\(\int \frac {-4608-9216 x-4608 x^2+(-384-768 x-384 x^2) \log (x) \log (4 x)+(384+768 x+384 x^2) \log ^2(4 x)+(-2 x^2-x^3+(32+64 x+32 x^2) \log (x)) \log ^3(4 x)}{(x+2 x^2+x^3) \log ^3(4 x)} \, dx\) [7243]

Optimal result

Integrand size = 94, antiderivative size = 26 \[ \int \frac {-4608-9216 x-4608 x^2+\left (-384-768 x-384 x^2\right ) \log (x) \log (4 x)+\left (384+768 x+384 x^2\right ) \log ^2(4 x)+\left (-2 x^2-x^3+\left (32+64 x+32 x^2\right ) \log (x)\right ) \log ^3(4 x)}{\left (x+2 x^2+x^3\right ) \log ^3(4 x)} \, dx=-x+\frac {x}{1+x}+16 \left (\log (x)+\frac {12}{\log (4 x)}\right )^2 \]

[Out]

x/(1+x)-x-4*(ln(x)+12/ln(4*x))*(-4*ln(x)-48/ln(4*x))

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.42, number of steps used = 16, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.106, Rules used = {1608, 27, 6820, 14, 75, 2338, 2339, 30, 2413, 29} \[ \int \frac {-4608-9216 x-4608 x^2+\left (-384-768 x-384 x^2\right ) \log (x) \log (4 x)+\left (384+768 x+384 x^2\right ) \log ^2(4 x)+\left (-2 x^2-x^3+\left (32+64 x+32 x^2\right ) \log (x)\right ) \log ^3(4 x)}{\left (x+2 x^2+x^3\right ) \log ^3(4 x)} \, dx=-\frac {(x+2)^2}{x+1}+16 \log ^2(x)+\frac {2304}{\log ^2(4 x)}+\frac {384 \log (x)}{\log (4 x)} \]

[In]

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]

[Out]

-((2 + x)^2/(1 + x)) + 16*Log[x]^2 + 2304/Log[4*x]^2 + (384*Log[x])/Log[4*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 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 75

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rule 1608

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

Rule 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2339

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2413

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 6820

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

Rubi steps \begin{align*} \text {integral}& = \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)}{x \left (1+2 x+x^2\right ) \log ^3(4 x)} \, dx \\ & = \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)}{x (1+x)^2 \log ^3(4 x)} \, dx \\ & = \int \frac {-\frac {x^2 (2+x)}{(1+x)^2}+\log (x) \left (32-\frac {384}{\log ^2(4 x)}\right )-\frac {4608}{\log ^3(4 x)}+\frac {384}{\log (4 x)}}{x} \, dx \\ & = \int \left (\frac {-2 x^2-x^3+32 \log (x)+64 x \log (x)+32 x^2 \log (x)}{x (1+x)^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 \\ & = -\left (384 \int \frac {\log (x)}{x \log ^2(4 x)} \, dx\right )+384 \int \frac {1}{x \log (4 x)} \, dx-4608 \int \frac {1}{x \log ^3(4 x)} \, dx+\int \frac {-2 x^2-x^3+32 \log (x)+64 x \log (x)+32 x^2 \log (x)}{x (1+x)^2} \, dx \\ & = \frac {384 \log (x)}{\log (4 x)}-384 \int \frac {1}{x \log (4 x)} \, dx+384 \text {Subst}\left (\int \frac {1}{x} \, dx,x,\log (4 x)\right )-4608 \text {Subst}\left (\int \frac {1}{x^3} \, dx,x,\log (4 x)\right )+\int \left (-\frac {x (2+x)}{(1+x)^2}+\frac {32 \log (x)}{x}\right ) \, dx \\ & = \frac {2304}{\log ^2(4 x)}+\frac {384 \log (x)}{\log (4 x)}+384 \log (\log (4 x))+32 \int \frac {\log (x)}{x} \, dx-384 \text {Subst}\left (\int \frac {1}{x} \, dx,x,\log (4 x)\right )-\int \frac {x (2+x)}{(1+x)^2} \, dx \\ & = -\frac {(2+x)^2}{1+x}+16 \log ^2(x)+\frac {2304}{\log ^2(4 x)}+\frac {384 \log (x)}{\log (4 x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38 \[ \int \frac {-4608-9216 x-4608 x^2+\left (-384-768 x-384 x^2\right ) \log (x) \log (4 x)+\left (384+768 x+384 x^2\right ) \log ^2(4 x)+\left (-2 x^2-x^3+\left (32+64 x+32 x^2\right ) \log (x)\right ) \log ^3(4 x)}{\left (x+2 x^2+x^3\right ) \log ^3(4 x)} \, dx=-384-x-\frac {1}{1+x}+16 \log ^2(x)+\frac {2304}{\log ^2(4 x)}+\frac {384 \log (x)}{\log (4 x)} \]

[In]

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]

[Out]

-384 - x - (1 + x)^(-1) + 16*Log[x]^2 + 2304/Log[4*x]^2 + (384*Log[x])/Log[4*x]

Maple [A] (verified)

Time = 1.25 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.85

method	result	size
risch	\(16 \ln \left (x \right )^{2}-\frac {x^{2}+x +1}{1+x}-\frac {384 \left (-24+16 \ln \left (2\right )^{2}+8 \ln \left (2\right ) \ln \left (x \right )\right )}{\left (2 \ln \left (x \right )+4 \ln \left (2\right )\right )^{2}}\)	\(48\)
parts	\(16 \ln \left (x \right )^{2}-384 \ln \left (\ln \left (x \right )+2 \ln \left (2\right )\right )-\frac {768 \ln \left (2\right )}{\ln \left (x \right )+2 \ln \left (2\right )}+\frac {2304}{\ln \left (4 x \right )^{2}}+384 \ln \left (\ln \left (4 x \right )\right )-x -\frac {1}{1+x}\)	\(56\)
parallelrisch	\(-\frac {-13824-2304 \ln \left (x \right ) \ln \left (4 x \right )-96 \ln \left (4 x \right )^{2} \ln \left (x \right )^{2}-13824 x +6 x^{2} \ln \left (4 x \right )^{2}-2304 \ln \left (x \right ) \ln \left (4 x \right ) x -96 \ln \left (4 x \right )^{2} x \ln \left (x \right )^{2}}{6 \ln \left (4 x \right )^{2} \left (1+x \right )}\)	\(72\)
default	\(\frac {\left (-256 \ln \left (2\right )^{3}-768 \ln \left (2\right )\right ) \ln \left (x \right )+\left (-256 \ln \left (2\right )^{4}-1536 \ln \left (2\right )^{2}+2304\right ) x +\left (-256 \ln \left (2\right )^{3}-768 \ln \left (2\right )\right ) \ln \left (x \right ) x +16 \ln \left (x \right )^{4}+16 x \ln \left (x \right )^{4}-x^{2} \ln \left (x \right )^{2}-4 x^{2} \ln \left (2\right )^{2}+64 \ln \left (2\right ) \ln \left (x \right )^{3}-4 x^{2} \ln \left (2\right ) \ln \left (x \right )+64 \ln \left (2\right ) x \ln \left (x \right )^{3}-256 \ln \left (2\right )^{4}[phone] \ln \left (2\right )^{2}}{\left (1+x \right ) \left (\ln \left (x \right )+2 \ln \left (2\right )\right )^{2}}\)	\(132\)

[In]

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,method=_RETURNVERBOSE)

[Out]

16*ln(x)^2-(x^2+x+1)/(1+x)-384*(-24+16*ln(2)^2+8*ln(2)*ln(x))/(2*ln(x)+4*ln(2))^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (26) = 52\).

Time = 0.26 (sec) , antiderivative size = 107, normalized size of antiderivative = 4.12 \[ \int \frac {-4608-9216 x-4608 x^2+\left (-384-768 x-384 x^2\right ) \log (x) \log (4 x)+\left (384+768 x+384 x^2\right ) \log ^2(4 x)+\left (-2 x^2-x^3+\left (32+64 x+32 x^2\right ) \log (x)\right ) \log ^3(4 x)}{\left (x+2 x^2+x^3\right ) \log ^3(4 x)} \, dx=\frac {64 \, {\left (x + 1\right )} \log \left (2\right ) \log \left (x\right )^{3} + 16 \, {\left (x + 1\right )} \log \left (x\right )^{4} - 4 \, {\left (x^{2} + 385 \, x + 385\right )} \log \left (2\right )^{2} - 4 \, {\left (x^{2} + 193 \, x + 193\right )} \log \left (2\right ) \log \left (x\right ) + {\left (64 \, {\left (x + 1\right )} \log \left (2\right )^{2} - x^{2} - x - 1\right )} \log \left (x\right )^{2} + 2304 \, x + 2304}{4 \, {\left (x + 1\right )} \log \left (2\right )^{2} + 4 \, {\left (x + 1\right )} \log \left (2\right ) \log \left (x\right ) + {\left (x + 1\right )} \log \left (x\right )^{2}} \]

[In]

integrate((((32*x^2+64*x+32)*log(x)-x^3-2*x^2)*log(4*x)^3+(384*x^2+768*x+384)*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")

[Out]

(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)

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.88 \[ \int \frac {-4608-9216 x-4608 x^2+\left (-384-768 x-384 x^2\right ) \log (x) \log (4 x)+\left (384+768 x+384 x^2\right ) \log ^2(4 x)+\left (-2 x^2-x^3+\left (32+64 x+32 x^2\right ) \log (x)\right ) \log ^3(4 x)}{\left (x+2 x^2+x^3\right ) \log ^3(4 x)} \, dx=- x + \frac {- 768 \log {\left (2 \right )} \log {\left (x \right )} - 1536 \log {\left (2 \right )}^{2} + 2304}{\log {\left (x \right )}^{2} + 4 \log {\left (2 \right )} \log {\left (x \right )} + 4 \log {\left (2 \right )}^{2}} + 16 \log {\left (x \right )}^{2} - \frac {1}{x + 1} \]

[In]

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-3
84)*ln(x)*ln(4*x)-4608*x**2-9216*x-4608)/(x**3+2*x**2+x)/ln(4*x)**3,x)

[Out]

-x + (-768*log(2)*log(x) - 1536*log(2)**2 + 2304)/(log(x)**2 + 4*log(2)*log(x) + 4*log(2)**2) + 16*log(x)**2 -
1/(x + 1)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (26) = 52\).

Time = 0.31 (sec) , antiderivative size = 134, normalized size of antiderivative = 5.15 \[ \int \frac {-4608-9216 x-4608 x^2+\left (-384-768 x-384 x^2\right ) \log (x) \log (4 x)+\left (384+768 x+384 x^2\right ) \log ^2(4 x)+\left (-2 x^2-x^3+\left (32+64 x+32 x^2\right ) \log (x)\right ) \log ^3(4 x)}{\left (x+2 x^2+x^3\right ) \log ^3(4 x)} \, dx=\frac {16 \, {\left (x + 1\right )} \log \left (x\right )^{4} - 4 \, x^{2} \log \left (2\right )^{2} + 64 \, {\left (x \log \left (2\right ) + \log \left (2\right )\right )} \log \left (x\right )^{3} + {\left ({\left (64 \, \log \left (2\right )^{2} - 1\right )} x - x^{2} + 64 \, \log \left (2\right )^{2} - 1\right )} \log \left (x\right )^{2} - 4 \, {\left (385 \, \log \left (2\right )^{2} - 576\right )} x - 1540 \, \log \left (2\right )^{2} - 4 \, {\left (x^{2} \log \left (2\right ) + 193 \, x \log \left (2\right ) + 193 \, \log \left (2\right )\right )} \log \left (x\right ) + 2304}{4 \, x \log \left (2\right )^{2} + {\left (x + 1\right )} \log \left (x\right )^{2} + 4 \, \log \left (2\right )^{2} + 4 \, {\left (x \log \left (2\right ) + \log \left (2\right )\right )} \log \left (x\right )} \]

[In]

integrate((((32*x^2+64*x+32)*log(x)-x^3-2*x^2)*log(4*x)^3+(384*x^2+768*x+384)*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")

[Out]

(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) + log(2))*log(x))

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.96 \[ \int \frac {-4608-9216 x-4608 x^2+\left (-384-768 x-384 x^2\right ) \log (x) \log (4 x)+\left (384+768 x+384 x^2\right ) \log ^2(4 x)+\left (-2 x^2-x^3+\left (32+64 x+32 x^2\right ) \log (x)\right ) \log ^3(4 x)}{\left (x+2 x^2+x^3\right ) \log ^3(4 x)} \, dx=16 \, \log \left (x\right )^{2} - x - \frac {768 \, {\left (2 \, \log \left (2\right )^{2} + \log \left (2\right ) \log \left (x\right ) - 3\right )}}{4 \, \log \left (2\right )^{2} + 4 \, \log \left (2\right ) \log \left (x\right ) + \log \left (x\right )^{2}} - \frac {1}{x + 1} \]

[In]

integrate((((32*x^2+64*x+32)*log(x)-x^3-2*x^2)*log(4*x)^3+(384*x^2+768*x+384)*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")

[Out]

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)

Mupad [B] (verification not implemented)

Time = 12.13 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.65 \[ \int \frac {-4608-9216 x-4608 x^2+\left (-384-768 x-384 x^2\right ) \log (x) \log (4 x)+\left (384+768 x+384 x^2\right ) \log ^2(4 x)+\left (-2 x^2-x^3+\left (32+64 x+32 x^2\right ) \log (x)\right ) \log ^3(4 x)}{\left (x+2 x^2+x^3\right ) \log ^3(4 x)} \, dx=16\,{\ln \left (x\right )}^2-\frac {192\,\ln \left (x\right )\,\left (\ln \left (4\,x\right )-\ln \left (x\right )\right )+192\,{\left (\ln \left (4\,x\right )-\ln \left (x\right )\right )}^2-2304}{2\,\ln \left (x\right )\,\left (\ln \left (4\,x\right )-\ln \left (x\right )\right )+{\ln \left (x\right )}^2+{\left (\ln \left (4\,x\right )-\ln \left (x\right )\right )}^2}-\frac {1}{x+1}-x-\frac {192\,\left (\ln \left (4\,x\right )-\ln \left (x\right )\right )}{\ln \left (4\,x\right )} \]

[In]

int(-(9216*x + log(4*x)^3*(2*x^2 - log(x)*(64*x + 32*x^2 + 32) + x^3) - log(4*x)^2*(768*x + 384*x^2 + 384) + 4
608*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)

[Out]

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)