\(\int \frac {-40-8 x-8 x^2+(-40+2 x+14 x^2+6 x^3+4 x^4) \log (x)+(8+(8-2 x-4 x^2) \log (x)) \log (x^4 \log ^4(x))}{x \log (x)} \, dx\) [7605]

Optimal result

Integrand size = 66, antiderivative size = 19 \[ \int \frac {-40-8 x-8 x^2+\left (-40+2 x+14 x^2+6 x^3+4 x^4\right ) \log (x)+\left (8+\left (8-2 x-4 x^2\right ) \log (x)\right ) \log \left (x^4 \log ^4(x)\right )}{x \log (x)} \, dx=\left (5+x+x^2-\log \left (x^4 \log ^4(x)\right )\right )^2 \]

[Out]

(5-ln(x^4*ln(x)^4)+x+x^2)^2

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {6820, 12, 6818} \[ \int \frac {-40-8 x-8 x^2+\left (-40+2 x+14 x^2+6 x^3+4 x^4\right ) \log (x)+\left (8+\left (8-2 x-4 x^2\right ) \log (x)\right ) \log \left (x^4 \log ^4(x)\right )}{x \log (x)} \, dx=\left (-\log \left (x^4 \log ^4(x)\right )+x^2+x+5\right )^2 \]

[In]

Int[(-40 - 8*x - 8*x^2 + (-40 + 2*x + 14*x^2 + 6*x^3 + 4*x^4)*Log[x] + (8 + (8 - 2*x - 4*x^2)*Log[x])*Log[x^4*
Log[x]^4])/(x*Log[x]),x]

[Out]

(5 + x + x^2 - Log[x^4*Log[x]^4])^2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6818

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*(y^(m + 1)/(m + 1)), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

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 {2 \left (4-\left (-4+x+2 x^2\right ) \log (x)\right ) \left (-5-x-x^2+\log \left (x^4 \log ^4(x)\right )\right )}{x \log (x)} \, dx \\ & = 2 \int \frac {\left (4-\left (-4+x+2 x^2\right ) \log (x)\right ) \left (-5-x-x^2+\log \left (x^4 \log ^4(x)\right )\right )}{x \log (x)} \, dx \\ & = \left (5+x+x^2-\log \left (x^4 \log ^4(x)\right )\right )^2 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {-40-8 x-8 x^2+\left (-40+2 x+14 x^2+6 x^3+4 x^4\right ) \log (x)+\left (8+\left (8-2 x-4 x^2\right ) \log (x)\right ) \log \left (x^4 \log ^4(x)\right )}{x \log (x)} \, dx=\left (5+x+x^2-\log \left (x^4 \log ^4(x)\right )\right )^2 \]

[In]

Integrate[(-40 - 8*x - 8*x^2 + (-40 + 2*x + 14*x^2 + 6*x^3 + 4*x^4)*Log[x] + (8 + (8 - 2*x - 4*x^2)*Log[x])*Lo
g[x^4*Log[x]^4])/(x*Log[x]),x]

[Out]

(5 + x + x^2 - Log[x^4*Log[x]^4])^2

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(63\) vs. \(2(19)=38\).

Time = 8.51 (sec) , antiderivative size = 64, normalized size of antiderivative = 3.37

method	result	size
parallelrisch	\(x^{4}+2 x^{3}-2 \ln \left (x^{4} \ln \left (x \right )^{4}\right ) x^{2}+11 x^{2}-2 x \ln \left (x^{4} \ln \left (x \right )^{4}\right )+\ln \left (x^{4} \ln \left (x \right )^{4}\right )^{2}-40 \ln \left (x \right )-40 \ln \left (\ln \left (x \right )\right )+10 x\)	\(64\)
risch	\(\text {Expression too large to display}\)	\(13424\)

[In]

int((((-4*x^2-2*x+8)*ln(x)+8)*ln(x^4*ln(x)^4)+(4*x^4+6*x^3+14*x^2+2*x-40)*ln(x)-8*x^2-8*x-40)/x/ln(x),x,method
=_RETURNVERBOSE)

[Out]

x^4+2*x^3-2*ln(x^4*ln(x)^4)*x^2+11*x^2-2*x*ln(x^4*ln(x)^4)+ln(x^4*ln(x)^4)^2-40*ln(x)-40*ln(ln(x))+10*x

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (19) = 38\).

Time = 0.28 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.37 \[ \int \frac {-40-8 x-8 x^2+\left (-40+2 x+14 x^2+6 x^3+4 x^4\right ) \log (x)+\left (8+\left (8-2 x-4 x^2\right ) \log (x)\right ) \log \left (x^4 \log ^4(x)\right )}{x \log (x)} \, dx=x^{4} + 2 \, x^{3} + 11 \, x^{2} - 2 \, {\left (x^{2} + x + 5\right )} \log \left (x^{4} \log \left (x\right )^{4}\right ) + \log \left (x^{4} \log \left (x\right )^{4}\right )^{2} + 10 \, x \]

[In]

integrate((((-4*x^2-2*x+8)*log(x)+8)*log(x^4*log(x)^4)+(4*x^4+6*x^3+14*x^2+2*x-40)*log(x)-8*x^2-8*x-40)/x/log(
x),x, algorithm="fricas")

[Out]

x^4 + 2*x^3 + 11*x^2 - 2*(x^2 + x + 5)*log(x^4*log(x)^4) + log(x^4*log(x)^4)^2 + 10*x

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (17) = 34\).

Time = 0.17 (sec) , antiderivative size = 60, normalized size of antiderivative = 3.16 \[ \int \frac {-40-8 x-8 x^2+\left (-40+2 x+14 x^2+6 x^3+4 x^4\right ) \log (x)+\left (8+\left (8-2 x-4 x^2\right ) \log (x)\right ) \log \left (x^4 \log ^4(x)\right )}{x \log (x)} \, dx=x^{4} + 2 x^{3} + 11 x^{2} + 10 x + \left (- 2 x^{2} - 2 x\right ) \log {\left (x^{4} \log {\left (x \right )}^{4} \right )} - 40 \log {\left (x \right )} + \log {\left (x^{4} \log {\left (x \right )}^{4} \right )}^{2} - 40 \log {\left (\log {\left (x \right )} \right )} \]

[In]

integrate((((-4*x**2-2*x+8)*ln(x)+8)*ln(x**4*ln(x)**4)+(4*x**4+6*x**3+14*x**2+2*x-40)*ln(x)-8*x**2-8*x-40)/x/l
n(x),x)

[Out]

x**4 + 2*x**3 + 11*x**2 + 10*x + (-2*x**2 - 2*x)*log(x**4*log(x)**4) - 40*log(x) + log(x**4*log(x)**4)**2 - 40
*log(log(x))

Maxima [F]

\[ \int \frac {-40-8 x-8 x^2+\left (-40+2 x+14 x^2+6 x^3+4 x^4\right ) \log (x)+\left (8+\left (8-2 x-4 x^2\right ) \log (x)\right ) \log \left (x^4 \log ^4(x)\right )}{x \log (x)} \, dx=\int { -\frac {2 \, {\left (4 \, x^{2} + {\left ({\left (2 \, x^{2} + x - 4\right )} \log \left (x\right ) - 4\right )} \log \left (x^{4} \log \left (x\right )^{4}\right ) - {\left (2 \, x^{4} + 3 \, x^{3} + 7 \, x^{2} + x - 20\right )} \log \left (x\right ) + 4 \, x + 20\right )}}{x \log \left (x\right )} \,d x } \]

[In]

integrate((((-4*x^2-2*x+8)*log(x)+8)*log(x^4*log(x)^4)+(4*x^4+6*x^3+14*x^2+2*x-40)*log(x)-8*x^2-8*x-40)/x/log(
x),x, algorithm="maxima")

[Out]

x^4 + 2*x^3 + 11*x^2 - 8*(x^2 + x)*log(x) + 16*log(x)^2 - 8*(x^2 + x - 4*log(x))*log(log(x)) + 16*log(log(x))^
2 + 10*x - 8*Ei(2*log(x)) - 8*Ei(log(x)) + 2*integrate(4*(x + 1)/log(x), x) - 40*log(x) - 40*log(log(x))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (19) = 38\).

Time = 0.29 (sec) , antiderivative size = 64, normalized size of antiderivative = 3.37 \[ \int \frac {-40-8 x-8 x^2+\left (-40+2 x+14 x^2+6 x^3+4 x^4\right ) \log (x)+\left (8+\left (8-2 x-4 x^2\right ) \log (x)\right ) \log \left (x^4 \log ^4(x)\right )}{x \log (x)} \, dx=x^{4} + 2 \, x^{3} + 11 \, x^{2} - 2 \, {\left (x^{2} + x - 4 \, \log \left (x\right )\right )} \log \left (\log \left (x\right )^{4}\right ) + \log \left (\log \left (x\right )^{4}\right )^{2} - 8 \, {\left (x^{2} + x\right )} \log \left (x\right ) + 16 \, \log \left (x\right )^{2} + 10 \, x - 40 \, \log \left (x\right ) - 40 \, \log \left (\log \left (x\right )\right ) \]

[In]

integrate((((-4*x^2-2*x+8)*log(x)+8)*log(x^4*log(x)^4)+(4*x^4+6*x^3+14*x^2+2*x-40)*log(x)-8*x^2-8*x-40)/x/log(
x),x, algorithm="giac")

[Out]

x^4 + 2*x^3 + 11*x^2 - 2*(x^2 + x - 4*log(x))*log(log(x)^4) + log(log(x)^4)^2 - 8*(x^2 + x)*log(x) + 16*log(x)
^2 + 10*x - 40*log(x) - 40*log(log(x))

Mupad [B] (verification not implemented)

Time = 14.43 (sec) , antiderivative size = 57, normalized size of antiderivative = 3.00 \[ \int \frac {-40-8 x-8 x^2+\left (-40+2 x+14 x^2+6 x^3+4 x^4\right ) \log (x)+\left (8+\left (8-2 x-4 x^2\right ) \log (x)\right ) \log \left (x^4 \log ^4(x)\right )}{x \log (x)} \, dx=10\,x-40\,\ln \left (\ln \left (x\right )\right )-40\,\ln \left (x\right )+{\ln \left (x^4\,{\ln \left (x\right )}^4\right )}^2-\ln \left (x^4\,{\ln \left (x\right )}^4\right )\,\left (2\,x^2+2\,x\right )+11\,x^2+2\,x^3+x^4 \]

[In]

int(-(8*x - log(x)*(2*x + 14*x^2 + 6*x^3 + 4*x^4 - 40) + log(x^4*log(x)^4)*(log(x)*(2*x + 4*x^2 - 8) - 8) + 8*
x^2 + 40)/(x*log(x)),x)

[Out]

10*x - 40*log(log(x)) - 40*log(x) + log(x^4*log(x)^4)^2 - log(x^4*log(x)^4)*(2*x + 2*x^2) + 11*x^2 + 2*x^3 + x
^4