ln(5)*ln(x)-128*x*ln(5))*ln(4*x)**3+256*x*ln(5)*l n(x)*ln(4*x)**2+(8*x**3*ln(5)**2*ln(x)**2+4*x**3*ln(5)**2*ln(x))*ln(4*x)-8 *x**3*ln(5)**2*ln(x)**2)/ln(4*x)**5,x)
(-64*x**2*log(5)*log(x)**3 - 256*x**2*log(2)**2*log(5)*log(x) + (x**4*log( 5)**2 - 256*x**2*log(2)*log(5))*log(x)**2)/(64*log(x)**4 + 512*log(2)*log( x)**3 + 1536*log(2)**2*log(x)**2 + 2048*log(2)**3*log(x) + 1024*log(2)**4)
Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (20) = 40\).
Time = 0.33 (sec) , antiderivative size = 90, normalized size of antiderivative = 4.29 \[ \int \frac {-2 x^3 \log ^2(25) \log ^2(x)+\left (x^3 \log ^2(25) \log (x)+2 x^3 \log ^2(25) \log ^2(x)\right ) \log (4 x)+128 x \log (25) \log (x) \log ^2(4 x)+(-64 x \log (25)-128 x \log (25) \log (x)) \log ^3(4 x)}{128 \log ^5(4 x)} \, dx=-\frac {256 \, x^{2} \log \left (5\right ) \log \left (2\right )^{2} \log \left (x\right ) + 64 \, x^{2} \log \left (5\right ) \log \left (x\right )^{3} - {\left (x^{4} \log \left (5\right )^{2} - 256 \, x^{2} \log \left (5\right ) \log \left (2\right )\right )} \log \left (x\right )^{2}}{64 \, {\left (16 \, \log \left (2\right )^{4} + 32 \, \log \left (2\right )^{3} \log \left (x\right ) + 24 \, \log \left (2\right )^{2} \log \left (x\right )^{2} + 8 \, \log \left (2\right ) \log \left (x\right )^{3} + \log \left (x\right )^{4}\right )}} \]
integrate(1/128*((-256*x*log(5)*log(x)-128*x*log(5))*log(4*x)^3+256*x*log( 5)*log(x)*log(4*x)^2+(8*x^3*log(5)^2*log(x)^2+4*x^3*log(5)^2*log(x))*log(4 *x)-8*x^3*log(5)^2*log(x)^2)/log(4*x)^5,x, algorithm=\
-1/64*(256*x^2*log(5)*log(2)^2*log(x) + 64*x^2*log(5)*log(x)^3 - (x^4*log( 5)^2 - 256*x^2*log(5)*log(2))*log(x)^2)/(16*log(2)^4 + 32*log(2)^3*log(x) + 24*log(2)^2*log(x)^2 + 8*log(2)*log(x)^3 + log(x)^4)
Leaf count of result is larger than twice the leaf count of optimal. 207 vs. \(2 (20) = 40\).
Time = 0.29 (sec) , antiderivative size = 207, normalized size of antiderivative = 9.86 \[ \int \frac {-2 x^3 \log ^2(25) \log ^2(x)+\left (x^3 \log ^2(25) \log (x)+2 x^3 \log ^2(25) \log ^2(x)\right ) \log (4 x)+128 x \log (25) \log (x) \log ^2(4 x)+(-64 x \log (25)-128 x \log (25) \log (x)) \log ^3(4 x)}{128 \log ^5(4 x)} \, dx=\frac {x^{4} \log \left (5\right )^{2} \log \left (x\right )^{2}}{64 \, {\left (16 \, \log \left (2\right )^{4} + 32 \, \log \left (2\right )^{3} \log \left (x\right ) + 24 \, \log \left (2\right )^{2} \log \left (x\right )^{2} + 8 \, \log \left (2\right ) \log \left (x\right )^{3} + \log \left (x\right )^{4}\right )}} - \frac {4 \, x^{2} \log \left (5\right ) \log \left (2\right )^{2} \log \left (x\right )}{16 \, \log \left (2\right )^{4} + 32 \, \log \left (2\right )^{3} \log \left (x\right ) + 24 \, \log \left (2\right )^{2} \log \left (x\right )^{2} + 8 \, \log \left (2\right ) \log \left (x\right )^{3} + \log \left (x\right )^{4}} - \frac {4 \, x^{2} \log \left (5\right ) \log \left (2\right ) \log \left (x\right )^{2}}{16 \, \log \left (2\right )^{4} + 32 \, \log \left (2\right )^{3} \log \left (x\right ) + 24 \, \log \left (2\right )^{2} \log \left (x\right )^{2} + 8 \, \log \left (2\right ) \log \left (x\right )^{3} + \log \left (x\right )^{4}} - \frac {x^{2} \log \left (5\right ) \log \left (x\right )^{3}}{16 \, \log \left (2\right )^{4} + 32 \, \log \left (2\right )^{3} \log \left (x\right ) + 24 \, \log \left (2\right )^{2} \log \left (x\right )^{2} + 8 \, \log \left (2\right ) \log \left (x\right )^{3} + \log \left (x\right )^{4}} \]
integrate(1/128*((-256*x*log(5)*log(x)-128*x*log(5))*log(4*x)^3+256*x*log( 5)*log(x)*log(4*x)^2+(8*x^3*log(5)^2*log(x)^2+4*x^3*log(5)^2*log(x))*log(4 *x)-8*x^3*log(5)^2*log(x)^2)/log(4*x)^5,x, algorithm=\
1/64*x^4*log(5)^2*log(x)^2/(16*log(2)^4 + 32*log(2)^3*log(x) + 24*log(2)^2 *log(x)^2 + 8*log(2)*log(x)^3 + log(x)^4) - 4*x^2*log(5)*log(2)^2*log(x)/( 16*log(2)^4 + 32*log(2)^3*log(x) + 24*log(2)^2*log(x)^2 + 8*log(2)*log(x)^ 3 + log(x)^4) - 4*x^2*log(5)*log(2)*log(x)^2/(16*log(2)^4 + 32*log(2)^3*lo g(x) + 24*log(2)^2*log(x)^2 + 8*log(2)*log(x)^3 + log(x)^4) - x^2*log(5)*l og(x)^3/(16*log(2)^4 + 32*log(2)^3*log(x) + 24*log(2)^2*log(x)^2 + 8*log(2 )*log(x)^3 + log(x)^4)
Time = 12.90 (sec) , antiderivative size = 1277, normalized size of antiderivative = 60.81 \[ \int \frac {-2 x^3 \log ^2(25) \log ^2(x)+\left (x^3 \log ^2(25) \log (x)+2 x^3 \log ^2(25) \log ^2(x)\right ) \log (4 x)+128 x \log (25) \log (x) \log ^2(4 x)+(-64 x \log (25)-128 x \log (25) \log (x)) \log ^3(4 x)}{128 \log ^5(4 x)} \, dx=\text {Too large to display} \]
int(((log(4*x)*(8*x^3*log(5)^2*log(x)^2 + 4*x^3*log(5)^2*log(x)))/128 - (l og(4*x)^3*(128*x*log(5) + 256*x*log(5)*log(x)))/128 - (x^3*log(5)^2*log(x) ^2)/16 + 2*x*log(4*x)^2*log(5)*log(x))/log(4*x)^5,x)
(x^4*log(5)^2)/8 + log(x)^2*((x^4*log(5)^2)/6 - x^2*(4*log(5) + (4*log(5)* (log(4*x) - log(x)))/3)) - x^2*(2*log(5) + (4*log(5)*(log(4*x) - log(x))^2 )/3 + 4*log(5)*(log(4*x) - log(x))) + ((x*(x^3*log(5)^2 - 32*x*log(5)*(log (4*x) - log(x)) + 256*x*log(5)*(log(4*x) - log(x))^2 + 192*x*log(5)*(log(4 *x) - log(x))^3 - 6*x^3*log(5)^2*(log(4*x) - log(x))))/384 + (x*log(x)^2*( 12*x*log(5) - x^3*log(5)^2 + 68*x*log(5)*(log(4*x) - log(x)) + 24*x*log(5) *(log(4*x) - log(x))^2 - x^3*log(5)^2*(log(4*x) - log(x))))/24 + (x*log(x) *(x^3*log(5)^2 - 48*x*log(5) + 224*x*log(5)*(log(4*x) - log(x)) + 416*x*lo g(5)*(log(4*x) - log(x))^2 + 64*x*log(5)*(log(4*x) - log(x))^3 - 12*x^3*lo g(5)^2*(log(4*x) - log(x))))/192 + (x^2*log(5)*log(x)^4)/3 + (x*log(x)^3*( 28*x*log(5) - x^3*log(5)^2 + 24*x*log(5)*(log(4*x) - log(x))))/24)/(2*log( x)*(log(4*x) - log(x)) + log(x)^2 + (log(4*x) - log(x))^2) + ((x^2*log(5)* log(x)^4)/2 + (x^2*log(5)*(log(4*x) - log(x))^3)/4 + (x^2*log(5)*log(x)^2* (32*log(x) - 32*log(4*x) + 192*(log(4*x) - log(x))^2 + x^2*log(5) - 2*x^2* log(5)*(log(4*x) - log(x))))/128 - (x^2*log(5)*log(x)^3*(96*log(x) - 96*lo g(4*x) + x^2*log(5) + 16))/64 + (x^2*log(5)*log(x)*(log(4*x) - log(x))*(32 *log(4*x) - 32*log(x) + 64*(log(4*x) - log(x))^2 - x^2*log(5)))/128)/(log( x)^4 + (log(4*x) - log(x))^4 + 4*log(x)*(log(4*x) - log(x))^3 + 4*log(x)^3 *(log(4*x) - log(x)) + 6*log(x)^2*(log(4*x) - log(x))^2) + ((x*(3*x^3*log( 5)^2 - 48*x*log(5) + 192*x*log(5)*(log(4*x) - log(x)) + 672*x*log(5)*(l...