Integrand size = 94, antiderivative size = 26 \[ \int \frac {-4608-9216 x-4608 x^2+\left (-384-768 x-384 x^2\right ) \log (x) \log (4 x)+\left (384+768 x+384 x^2\right ) \log ^2(4 x)+\left (-2 x^2-x^3+\left (32+64 x+32 x^2\right ) \log (x)\right ) \log ^3(4 x)}{\left (x+2 x^2+x^3\right ) \log ^3(4 x)} \, dx=-x+\frac {x}{1+x}+16 \left (\log (x)+\frac {12}{\log (4 x)}\right )^2 \] Output:
x/(1+x)-x-4*(ln(x)+12/ln(4*x))*(-4*ln(x)-48/ln(4*x))
Time = 0.23 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38 \[ \int \frac {-4608-9216 x-4608 x^2+\left (-384-768 x-384 x^2\right ) \log (x) \log (4 x)+\left (384+768 x+384 x^2\right ) \log ^2(4 x)+\left (-2 x^2-x^3+\left (32+64 x+32 x^2\right ) \log (x)\right ) \log ^3(4 x)}{\left (x+2 x^2+x^3\right ) \log ^3(4 x)} \, dx=-384-x-\frac {1}{1+x}+16 \log ^2(x)+\frac {2304}{\log ^2(4 x)}+\frac {384 \log (x)}{\log (4 x)} \] Input:
Integrate[(-4608 - 9216*x - 4608*x^2 + (-384 - 768*x - 384*x^2)*Log[x]*Log [4*x] + (384 + 768*x + 384*x^2)*Log[4*x]^2 + (-2*x^2 - x^3 + (32 + 64*x + 32*x^2)*Log[x])*Log[4*x]^3)/((x + 2*x^2 + x^3)*Log[4*x]^3),x]
Output:
-384 - x - (1 + x)^(-1) + 16*Log[x]^2 + 2304/Log[4*x]^2 + (384*Log[x])/Log [4*x]
Time = 1.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.42, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2026, 2007, 7239, 2010, 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.
\(\displaystyle \int \frac {-4608 x^2+\left (384 x^2+768 x+384\right ) \log ^2(4 x)+\left (-384 x^2-768 x-384\right ) \log (x) \log (4 x)+\left (-x^3-2 x^2+\left (32 x^2+64 x+32\right ) \log (x)\right ) \log ^3(4 x)-9216 x-4608}{\left (x^3+2 x^2+x\right ) \log ^3(4 x)} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {-4608 x^2+\left (384 x^2+768 x+384\right ) \log ^2(4 x)+\left (-384 x^2-768 x-384\right ) \log (x) \log (4 x)+\left (-x^3-2 x^2+\left (32 x^2+64 x+32\right ) \log (x)\right ) \log ^3(4 x)-9216 x-4608}{x \left (x^2+2 x+1\right ) \log ^3(4 x)}dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {-4608 x^2+\left (384 x^2+768 x+384\right ) \log ^2(4 x)+\left (-384 x^2-768 x-384\right ) \log (x) \log (4 x)+\left (-x^3-2 x^2+\left (32 x^2+64 x+32\right ) \log (x)\right ) \log ^3(4 x)-9216 x-4608}{x (x+1)^2 \log ^3(4 x)}dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {-\frac {(x+2) x^2}{(x+1)^2}-\frac {4608}{\log ^3(4 x)}+\log (x) \left (32-\frac {384}{\log ^2(4 x)}\right )+\frac {384}{\log (4 x)}}{x}dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (\frac {-x^3-2 x^2+32 x^2 \log (x)+64 x \log (x)+32 \log (x)}{x (x+1)^2}-\frac {4608}{x \log ^3(4 x)}-\frac {384 \log (x)}{x \log ^2(4 x)}+\frac {384}{x \log (4 x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {(x+2)^2}{x+1}+16 \log ^2(x)+\frac {2304}{\log ^2(4 x)}+\frac {384 \log (x)}{\log (4 x)}\) |
Input:
Int[(-4608 - 9216*x - 4608*x^2 + (-384 - 768*x - 384*x^2)*Log[x]*Log[4*x] + (384 + 768*x + 384*x^2)*Log[4*x]^2 + (-2*x^2 - x^3 + (32 + 64*x + 32*x^2 )*Log[x])*Log[4*x]^3)/((x + 2*x^2 + x^3)*Log[4*x]^3),x]
Output:
-((2 + x)^2/(1 + x)) + 16*Log[x]^2 + 2304/Log[4*x]^2 + (384*Log[x])/Log[4* x]
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
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]]
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(131\) vs. \(2(37)=74\).
Time = 0.12 (sec) , antiderivative size = 132, normalized size of antiderivative = 5.08
\[\frac {\left (-256 \ln \left (2\right )^{3}-768 \ln \left (2\right )\right ) \ln \left (x \right )+\left (-256 \ln \left (2\right )^{4}-1536 \ln \left (2\right )^{2}+2304\right ) x +\left (-256 \ln \left (2\right )^{3}-768 \ln \left (2\right )\right ) \ln \left (x \right ) x +16 \ln \left (x \right )^{4}+16 x \ln \left (x \right )^{4}-x^{2} \ln \left (x \right )^{2}-4 x^{2} \ln \left (2\right )^{2}+64 \ln \left (2\right ) \ln \left (x \right )^{3}+64 x \ln \left (x \right )^{3} \ln \left (2\right )-4 x^{2} \ln \left (2\right ) \ln \left (x \right )-256 \ln \left (2\right )^{4}+2304-1536 \ln \left (2\right )^{2}}{\left (1+x \right ) \left (\ln \left (x \right )+2 \ln \left (2\right )\right )^{2}}\]
Input:
int((((32*x^2+64*x+32)*ln(x)-x^3-2*x^2)*ln(4*x)^3+(384*x^2+768*x+384)*ln(4 *x)^2+(-384*x^2-768*x-384)*ln(x)*ln(4*x)-4608*x^2-9216*x-4608)/(x^3+2*x^2+ x)/ln(4*x)^3,x)
Output:
((-256*ln(2)^3-768*ln(2))*ln(x)+(-256*ln(2)^4-1536*ln(2)^2+2304)*x+(-256*l n(2)^3-768*ln(2))*ln(x)*x+16*ln(x)^4+16*x*ln(x)^4-x^2*ln(x)^2-4*x^2*ln(2)^ 2+64*ln(2)*ln(x)^3+64*x*ln(x)^3*ln(2)-4*x^2*ln(2)*ln(x)-256*ln(2)^4+2304-1 536*ln(2)^2)/(1+x)/(ln(x)+2*ln(2))^2
Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (26) = 52\).
Time = 0.08 (sec) , antiderivative size = 107, normalized size of antiderivative = 4.12 \[ \int \frac {-4608-9216 x-4608 x^2+\left (-384-768 x-384 x^2\right ) \log (x) \log (4 x)+\left (384+768 x+384 x^2\right ) \log ^2(4 x)+\left (-2 x^2-x^3+\left (32+64 x+32 x^2\right ) \log (x)\right ) \log ^3(4 x)}{\left (x+2 x^2+x^3\right ) \log ^3(4 x)} \, dx=\frac {64 \, {\left (x + 1\right )} \log \left (2\right ) \log \left (x\right )^{3} + 16 \, {\left (x + 1\right )} \log \left (x\right )^{4} - 4 \, {\left (x^{2} + 385 \, x + 385\right )} \log \left (2\right )^{2} - 4 \, {\left (x^{2} + 193 \, x + 193\right )} \log \left (2\right ) \log \left (x\right ) + {\left (64 \, {\left (x + 1\right )} \log \left (2\right )^{2} - x^{2} - x - 1\right )} \log \left (x\right )^{2} + 2304 \, x + 2304}{4 \, {\left (x + 1\right )} \log \left (2\right )^{2} + 4 \, {\left (x + 1\right )} \log \left (2\right ) \log \left (x\right ) + {\left (x + 1\right )} \log \left (x\right )^{2}} \] Input:
integrate((((32*x^2+64*x+32)*log(x)-x^3-2*x^2)*log(4*x)^3+(384*x^2+768*x+3 84)*log(4*x)^2+(-384*x^2-768*x-384)*log(x)*log(4*x)-4608*x^2-9216*x-4608)/ (x^3+2*x^2+x)/log(4*x)^3,x, algorithm="fricas")
Output:
(64*(x + 1)*log(2)*log(x)^3 + 16*(x + 1)*log(x)^4 - 4*(x^2 + 385*x + 385)* log(2)^2 - 4*(x^2 + 193*x + 193)*log(2)*log(x) + (64*(x + 1)*log(2)^2 - x^ 2 - x - 1)*log(x)^2 + 2304*x + 2304)/(4*(x + 1)*log(2)^2 + 4*(x + 1)*log(2 )*log(x) + (x + 1)*log(x)^2)
Time = 0.14 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.88 \[ \int \frac {-4608-9216 x-4608 x^2+\left (-384-768 x-384 x^2\right ) \log (x) \log (4 x)+\left (384+768 x+384 x^2\right ) \log ^2(4 x)+\left (-2 x^2-x^3+\left (32+64 x+32 x^2\right ) \log (x)\right ) \log ^3(4 x)}{\left (x+2 x^2+x^3\right ) \log ^3(4 x)} \, dx=- x + \frac {- 768 \log {\left (2 \right )} \log {\left (x \right )} - 1536 \log {\left (2 \right )}^{2} + 2304}{\log {\left (x \right )}^{2} + 4 \log {\left (2 \right )} \log {\left (x \right )} + 4 \log {\left (2 \right )}^{2}} + 16 \log {\left (x \right )}^{2} - \frac {1}{x + 1} \] Input:
integrate((((32*x**2+64*x+32)*ln(x)-x**3-2*x**2)*ln(4*x)**3+(384*x**2+768* x+384)*ln(4*x)**2+(-384*x**2-768*x-384)*ln(x)*ln(4*x)-4608*x**2-9216*x-460 8)/(x**3+2*x**2+x)/ln(4*x)**3,x)
Output:
-x + (-768*log(2)*log(x) - 1536*log(2)**2 + 2304)/(log(x)**2 + 4*log(2)*lo g(x) + 4*log(2)**2) + 16*log(x)**2 - 1/(x + 1)
Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (26) = 52\).
Time = 0.16 (sec) , antiderivative size = 134, normalized size of antiderivative = 5.15 \[ \int \frac {-4608-9216 x-4608 x^2+\left (-384-768 x-384 x^2\right ) \log (x) \log (4 x)+\left (384+768 x+384 x^2\right ) \log ^2(4 x)+\left (-2 x^2-x^3+\left (32+64 x+32 x^2\right ) \log (x)\right ) \log ^3(4 x)}{\left (x+2 x^2+x^3\right ) \log ^3(4 x)} \, dx=\frac {16 \, {\left (x + 1\right )} \log \left (x\right )^{4} - 4 \, x^{2} \log \left (2\right )^{2} + 64 \, {\left (x \log \left (2\right ) + \log \left (2\right )\right )} \log \left (x\right )^{3} + {\left ({\left (64 \, \log \left (2\right )^{2} - 1\right )} x - x^{2} + 64 \, \log \left (2\right )^{2} - 1\right )} \log \left (x\right )^{2} - 4 \, {\left (385 \, \log \left (2\right )^{2} - 576\right )} x - 1540 \, \log \left (2\right )^{2} - 4 \, {\left (x^{2} \log \left (2\right ) + 193 \, x \log \left (2\right ) + 193 \, \log \left (2\right )\right )} \log \left (x\right ) + 2304}{4 \, x \log \left (2\right )^{2} + {\left (x + 1\right )} \log \left (x\right )^{2} + 4 \, \log \left (2\right )^{2} + 4 \, {\left (x \log \left (2\right ) + \log \left (2\right )\right )} \log \left (x\right )} \] Input:
integrate((((32*x^2+64*x+32)*log(x)-x^3-2*x^2)*log(4*x)^3+(384*x^2+768*x+3 84)*log(4*x)^2+(-384*x^2-768*x-384)*log(x)*log(4*x)-4608*x^2-9216*x-4608)/ (x^3+2*x^2+x)/log(4*x)^3,x, algorithm="maxima")
Output:
(16*(x + 1)*log(x)^4 - 4*x^2*log(2)^2 + 64*(x*log(2) + log(2))*log(x)^3 + ((64*log(2)^2 - 1)*x - x^2 + 64*log(2)^2 - 1)*log(x)^2 - 4*(385*log(2)^2 - 576)*x - 1540*log(2)^2 - 4*(x^2*log(2) + 193*x*log(2) + 193*log(2))*log(x ) + 2304)/(4*x*log(2)^2 + (x + 1)*log(x)^2 + 4*log(2)^2 + 4*(x*log(2) + lo g(2))*log(x))
Time = 0.14 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.96 \[ \int \frac {-4608-9216 x-4608 x^2+\left (-384-768 x-384 x^2\right ) \log (x) \log (4 x)+\left (384+768 x+384 x^2\right ) \log ^2(4 x)+\left (-2 x^2-x^3+\left (32+64 x+32 x^2\right ) \log (x)\right ) \log ^3(4 x)}{\left (x+2 x^2+x^3\right ) \log ^3(4 x)} \, dx=16 \, \log \left (x\right )^{2} - x - \frac {768 \, {\left (2 \, \log \left (2\right )^{2} + \log \left (2\right ) \log \left (x\right ) - 3\right )}}{4 \, \log \left (2\right )^{2} + 4 \, \log \left (2\right ) \log \left (x\right ) + \log \left (x\right )^{2}} - \frac {1}{x + 1} \] Input:
integrate((((32*x^2+64*x+32)*log(x)-x^3-2*x^2)*log(4*x)^3+(384*x^2+768*x+3 84)*log(4*x)^2+(-384*x^2-768*x-384)*log(x)*log(4*x)-4608*x^2-9216*x-4608)/ (x^3+2*x^2+x)/log(4*x)^3,x, algorithm="giac")
Output:
16*log(x)^2 - x - 768*(2*log(2)^2 + log(2)*log(x) - 3)/(4*log(2)^2 + 4*log (2)*log(x) + log(x)^2) - 1/(x + 1)
Time = 3.13 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.65 \[ \int \frac {-4608-9216 x-4608 x^2+\left (-384-768 x-384 x^2\right ) \log (x) \log (4 x)+\left (384+768 x+384 x^2\right ) \log ^2(4 x)+\left (-2 x^2-x^3+\left (32+64 x+32 x^2\right ) \log (x)\right ) \log ^3(4 x)}{\left (x+2 x^2+x^3\right ) \log ^3(4 x)} \, dx=16\,{\ln \left (x\right )}^2-\frac {192\,\ln \left (x\right )\,\left (\ln \left (4\,x\right )-\ln \left (x\right )\right )+192\,{\left (\ln \left (4\,x\right )-\ln \left (x\right )\right )}^2-2304}{2\,\ln \left (x\right )\,\left (\ln \left (4\,x\right )-\ln \left (x\right )\right )+{\ln \left (x\right )}^2+{\left (\ln \left (4\,x\right )-\ln \left (x\right )\right )}^2}-\frac {1}{x+1}-x-\frac {192\,\left (\ln \left (4\,x\right )-\ln \left (x\right )\right )}{\ln \left (4\,x\right )} \] Input:
int(-(9216*x + log(4*x)^3*(2*x^2 - log(x)*(64*x + 32*x^2 + 32) + x^3) - lo g(4*x)^2*(768*x + 384*x^2 + 384) + 4608*x^2 + log(4*x)*log(x)*(768*x + 384 *x^2 + 384) + 4608)/(log(4*x)^3*(x + 2*x^2 + x^3)),x)
Output:
16*log(x)^2 - (192*log(x)*(log(4*x) - log(x)) + 192*(log(4*x) - log(x))^2 - 2304)/(2*log(x)*(log(4*x) - log(x)) + log(x)^2 + (log(4*x) - log(x))^2) - 1/(x + 1) - x - (192*(log(4*x) - log(x)))/log(4*x)
Time = 0.21 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.69 \[ \int \frac {-4608-9216 x-4608 x^2+\left (-384-768 x-384 x^2\right ) \log (x) \log (4 x)+\left (384+768 x+384 x^2\right ) \log ^2(4 x)+\left (-2 x^2-x^3+\left (32+64 x+32 x^2\right ) \log (x)\right ) \log ^3(4 x)}{\left (x+2 x^2+x^3\right ) \log ^3(4 x)} \, dx=\frac {16 \mathrm {log}\left (4 x \right )^{2} \mathrm {log}\left (x \right )^{2} x +16 \mathrm {log}\left (4 x \right )^{2} \mathrm {log}\left (x \right )^{2}-\mathrm {log}\left (4 x \right )^{2} x^{2}+384 \,\mathrm {log}\left (4 x \right ) \mathrm {log}\left (x \right ) x +384 \,\mathrm {log}\left (4 x \right ) \mathrm {log}\left (x \right )+2304 x +2304}{\mathrm {log}\left (4 x \right )^{2} \left (x +1\right )} \] Input:
int((((32*x^2+64*x+32)*log(x)-x^3-2*x^2)*log(4*x)^3+(384*x^2+768*x+384)*lo g(4*x)^2+(-384*x^2-768*x-384)*log(x)*log(4*x)-4608*x^2-9216*x-4608)/(x^3+2 *x^2+x)/log(4*x)^3,x)
Output:
(16*log(4*x)**2*log(x)**2*x + 16*log(4*x)**2*log(x)**2 - log(4*x)**2*x**2 + 384*log(4*x)*log(x)*x + 384*log(4*x)*log(x) + 2304*x + 2304)/(log(4*x)** 2*(x + 1))