\(\int \frac {47+(10+47 x) \log (\frac {1}{2} (10+47 x)) \log (\frac {2}{\log (\frac {1}{2} (10+47 x))})}{(10 x^2+47 x^3) \log (\frac {1}{2} (10+47 x)) \log (\frac {2}{\log (\frac {1}{2} (10+47 x))})+(-20 x-94 x^2) \log (\frac {1}{2} (10+47 x)) \log (\frac {2}{\log (\frac {1}{2} (10+47 x))}) \log (\log (\frac {2}{\log (\frac {1}{2} (10+47 x))}))+(10+47 x) \log (\frac {1}{2} (10+47 x)) \log (\frac {2}{\log (\frac {1}{2} (10+47 x))}) \log ^2(\log (\frac {2}{\log (\frac {1}{2} (10+47 x))}))} \, dx\) [6780]

Optimal result

Integrand size = 174, antiderivative size = 20 \[ \int \frac {47+(10+47 x) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )}{\left (10 x^2+47 x^3\right ) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )+\left (-20 x-94 x^2\right ) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right ) \log \left (\log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )\right )+(10+47 x) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right ) \log ^2\left (\log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )\right )} \, dx=\frac {1}{-x+\log \left (\log \left (\frac {2}{\log \left (5+\frac {47 x}{2}\right )}\right )\right )} \]

[Out]

1/(ln(ln(2/ln(47/2*x+5)))-x)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.011, Rules used = {6820, 6818} \[ \int \frac {47+(10+47 x) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )}{\left (10 x^2+47 x^3\right ) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )+\left (-20 x-94 x^2\right ) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right ) \log \left (\log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )\right )+(10+47 x) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right ) \log ^2\left (\log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )\right )} \, dx=-\frac {1}{x-\log \left (\log \left (\frac {2}{\log \left (\frac {47 x}{2}+5\right )}\right )\right )} \]

[In]

Int[(47 + (10 + 47*x)*Log[(10 + 47*x)/2]*Log[2/Log[(10 + 47*x)/2]])/((10*x^2 + 47*x^3)*Log[(10 + 47*x)/2]*Log[
2/Log[(10 + 47*x)/2]] + (-20*x - 94*x^2)*Log[(10 + 47*x)/2]*Log[2/Log[(10 + 47*x)/2]]*Log[Log[2/Log[(10 + 47*x
)/2]]] + (10 + 47*x)*Log[(10 + 47*x)/2]*Log[2/Log[(10 + 47*x)/2]]*Log[Log[2/Log[(10 + 47*x)/2]]]^2),x]

[Out]

-(x - Log[Log[2/Log[5 + (47*x)/2]]])^(-1)

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 {1+\frac {47}{(10+47 x) \log \left (5+\frac {47 x}{2}\right ) \log \left (\frac {2}{\log \left (5+\frac {47 x}{2}\right )}\right )}}{\left (x-\log \left (\log \left (\frac {2}{\log \left (5+\frac {47 x}{2}\right )}\right )\right )\right )^2} \, dx \\ & = -\frac {1}{x-\log \left (\log \left (\frac {2}{\log \left (5+\frac {47 x}{2}\right )}\right )\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {47+(10+47 x) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )}{\left (10 x^2+47 x^3\right ) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )+\left (-20 x-94 x^2\right ) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right ) \log \left (\log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )\right )+(10+47 x) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right ) \log ^2\left (\log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )\right )} \, dx=-\frac {1}{x-\log \left (\log \left (\frac {2}{\log \left (5+\frac {47 x}{2}\right )}\right )\right )} \]

[In]

Integrate[(47 + (10 + 47*x)*Log[(10 + 47*x)/2]*Log[2/Log[(10 + 47*x)/2]])/((10*x^2 + 47*x^3)*Log[(10 + 47*x)/2
]*Log[2/Log[(10 + 47*x)/2]] + (-20*x - 94*x^2)*Log[(10 + 47*x)/2]*Log[2/Log[(10 + 47*x)/2]]*Log[Log[2/Log[(10
+ 47*x)/2]]] + (10 + 47*x)*Log[(10 + 47*x)/2]*Log[2/Log[(10 + 47*x)/2]]*Log[Log[2/Log[(10 + 47*x)/2]]]^2),x]

[Out]

-(x - Log[Log[2/Log[5 + (47*x)/2]]])^(-1)

Maple [A] (verified)

Time = 8.21 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10

method	result	size
risch	\(-\frac {1}{x -\ln \left (\ln \left (2\right )-\ln \left (\ln \left (\frac {47 x}{2}+5\right )\right )\right )}\)	\(22\)
parallelrisch	\(\frac {-940+2209 x -2209 \ln \left (\ln \left (\frac {2}{\ln \left (\frac {47 x}{2}+5\right )}\right )\right )}{940 x -940 \ln \left (\ln \left (\frac {2}{\ln \left (\frac {47 x}{2}+5\right )}\right )\right )}\)	\(40\)
default	\(-\frac {47}{47 x -47 \ln \left (\ln \left (2\right )+i \pi -\ln \left (\ln \left (2\right )-\ln \left (47 x +10\right )\right )+i \pi \operatorname {csgn}\left (\frac {i}{\ln \left (2\right )-\ln \left (47 x +10\right )}\right )^{2} \left (\operatorname {csgn}\left (\frac {i}{\ln \left (2\right )-\ln \left (47 x +10\right )}\right )-1\right )\right )}\)	\(75\)
parts	\(\text {Expression too large to display}\)	\(695\)

[In]

int(((47*x+10)*ln(47/2*x+5)*ln(2/ln(47/2*x+5))+47)/((47*x+10)*ln(47/2*x+5)*ln(2/ln(47/2*x+5))*ln(ln(2/ln(47/2*
x+5)))^2+(-94*x^2-20*x)*ln(47/2*x+5)*ln(2/ln(47/2*x+5))*ln(ln(2/ln(47/2*x+5)))+(47*x^3+10*x^2)*ln(47/2*x+5)*ln
(2/ln(47/2*x+5))),x,method=_RETURNVERBOSE)

[Out]

-1/(x-ln(ln(2)-ln(ln(47/2*x+5))))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {47+(10+47 x) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )}{\left (10 x^2+47 x^3\right ) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )+\left (-20 x-94 x^2\right ) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right ) \log \left (\log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )\right )+(10+47 x) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right ) \log ^2\left (\log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )\right )} \, dx=-\frac {1}{x - \log \left (\log \left (\frac {2}{\log \left (\frac {47}{2} \, x + 5\right )}\right )\right )} \]

[In]

integrate(((47*x+10)*log(47/2*x+5)*log(2/log(47/2*x+5))+47)/((47*x+10)*log(47/2*x+5)*log(2/log(47/2*x+5))*log(
log(2/log(47/2*x+5)))^2+(-94*x^2-20*x)*log(47/2*x+5)*log(2/log(47/2*x+5))*log(log(2/log(47/2*x+5)))+(47*x^3+10
*x^2)*log(47/2*x+5)*log(2/log(47/2*x+5))),x, algorithm="fricas")

[Out]

-1/(x - log(log(2/log(47/2*x + 5))))

Sympy [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \[ \int \frac {47+(10+47 x) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )}{\left (10 x^2+47 x^3\right ) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )+\left (-20 x-94 x^2\right ) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right ) \log \left (\log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )\right )+(10+47 x) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right ) \log ^2\left (\log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )\right )} \, dx=\frac {1}{- x + \log {\left (\log {\left (\frac {2}{\log {\left (\frac {47 x}{2} + 5 \right )}} \right )} \right )}} \]

[In]

integrate(((47*x+10)*ln(47/2*x+5)*ln(2/ln(47/2*x+5))+47)/((47*x+10)*ln(47/2*x+5)*ln(2/ln(47/2*x+5))*ln(ln(2/ln
(47/2*x+5)))**2+(-94*x**2-20*x)*ln(47/2*x+5)*ln(2/ln(47/2*x+5))*ln(ln(2/ln(47/2*x+5)))+(47*x**3+10*x**2)*ln(47
/2*x+5)*ln(2/ln(47/2*x+5))),x)

[Out]

1/(-x + log(log(2/log(47*x/2 + 5))))

Maxima [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.30 \[ \int \frac {47+(10+47 x) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )}{\left (10 x^2+47 x^3\right ) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )+\left (-20 x-94 x^2\right ) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right ) \log \left (\log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )\right )+(10+47 x) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right ) \log ^2\left (\log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )\right )} \, dx=-\frac {1}{x - \log \left (\log \left (2\right ) - \log \left (-\log \left (2\right ) + \log \left (47 \, x + 10\right )\right )\right )} \]

[In]

integrate(((47*x+10)*log(47/2*x+5)*log(2/log(47/2*x+5))+47)/((47*x+10)*log(47/2*x+5)*log(2/log(47/2*x+5))*log(
log(2/log(47/2*x+5)))^2+(-94*x^2-20*x)*log(47/2*x+5)*log(2/log(47/2*x+5))*log(log(2/log(47/2*x+5)))+(47*x^3+10
*x^2)*log(47/2*x+5)*log(2/log(47/2*x+5))),x, algorithm="maxima")

[Out]

-1/(x - log(log(2) - log(-log(2) + log(47*x + 10))))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 590 vs. \(2 (20) = 40\).

Time = 2.73 (sec) , antiderivative size = 590, normalized size of antiderivative = 29.50 \[ \int \frac {47+(10+47 x) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )}{\left (10 x^2+47 x^3\right ) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )+\left (-20 x-94 x^2\right ) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right ) \log \left (\log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )\right )+(10+47 x) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right ) \log ^2\left (\log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )\right )} \, dx=\text {Too large to display} \]

[In]

integrate(((47*x+10)*log(47/2*x+5)*log(2/log(47/2*x+5))+47)/((47*x+10)*log(47/2*x+5)*log(2/log(47/2*x+5))*log(
log(2/log(47/2*x+5)))^2+(-94*x^2-20*x)*log(47/2*x+5)*log(2/log(47/2*x+5))*log(log(2/log(47/2*x+5)))+(47*x^3+10
*x^2)*log(47/2*x+5)*log(2/log(47/2*x+5))),x, algorithm="giac")

[Out]

-(47*x*log(2)^2*log(47/2*x + 5) - 47*x*log(2)*log(47*x + 10)*log(47/2*x + 5) - 47*x*log(2)*log(47/2*x + 5)*log
(log(47/2*x + 5)) + 47*x*log(47*x + 10)*log(47/2*x + 5)*log(log(47/2*x + 5)) + 10*log(2)^2*log(47/2*x + 5) - 1
0*log(2)*log(47*x + 10)*log(47/2*x + 5) - 10*log(2)*log(47/2*x + 5)*log(log(47/2*x + 5)) + 10*log(47*x + 10)*l
og(47/2*x + 5)*log(log(47/2*x + 5)) - 47*log(47/2*x + 5))/(47*x^2*log(2)^2*log(47/2*x + 5) - 47*x^2*log(2)*log
(47*x + 10)*log(47/2*x + 5) - 47*x*log(2)^2*log(47/2*x + 5)*log(log(2) - log(log(47/2*x + 5))) + 47*x*log(2)*l
og(47*x + 10)*log(47/2*x + 5)*log(log(2) - log(log(47/2*x + 5))) - 47*x^2*log(2)*log(47/2*x + 5)*log(log(47/2*
x + 5)) + 47*x^2*log(47*x + 10)*log(47/2*x + 5)*log(log(47/2*x + 5)) + 47*x*log(2)*log(47/2*x + 5)*log(log(2)
- log(log(47/2*x + 5)))*log(log(47/2*x + 5)) - 47*x*log(47*x + 10)*log(47/2*x + 5)*log(log(2) - log(log(47/2*x
+ 5)))*log(log(47/2*x + 5)) + 10*x*log(2)^2*log(47/2*x + 5) - 10*x*log(2)*log(47*x + 10)*log(47/2*x + 5) - 10
*log(2)^2*log(47/2*x + 5)*log(log(2) - log(log(47/2*x + 5))) + 10*log(2)*log(47*x + 10)*log(47/2*x + 5)*log(lo
g(2) - log(log(47/2*x + 5))) - 10*x*log(2)*log(47/2*x + 5)*log(log(47/2*x + 5)) + 10*x*log(47*x + 10)*log(47/2
*x + 5)*log(log(47/2*x + 5)) + 10*log(2)*log(47/2*x + 5)*log(log(2) - log(log(47/2*x + 5)))*log(log(47/2*x + 5
)) - 10*log(47*x + 10)*log(47/2*x + 5)*log(log(2) - log(log(47/2*x + 5)))*log(log(47/2*x + 5)) + 47*x*log(2) -
47*x*log(47*x + 10) - 47*log(2)*log(log(2) - log(log(47/2*x + 5))) + 47*log(47*x + 10)*log(log(2) - log(log(4
7/2*x + 5))))

Mupad [B] (verification not implemented)

Time = 12.92 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {47+(10+47 x) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )}{\left (10 x^2+47 x^3\right ) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )+\left (-20 x-94 x^2\right ) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right ) \log \left (\log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )\right )+(10+47 x) \log \left (\frac {1}{2} (10+47 x)\right ) \log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right ) \log ^2\left (\log \left (\frac {2}{\log \left (\frac {1}{2} (10+47 x)\right )}\right )\right )} \, dx=-\frac {1}{x-\ln \left (\ln \left (\frac {2}{\ln \left (\frac {47\,x}{2}+5\right )}\right )\right )} \]

[In]

int((log((47*x)/2 + 5)*log(2/log((47*x)/2 + 5))*(47*x + 10) + 47)/(log((47*x)/2 + 5)*log(2/log((47*x)/2 + 5))*
(10*x^2 + 47*x^3) - log(log(2/log((47*x)/2 + 5)))*log((47*x)/2 + 5)*log(2/log((47*x)/2 + 5))*(20*x + 94*x^2) +
log(log(2/log((47*x)/2 + 5)))^2*log((47*x)/2 + 5)*log(2/log((47*x)/2 + 5))*(47*x + 10)),x)

[Out]

-1/(x - log(log(2/log((47*x)/2 + 5))))