\(\int \frac {-960 x+1152 x \log (150 x \log (5))+(592 x-960 x \log (150 x \log (5))) \log (\log ^2(150 x \log (5)))+(-120 x+296 x \log (150 x \log (5))) \log ^2(\log ^2(150 x \log (5)))+(8 x-40 x \log (150 x \log (5))) \log ^3(\log ^2(150 x \log (5)))+2 x \log (150 x \log (5)) \log ^4(\log ^2(150 x \log (5)))}{\log (150 x \log (5))} \, dx\) [2303]

Optimal result

Integrand size = 115, antiderivative size = 23 \[ \int \frac {-960 x+1152 x \log (150 x \log (5))+(592 x-960 x \log (150 x \log (5))) \log \left (\log ^2(150 x \log (5))\right )+(-120 x+296 x \log (150 x \log (5))) \log ^2\left (\log ^2(150 x \log (5))\right )+(8 x-40 x \log (150 x \log (5))) \log ^3\left (\log ^2(150 x \log (5))\right )+2 x \log (150 x \log (5)) \log ^4\left (\log ^2(150 x \log (5))\right )}{\log (150 x \log (5))} \, dx=\left (x+x \left (-2+\left (5-\log \left (\log ^2(150 x \log (5))\right )\right )^2\right )\right )^2 \]

[Out]

(x+((5-ln(ln(150*x*ln(5))^2))^2-2)*x)^2

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.30, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {6820, 12, 6819} \[ \int \frac {-960 x+1152 x \log (150 x \log (5))+(592 x-960 x \log (150 x \log (5))) \log \left (\log ^2(150 x \log (5))\right )+(-120 x+296 x \log (150 x \log (5))) \log ^2\left (\log ^2(150 x \log (5))\right )+(8 x-40 x \log (150 x \log (5))) \log ^3\left (\log ^2(150 x \log (5))\right )+2 x \log (150 x \log (5)) \log ^4\left (\log ^2(150 x \log (5))\right )}{\log (150 x \log (5))} \, dx=x^2 \left (\log ^2\left (\log ^2(150 x \log (5))\right )-10 \log \left (\log ^2(150 x \log (5))\right )+24\right )^2 \]

[In]

Int[(-960*x + 1152*x*Log[150*x*Log[5]] + (592*x - 960*x*Log[150*x*Log[5]])*Log[Log[150*x*Log[5]]^2] + (-120*x
+ 296*x*Log[150*x*Log[5]])*Log[Log[150*x*Log[5]]^2]^2 + (8*x - 40*x*Log[150*x*Log[5]])*Log[Log[150*x*Log[5]]^2
]^3 + 2*x*Log[150*x*Log[5]]*Log[Log[150*x*Log[5]]^2]^4)/Log[150*x*Log[5]],x]

[Out]

x^2*(24 - 10*Log[Log[150*x*Log[5]]^2] + Log[Log[150*x*Log[5]]^2]^2)^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 6819

Int[(u_)*(y_)^(m_.)*(z_)^(n_.), x_Symbol] :> With[{q = DerivativeDivides[y*z, u*z^(n - m), x]}, Simp[q*y^(m +
1)*(z^(m + 1)/(m + 1)), x] /;  !FalseQ[q]] /; FreeQ[{m, n}, 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 x \left (24-10 \log \left (\log ^2(150 x \log (5))\right )+\log ^2\left (\log ^2(150 x \log (5))\right )\right ) \left (4 \left (-5+\log \left (\log ^2(150 x \log (5))\right )\right )+\log (150 x \log (5)) \left (24-10 \log \left (\log ^2(150 x \log (5))\right )+\log ^2\left (\log ^2(150 x \log (5))\right )\right )\right )}{\log (150 x \log (5))} \, dx \\ & = 2 \int \frac {x \left (24-10 \log \left (\log ^2(150 x \log (5))\right )+\log ^2\left (\log ^2(150 x \log (5))\right )\right ) \left (4 \left (-5+\log \left (\log ^2(150 x \log (5))\right )\right )+\log (150 x \log (5)) \left (24-10 \log \left (\log ^2(150 x \log (5))\right )+\log ^2\left (\log ^2(150 x \log (5))\right )\right )\right )}{\log (150 x \log (5))} \, dx \\ & = x^2 \left (24-10 \log \left (\log ^2(150 x \log (5))\right )+\log ^2\left (\log ^2(150 x \log (5))\right )\right )^2 \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(67\) vs. \(2(23)=46\).

Time = 0.51 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.91 \[ \int \frac {-960 x+1152 x \log (150 x \log (5))+(592 x-960 x \log (150 x \log (5))) \log \left (\log ^2(150 x \log (5))\right )+(-120 x+296 x \log (150 x \log (5))) \log ^2\left (\log ^2(150 x \log (5))\right )+(8 x-40 x \log (150 x \log (5))) \log ^3\left (\log ^2(150 x \log (5))\right )+2 x \log (150 x \log (5)) \log ^4\left (\log ^2(150 x \log (5))\right )}{\log (150 x \log (5))} \, dx=576 x^2-480 x^2 \log \left (\log ^2(150 x \log (5))\right )+148 x^2 \log ^2\left (\log ^2(150 x \log (5))\right )-20 x^2 \log ^3\left (\log ^2(150 x \log (5))\right )+x^2 \log ^4\left (\log ^2(150 x \log (5))\right ) \]

[In]

Integrate[(-960*x + 1152*x*Log[150*x*Log[5]] + (592*x - 960*x*Log[150*x*Log[5]])*Log[Log[150*x*Log[5]]^2] + (-
120*x + 296*x*Log[150*x*Log[5]])*Log[Log[150*x*Log[5]]^2]^2 + (8*x - 40*x*Log[150*x*Log[5]])*Log[Log[150*x*Log
[5]]^2]^3 + 2*x*Log[150*x*Log[5]]*Log[Log[150*x*Log[5]]^2]^4)/Log[150*x*Log[5]],x]

[Out]

576*x^2 - 480*x^2*Log[Log[150*x*Log[5]]^2] + 148*x^2*Log[Log[150*x*Log[5]]^2]^2 - 20*x^2*Log[Log[150*x*Log[5]]
^2]^3 + x^2*Log[Log[150*x*Log[5]]^2]^4

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(67\) vs. \(2(23)=46\).

Time = 5.56 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.96

method	result	size
parallelrisch	\(x^{2} \ln \left (\ln \left (150 x \ln \left (5\right )\right )^{2}\right )^{4}-20 \ln \left (\ln \left (150 x \ln \left (5\right )\right )^{2}\right )^{3} x^{2}+148 \ln \left (\ln \left (150 x \ln \left (5\right )\right )^{2}\right )^{2} x^{2}-480 x^{2} \ln \left (\ln \left (150 x \ln \left (5\right )\right )^{2}\right )+576 x^{2}\)	\(68\)
risch	\(\text {Expression too large to display}\)	\(1463\)

[In]

int((2*x*ln(150*x*ln(5))*ln(ln(150*x*ln(5))^2)^4+(-40*x*ln(150*x*ln(5))+8*x)*ln(ln(150*x*ln(5))^2)^3+(296*x*ln
(150*x*ln(5))-120*x)*ln(ln(150*x*ln(5))^2)^2+(-960*x*ln(150*x*ln(5))+592*x)*ln(ln(150*x*ln(5))^2)+1152*x*ln(15
0*x*ln(5))-960*x)/ln(150*x*ln(5)),x,method=_RETURNVERBOSE)

[Out]

x^2*ln(ln(150*x*ln(5))^2)^4-20*ln(ln(150*x*ln(5))^2)^3*x^2+148*ln(ln(150*x*ln(5))^2)^2*x^2-480*x^2*ln(ln(150*x
*ln(5))^2)+576*x^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (21) = 42\).

Time = 0.24 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.91 \[ \int \frac {-960 x+1152 x \log (150 x \log (5))+(592 x-960 x \log (150 x \log (5))) \log \left (\log ^2(150 x \log (5))\right )+(-120 x+296 x \log (150 x \log (5))) \log ^2\left (\log ^2(150 x \log (5))\right )+(8 x-40 x \log (150 x \log (5))) \log ^3\left (\log ^2(150 x \log (5))\right )+2 x \log (150 x \log (5)) \log ^4\left (\log ^2(150 x \log (5))\right )}{\log (150 x \log (5))} \, dx=x^{2} \log \left (\log \left (150 \, x \log \left (5\right )\right )^{2}\right )^{4} - 20 \, x^{2} \log \left (\log \left (150 \, x \log \left (5\right )\right )^{2}\right )^{3} + 148 \, x^{2} \log \left (\log \left (150 \, x \log \left (5\right )\right )^{2}\right )^{2} - 480 \, x^{2} \log \left (\log \left (150 \, x \log \left (5\right )\right )^{2}\right ) + 576 \, x^{2} \]

[In]

integrate((2*x*log(150*x*log(5))*log(log(150*x*log(5))^2)^4+(-40*x*log(150*x*log(5))+8*x)*log(log(150*x*log(5)
)^2)^3+(296*x*log(150*x*log(5))-120*x)*log(log(150*x*log(5))^2)^2+(-960*x*log(150*x*log(5))+592*x)*log(log(150
*x*log(5))^2)+1152*x*log(150*x*log(5))-960*x)/log(150*x*log(5)),x, algorithm="fricas")

[Out]

x^2*log(log(150*x*log(5))^2)^4 - 20*x^2*log(log(150*x*log(5))^2)^3 + 148*x^2*log(log(150*x*log(5))^2)^2 - 480*
x^2*log(log(150*x*log(5))^2) + 576*x^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (20) = 40\).

Time = 0.31 (sec) , antiderivative size = 75, normalized size of antiderivative = 3.26 \[ \int \frac {-960 x+1152 x \log (150 x \log (5))+(592 x-960 x \log (150 x \log (5))) \log \left (\log ^2(150 x \log (5))\right )+(-120 x+296 x \log (150 x \log (5))) \log ^2\left (\log ^2(150 x \log (5))\right )+(8 x-40 x \log (150 x \log (5))) \log ^3\left (\log ^2(150 x \log (5))\right )+2 x \log (150 x \log (5)) \log ^4\left (\log ^2(150 x \log (5))\right )}{\log (150 x \log (5))} \, dx=x^{2} \log {\left (\log {\left (150 x \log {\left (5 \right )} \right )}^{2} \right )}^{4} - 20 x^{2} \log {\left (\log {\left (150 x \log {\left (5 \right )} \right )}^{2} \right )}^{3} + 148 x^{2} \log {\left (\log {\left (150 x \log {\left (5 \right )} \right )}^{2} \right )}^{2} - 480 x^{2} \log {\left (\log {\left (150 x \log {\left (5 \right )} \right )}^{2} \right )} + 576 x^{2} \]

[In]

integrate((2*x*ln(150*x*ln(5))*ln(ln(150*x*ln(5))**2)**4+(-40*x*ln(150*x*ln(5))+8*x)*ln(ln(150*x*ln(5))**2)**3
+(296*x*ln(150*x*ln(5))-120*x)*ln(ln(150*x*ln(5))**2)**2+(-960*x*ln(150*x*ln(5))+592*x)*ln(ln(150*x*ln(5))**2)
+1152*x*ln(150*x*ln(5))-960*x)/ln(150*x*ln(5)),x)

[Out]

x**2*log(log(150*x*log(5))**2)**4 - 20*x**2*log(log(150*x*log(5))**2)**3 + 148*x**2*log(log(150*x*log(5))**2)*
*2 - 480*x**2*log(log(150*x*log(5))**2) + 576*x**2

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (21) = 42\).

Time = 0.36 (sec) , antiderivative size = 86, normalized size of antiderivative = 3.74 \[ \int \frac {-960 x+1152 x \log (150 x \log (5))+(592 x-960 x \log (150 x \log (5))) \log \left (\log ^2(150 x \log (5))\right )+(-120 x+296 x \log (150 x \log (5))) \log ^2\left (\log ^2(150 x \log (5))\right )+(8 x-40 x \log (150 x \log (5))) \log ^3\left (\log ^2(150 x \log (5))\right )+2 x \log (150 x \log (5)) \log ^4\left (\log ^2(150 x \log (5))\right )}{\log (150 x \log (5))} \, dx=16 \, x^{2} \log \left (2 \, \log \left (5\right ) + \log \left (3\right ) + \log \left (2\right ) + \log \left (x\right ) + \log \left (\log \left (5\right )\right )\right )^{4} - 160 \, x^{2} \log \left (2 \, \log \left (5\right ) + \log \left (3\right ) + \log \left (2\right ) + \log \left (x\right ) + \log \left (\log \left (5\right )\right )\right )^{3} + 592 \, x^{2} \log \left (2 \, \log \left (5\right ) + \log \left (3\right ) + \log \left (2\right ) + \log \left (x\right ) + \log \left (\log \left (5\right )\right )\right )^{2} - 480 \, x^{2} \log \left (\log \left (150 \, x \log \left (5\right )\right )^{2}\right ) + 576 \, x^{2} \]

[In]

integrate((2*x*log(150*x*log(5))*log(log(150*x*log(5))^2)^4+(-40*x*log(150*x*log(5))+8*x)*log(log(150*x*log(5)
)^2)^3+(296*x*log(150*x*log(5))-120*x)*log(log(150*x*log(5))^2)^2+(-960*x*log(150*x*log(5))+592*x)*log(log(150
*x*log(5))^2)+1152*x*log(150*x*log(5))-960*x)/log(150*x*log(5)),x, algorithm="maxima")

[Out]

16*x^2*log(2*log(5) + log(3) + log(2) + log(x) + log(log(5)))^4 - 160*x^2*log(2*log(5) + log(3) + log(2) + log
(x) + log(log(5)))^3 + 592*x^2*log(2*log(5) + log(3) + log(2) + log(x) + log(log(5)))^2 - 480*x^2*log(log(150*
x*log(5))^2) + 576*x^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (21) = 42\).

Time = 1.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.78 \[ \int \frac {-960 x+1152 x \log (150 x \log (5))+(592 x-960 x \log (150 x \log (5))) \log \left (\log ^2(150 x \log (5))\right )+(-120 x+296 x \log (150 x \log (5))) \log ^2\left (\log ^2(150 x \log (5))\right )+(8 x-40 x \log (150 x \log (5))) \log ^3\left (\log ^2(150 x \log (5))\right )+2 x \log (150 x \log (5)) \log ^4\left (\log ^2(150 x \log (5))\right )}{\log (150 x \log (5))} \, dx=16 \, x^{2} \log \left ({\left | \log \left (150 \, x \log \left (5\right )\right ) \right |}\right )^{4} - 160 \, x^{2} \log \left ({\left | \log \left (150 \, x \log \left (5\right )\right ) \right |}\right )^{3} + 592 \, x^{2} \log \left ({\left | \log \left (150 \, x \log \left (5\right )\right ) \right |}\right )^{2} - 960 \, x^{2} \log \left ({\left | \log \left (150 \, x \log \left (5\right )\right ) \right |}\right ) + 576 \, x^{2} \]

[In]

integrate((2*x*log(150*x*log(5))*log(log(150*x*log(5))^2)^4+(-40*x*log(150*x*log(5))+8*x)*log(log(150*x*log(5)
)^2)^3+(296*x*log(150*x*log(5))-120*x)*log(log(150*x*log(5))^2)^2+(-960*x*log(150*x*log(5))+592*x)*log(log(150
*x*log(5))^2)+1152*x*log(150*x*log(5))-960*x)/log(150*x*log(5)),x, algorithm="giac")

[Out]

16*x^2*log(abs(log(150*x*log(5))))^4 - 160*x^2*log(abs(log(150*x*log(5))))^3 + 592*x^2*log(abs(log(150*x*log(5
))))^2 - 960*x^2*log(abs(log(150*x*log(5)))) + 576*x^2

Mupad [B] (verification not implemented)

Time = 7.90 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.91 \[ \int \frac {-960 x+1152 x \log (150 x \log (5))+(592 x-960 x \log (150 x \log (5))) \log \left (\log ^2(150 x \log (5))\right )+(-120 x+296 x \log (150 x \log (5))) \log ^2\left (\log ^2(150 x \log (5))\right )+(8 x-40 x \log (150 x \log (5))) \log ^3\left (\log ^2(150 x \log (5))\right )+2 x \log (150 x \log (5)) \log ^4\left (\log ^2(150 x \log (5))\right )}{\log (150 x \log (5))} \, dx=x^2\,{\ln \left ({\ln \left (150\,x\,\ln \left (5\right )\right )}^2\right )}^4-20\,x^2\,{\ln \left ({\ln \left (150\,x\,\ln \left (5\right )\right )}^2\right )}^3+148\,x^2\,{\ln \left ({\ln \left (150\,x\,\ln \left (5\right )\right )}^2\right )}^2-480\,x^2\,\ln \left ({\ln \left (150\,x\,\ln \left (5\right )\right )}^2\right )+576\,x^2 \]

[In]

int((1152*x*log(150*x*log(5)) - 960*x + log(log(150*x*log(5))^2)*(592*x - 960*x*log(150*x*log(5))) + log(log(1
50*x*log(5))^2)^3*(8*x - 40*x*log(150*x*log(5))) - log(log(150*x*log(5))^2)^2*(120*x - 296*x*log(150*x*log(5))
) + 2*x*log(log(150*x*log(5))^2)^4*log(150*x*log(5)))/log(150*x*log(5)),x)

[Out]

148*x^2*log(log(150*x*log(5))^2)^2 - 20*x^2*log(log(150*x*log(5))^2)^3 + x^2*log(log(150*x*log(5))^2)^4 - 480*
x^2*log(log(150*x*log(5))^2) + 576*x^2