\(\int \frac {-25 x^3-x^4 \log (\frac {4}{x})+(-2450 x^3+x^4-199 x^4 \log (\frac {4}{x})-4 x^5 \log ^2(\frac {4}{x})) \log (x)+(10000 x^3+100 x^4+(700 x^4+4 x^5) \log (\frac {4}{x})+12 x^5 \log ^2(\frac {4}{x})) \log ^2(x)+(500000 x^3+60000 x^4 \log (\frac {4}{x})+2400 x^5 \log ^2(\frac {4}{x})+32 x^6 \log ^3(\frac {4}{x})) \log ^3(x)}{(31250+3750 x \log (\frac {4}{x})+150 x^2 \log ^2(\frac {4}{x})+2 x^3 \log ^3(\frac {4}{x})) \log ^3(x)} \, dx\) [2939]

Optimal result

Integrand size = 192, antiderivative size = 30 \[ \int \frac {-25 x^3-x^4 \log \left (\frac {4}{x}\right )+\left (-2450 x^3+x^4-199 x^4 \log \left (\frac {4}{x}\right )-4 x^5 \log ^2\left (\frac {4}{x}\right )\right ) \log (x)+\left (10000 x^3+100 x^4+\left (700 x^4+4 x^5\right ) \log \left (\frac {4}{x}\right )+12 x^5 \log ^2\left (\frac {4}{x}\right )\right ) \log ^2(x)+\left (500000 x^3+60000 x^4 \log \left (\frac {4}{x}\right )+2400 x^5 \log ^2\left (\frac {4}{x}\right )+32 x^6 \log ^3\left (\frac {4}{x}\right )\right ) \log ^3(x)}{\left (31250+3750 x \log \left (\frac {4}{x}\right )+150 x^2 \log ^2\left (\frac {4}{x}\right )+2 x^3 \log ^3\left (\frac {4}{x}\right )\right ) \log ^3(x)} \, dx=4 x^2 \left (x+\frac {x}{4 \left (25+x \log \left (\frac {4}{x}\right )\right ) \log (x)}\right )^2 \]

[Out]

4*x^2*(1/4*x/(25+x*ln(4/x))/ln(x)+x)^2

Rubi [F]

\[ \int \frac {-25 x^3-x^4 \log \left (\frac {4}{x}\right )+\left (-2450 x^3+x^4-199 x^4 \log \left (\frac {4}{x}\right )-4 x^5 \log ^2\left (\frac {4}{x}\right )\right ) \log (x)+\left (10000 x^3+100 x^4+\left (700 x^4+4 x^5\right ) \log \left (\frac {4}{x}\right )+12 x^5 \log ^2\left (\frac {4}{x}\right )\right ) \log ^2(x)+\left (500000 x^3+60000 x^4 \log \left (\frac {4}{x}\right )+2400 x^5 \log ^2\left (\frac {4}{x}\right )+32 x^6 \log ^3\left (\frac {4}{x}\right )\right ) \log ^3(x)}{\left (31250+3750 x \log \left (\frac {4}{x}\right )+150 x^2 \log ^2\left (\frac {4}{x}\right )+2 x^3 \log ^3\left (\frac {4}{x}\right )\right ) \log ^3(x)} \, dx=\int \frac {-25 x^3-x^4 \log \left (\frac {4}{x}\right )+\left (-2450 x^3+x^4-199 x^4 \log \left (\frac {4}{x}\right )-4 x^5 \log ^2\left (\frac {4}{x}\right )\right ) \log (x)+\left (10000 x^3+100 x^4+\left (700 x^4+4 x^5\right ) \log \left (\frac {4}{x}\right )+12 x^5 \log ^2\left (\frac {4}{x}\right )\right ) \log ^2(x)+\left (500000 x^3+60000 x^4 \log \left (\frac {4}{x}\right )+2400 x^5 \log ^2\left (\frac {4}{x}\right )+32 x^6 \log ^3\left (\frac {4}{x}\right )\right ) \log ^3(x)}{\left (31250+3750 x \log \left (\frac {4}{x}\right )+150 x^2 \log ^2\left (\frac {4}{x}\right )+2 x^3 \log ^3\left (\frac {4}{x}\right )\right ) \log ^3(x)} \, dx \]

[In]

Int[(-25*x^3 - x^4*Log[4/x] + (-2450*x^3 + x^4 - 199*x^4*Log[4/x] - 4*x^5*Log[4/x]^2)*Log[x] + (10000*x^3 + 10
0*x^4 + (700*x^4 + 4*x^5)*Log[4/x] + 12*x^5*Log[4/x]^2)*Log[x]^2 + (500000*x^3 + 60000*x^4*Log[4/x] + 2400*x^5
*Log[4/x]^2 + 32*x^6*Log[4/x]^3)*Log[x]^3)/((31250 + 3750*x*Log[4/x] + 150*x^2*Log[4/x]^2 + 2*x^3*Log[4/x]^3)*
Log[x]^3),x]

[Out]

4*x^4 - Defer[Int][x^3/((25 + x*Log[4/x])^2*Log[x]^3), x]/2 - 1225*Defer[Int][x^3/((25 + x*Log[4/x])^3*Log[x]^
2), x] + Defer[Int][x^4/((25 + x*Log[4/x])^3*Log[x]^2), x]/2 - (199*Defer[Int][(x^4*Log[4/x])/((25 + x*Log[4/x
])^3*Log[x]^2), x])/2 - 2*Defer[Int][(x^5*Log[4/x]^2)/((25 + x*Log[4/x])^3*Log[x]^2), x] + 200*Defer[Int][x^3/
((25 + x*Log[4/x])^2*Log[x]), x] + 2*Defer[Int][x^4/((25 + x*Log[4/x])^2*Log[x]), x] + 6*Defer[Int][(x^4*Log[4
/x])/((25 + x*Log[4/x])^2*Log[x]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^3 \left (32 x^3 \log ^3\left (\frac {4}{x}\right ) \log ^3(x)+4 x^2 \log ^2\left (\frac {4}{x}\right ) \log (x) \left (-1+3 \log (x)+600 \log ^2(x)\right )+(1+100 \log (x)) \left (-25+(50+x) \log (x)+5000 \log ^2(x)\right )+x \log \left (\frac {4}{x}\right ) \left (-1-199 \log (x)+4 (175+x) \log ^2(x)+60000 \log ^3(x)\right )\right )}{2 \left (25+x \log \left (\frac {4}{x}\right )\right )^3 \log ^3(x)} \, dx \\ & = \frac {1}{2} \int \frac {x^3 \left (32 x^3 \log ^3\left (\frac {4}{x}\right ) \log ^3(x)+4 x^2 \log ^2\left (\frac {4}{x}\right ) \log (x) \left (-1+3 \log (x)+600 \log ^2(x)\right )+(1+100 \log (x)) \left (-25+(50+x) \log (x)+5000 \log ^2(x)\right )+x \log \left (\frac {4}{x}\right ) \left (-1-199 \log (x)+4 (175+x) \log ^2(x)+60000 \log ^3(x)\right )\right )}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3 \log ^3(x)} \, dx \\ & = \frac {1}{2} \int \left (\frac {500000 x^3}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3}+\frac {60000 x^4 \log \left (\frac {4}{x}\right )}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3}+\frac {2400 x^5 \log ^2\left (\frac {4}{x}\right )}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3}+\frac {32 x^6 \log ^3\left (\frac {4}{x}\right )}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3}-\frac {x^3}{\left (25+x \log \left (\frac {4}{x}\right )\right )^2 \log ^3(x)}-\frac {x^3 \left (2450-x+199 x \log \left (\frac {4}{x}\right )+4 x^2 \log ^2\left (\frac {4}{x}\right )\right )}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3 \log ^2(x)}+\frac {4 x^3 \left (100+x+3 x \log \left (\frac {4}{x}\right )\right )}{\left (25+x \log \left (\frac {4}{x}\right )\right )^2 \log (x)}\right ) \, dx \\ & = -\left (\frac {1}{2} \int \frac {x^3}{\left (25+x \log \left (\frac {4}{x}\right )\right )^2 \log ^3(x)} \, dx\right )-\frac {1}{2} \int \frac {x^3 \left (2450-x+199 x \log \left (\frac {4}{x}\right )+4 x^2 \log ^2\left (\frac {4}{x}\right )\right )}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3 \log ^2(x)} \, dx+2 \int \frac {x^3 \left (100+x+3 x \log \left (\frac {4}{x}\right )\right )}{\left (25+x \log \left (\frac {4}{x}\right )\right )^2 \log (x)} \, dx+16 \int \frac {x^6 \log ^3\left (\frac {4}{x}\right )}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3} \, dx+1200 \int \frac {x^5 \log ^2\left (\frac {4}{x}\right )}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3} \, dx+30000 \int \frac {x^4 \log \left (\frac {4}{x}\right )}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3} \, dx+250000 \int \frac {x^3}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3} \, dx \\ & = -\left (\frac {1}{2} \int \left (\frac {2450 x^3}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3 \log ^2(x)}-\frac {x^4}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3 \log ^2(x)}+\frac {199 x^4 \log \left (\frac {4}{x}\right )}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3 \log ^2(x)}+\frac {4 x^5 \log ^2\left (\frac {4}{x}\right )}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3 \log ^2(x)}\right ) \, dx\right )-\frac {1}{2} \int \frac {x^3}{\left (25+x \log \left (\frac {4}{x}\right )\right )^2 \log ^3(x)} \, dx+2 \int \left (\frac {100 x^3}{\left (25+x \log \left (\frac {4}{x}\right )\right )^2 \log (x)}+\frac {x^4}{\left (25+x \log \left (\frac {4}{x}\right )\right )^2 \log (x)}+\frac {3 x^4 \log \left (\frac {4}{x}\right )}{\left (25+x \log \left (\frac {4}{x}\right )\right )^2 \log (x)}\right ) \, dx+16 \int \left (x^3-\frac {15625 x^3}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3}+\frac {1875 x^3}{\left (25+x \log \left (\frac {4}{x}\right )\right )^2}-\frac {75 x^3}{25+x \log \left (\frac {4}{x}\right )}\right ) \, dx+1200 \int \left (\frac {625 x^3}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3}-\frac {50 x^3}{\left (25+x \log \left (\frac {4}{x}\right )\right )^2}+\frac {x^3}{25+x \log \left (\frac {4}{x}\right )}\right ) \, dx+30000 \int \left (-\frac {25 x^3}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3}+\frac {x^3}{\left (25+x \log \left (\frac {4}{x}\right )\right )^2}\right ) \, dx+250000 \int \frac {x^3}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3} \, dx \\ & = 4 x^4-\frac {1}{2} \int \frac {x^3}{\left (25+x \log \left (\frac {4}{x}\right )\right )^2 \log ^3(x)} \, dx+\frac {1}{2} \int \frac {x^4}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3 \log ^2(x)} \, dx-2 \int \frac {x^5 \log ^2\left (\frac {4}{x}\right )}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3 \log ^2(x)} \, dx+2 \int \frac {x^4}{\left (25+x \log \left (\frac {4}{x}\right )\right )^2 \log (x)} \, dx+6 \int \frac {x^4 \log \left (\frac {4}{x}\right )}{\left (25+x \log \left (\frac {4}{x}\right )\right )^2 \log (x)} \, dx-\frac {199}{2} \int \frac {x^4 \log \left (\frac {4}{x}\right )}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3 \log ^2(x)} \, dx+200 \int \frac {x^3}{\left (25+x \log \left (\frac {4}{x}\right )\right )^2 \log (x)} \, dx-1225 \int \frac {x^3}{\left (25+x \log \left (\frac {4}{x}\right )\right )^3 \log ^2(x)} \, dx+2 \left (30000 \int \frac {x^3}{\left (25+x \log \left (\frac {4}{x}\right )\right )^2} \, dx\right )-60000 \int \frac {x^3}{\left (25+x \log \left (\frac {4}{x}\right )\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.37 \[ \int \frac {-25 x^3-x^4 \log \left (\frac {4}{x}\right )+\left (-2450 x^3+x^4-199 x^4 \log \left (\frac {4}{x}\right )-4 x^5 \log ^2\left (\frac {4}{x}\right )\right ) \log (x)+\left (10000 x^3+100 x^4+\left (700 x^4+4 x^5\right ) \log \left (\frac {4}{x}\right )+12 x^5 \log ^2\left (\frac {4}{x}\right )\right ) \log ^2(x)+\left (500000 x^3+60000 x^4 \log \left (\frac {4}{x}\right )+2400 x^5 \log ^2\left (\frac {4}{x}\right )+32 x^6 \log ^3\left (\frac {4}{x}\right )\right ) \log ^3(x)}{\left (31250+3750 x \log \left (\frac {4}{x}\right )+150 x^2 \log ^2\left (\frac {4}{x}\right )+2 x^3 \log ^3\left (\frac {4}{x}\right )\right ) \log ^3(x)} \, dx=\frac {x^4 \left (1+4 \left (25+x \log \left (\frac {4}{x}\right )\right ) \log (x)\right )^2}{4 \left (25+x \log \left (\frac {4}{x}\right )\right )^2 \log ^2(x)} \]

[In]

Integrate[(-25*x^3 - x^4*Log[4/x] + (-2450*x^3 + x^4 - 199*x^4*Log[4/x] - 4*x^5*Log[4/x]^2)*Log[x] + (10000*x^
3 + 100*x^4 + (700*x^4 + 4*x^5)*Log[4/x] + 12*x^5*Log[4/x]^2)*Log[x]^2 + (500000*x^3 + 60000*x^4*Log[4/x] + 24
00*x^5*Log[4/x]^2 + 32*x^6*Log[4/x]^3)*Log[x]^3)/((31250 + 3750*x*Log[4/x] + 150*x^2*Log[4/x]^2 + 2*x^3*Log[4/
x]^3)*Log[x]^3),x]

[Out]

(x^4*(1 + 4*(25 + x*Log[4/x])*Log[x])^2)/(4*(25 + x*Log[4/x])^2*Log[x]^2)

Maple [A] (verified)

Time = 98.23 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.67

method	result	size
risch	\(4 x^{4}-\frac {x^{4} \left (-1-16 x \ln \left (2\right ) \ln \left (x \right )+8 x \ln \left (x \right )^{2}-200 \ln \left (x \right )\right )}{\ln \left (x \right )^{2} \left (-50-4 x \ln \left (2\right )+2 x \ln \left (x \right )\right )^{2}}\)	\(50\)
parallelrisch	\(\frac {21000000 x^{4} \ln \left (x \right )^{2}+420000 x^{4} \ln \left (x \right )+2100 x^{4}+33600 \ln \left (\frac {4}{x}\right )^{2} x^{6} \ln \left (x \right )^{2}+16800 \ln \left (\frac {4}{x}\right ) x^{5} \ln \left (x \right )+1680000 \ln \left (\frac {4}{x}\right ) x^{5} \ln \left (x \right )^{2}}{8400 \ln \left (x \right )^{2} \left (x^{2} \ln \left (\frac {4}{x}\right )^{2}+50 x \ln \left (\frac {4}{x}\right )+625\right )}\)	\(99\)

[In]

int(((32*x^6*ln(4/x)^3+2400*x^5*ln(4/x)^2+60000*x^4*ln(4/x)+500000*x^3)*ln(x)^3+(12*x^5*ln(4/x)^2+(4*x^5+700*x
^4)*ln(4/x)+100*x^4+10000*x^3)*ln(x)^2+(-4*x^5*ln(4/x)^2-199*x^4*ln(4/x)+x^4-2450*x^3)*ln(x)-x^4*ln(4/x)-25*x^
3)/(2*x^3*ln(4/x)^3+150*x^2*ln(4/x)^2+3750*x*ln(4/x)+31250)/ln(x)^3,x,method=_RETURNVERBOSE)

[Out]

4*x^4-x^4*(-1-16*x*ln(2)*ln(x)+8*x*ln(x)^2-200*ln(x))/ln(x)^2/(-50-4*x*ln(2)+2*x*ln(x))^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (29) = 58\).

Time = 0.25 (sec) , antiderivative size = 218, normalized size of antiderivative = 7.27 \[ \int \frac {-25 x^3-x^4 \log \left (\frac {4}{x}\right )+\left (-2450 x^3+x^4-199 x^4 \log \left (\frac {4}{x}\right )-4 x^5 \log ^2\left (\frac {4}{x}\right )\right ) \log (x)+\left (10000 x^3+100 x^4+\left (700 x^4+4 x^5\right ) \log \left (\frac {4}{x}\right )+12 x^5 \log ^2\left (\frac {4}{x}\right )\right ) \log ^2(x)+\left (500000 x^3+60000 x^4 \log \left (\frac {4}{x}\right )+2400 x^5 \log ^2\left (\frac {4}{x}\right )+32 x^6 \log ^3\left (\frac {4}{x}\right )\right ) \log ^3(x)}{\left (31250+3750 x \log \left (\frac {4}{x}\right )+150 x^2 \log ^2\left (\frac {4}{x}\right )+2 x^3 \log ^3\left (\frac {4}{x}\right )\right ) \log ^3(x)} \, dx=\frac {16 \, x^{6} \log \left (\frac {4}{x}\right )^{4} + 40000 \, x^{4} \log \left (2\right )^{2} + 400 \, x^{4} \log \left (2\right ) + x^{4} - 32 \, {\left (2 \, x^{6} \log \left (2\right ) - 25 \, x^{5}\right )} \log \left (\frac {4}{x}\right )^{3} + 8 \, {\left (8 \, x^{6} \log \left (2\right )^{2} - 400 \, x^{5} \log \left (2\right ) - x^{5} + 1250 \, x^{4}\right )} \log \left (\frac {4}{x}\right )^{2} + 8 \, {\left (400 \, x^{5} \log \left (2\right )^{2} - 25 \, x^{4} + 2 \, {\left (x^{5} - 2500 \, x^{4}\right )} \log \left (2\right )\right )} \log \left (\frac {4}{x}\right )}{4 \, {\left (x^{2} \log \left (\frac {4}{x}\right )^{4} - 2 \, {\left (2 \, x^{2} \log \left (2\right ) - 25 \, x\right )} \log \left (\frac {4}{x}\right )^{3} + {\left (4 \, x^{2} \log \left (2\right )^{2} - 200 \, x \log \left (2\right ) + 625\right )} \log \left (\frac {4}{x}\right )^{2} + 2500 \, \log \left (2\right )^{2} + 100 \, {\left (2 \, x \log \left (2\right )^{2} - 25 \, \log \left (2\right )\right )} \log \left (\frac {4}{x}\right )\right )}} \]

[In]

integrate(((32*x^6*log(4/x)^3+2400*x^5*log(4/x)^2+60000*x^4*log(4/x)+500000*x^3)*log(x)^3+(12*x^5*log(4/x)^2+(
4*x^5+700*x^4)*log(4/x)+100*x^4+10000*x^3)*log(x)^2+(-4*x^5*log(4/x)^2-199*x^4*log(4/x)+x^4-2450*x^3)*log(x)-x
^4*log(4/x)-25*x^3)/(2*x^3*log(4/x)^3+150*x^2*log(4/x)^2+3750*x*log(4/x)+31250)/log(x)^3,x, algorithm="fricas"
)

[Out]

1/4*(16*x^6*log(4/x)^4 + 40000*x^4*log(2)^2 + 400*x^4*log(2) + x^4 - 32*(2*x^6*log(2) - 25*x^5)*log(4/x)^3 + 8
*(8*x^6*log(2)^2 - 400*x^5*log(2) - x^5 + 1250*x^4)*log(4/x)^2 + 8*(400*x^5*log(2)^2 - 25*x^4 + 2*(x^5 - 2500*
x^4)*log(2))*log(4/x))/(x^2*log(4/x)^4 - 2*(2*x^2*log(2) - 25*x)*log(4/x)^3 + (4*x^2*log(2)^2 - 200*x*log(2) +
625)*log(4/x)^2 + 2500*log(2)^2 + 100*(2*x*log(2)^2 - 25*log(2))*log(4/x))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (22) = 44\).

Time = 0.24 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.90 \[ \int \frac {-25 x^3-x^4 \log \left (\frac {4}{x}\right )+\left (-2450 x^3+x^4-199 x^4 \log \left (\frac {4}{x}\right )-4 x^5 \log ^2\left (\frac {4}{x}\right )\right ) \log (x)+\left (10000 x^3+100 x^4+\left (700 x^4+4 x^5\right ) \log \left (\frac {4}{x}\right )+12 x^5 \log ^2\left (\frac {4}{x}\right )\right ) \log ^2(x)+\left (500000 x^3+60000 x^4 \log \left (\frac {4}{x}\right )+2400 x^5 \log ^2\left (\frac {4}{x}\right )+32 x^6 \log ^3\left (\frac {4}{x}\right )\right ) \log ^3(x)}{\left (31250+3750 x \log \left (\frac {4}{x}\right )+150 x^2 \log ^2\left (\frac {4}{x}\right )+2 x^3 \log ^3\left (\frac {4}{x}\right )\right ) \log ^3(x)} \, dx=4 x^{4} + \frac {- 8 x^{5} \log {\left (x \right )}^{2} + x^{4} + \left (16 x^{5} \log {\left (2 \right )} + 200 x^{4}\right ) \log {\left (x \right )}}{4 x^{2} \log {\left (x \right )}^{4} + \left (- 16 x^{2} \log {\left (2 \right )} - 200 x\right ) \log {\left (x \right )}^{3} + \left (16 x^{2} \log {\left (2 \right )}^{2} + 400 x \log {\left (2 \right )} + 2500\right ) \log {\left (x \right )}^{2}} \]

[In]

integrate(((32*x**6*ln(4/x)**3+2400*x**5*ln(4/x)**2+60000*x**4*ln(4/x)+500000*x**3)*ln(x)**3+(12*x**5*ln(4/x)*
*2+(4*x**5+700*x**4)*ln(4/x)+100*x**4+10000*x**3)*ln(x)**2+(-4*x**5*ln(4/x)**2-199*x**4*ln(4/x)+x**4-2450*x**3
)*ln(x)-x**4*ln(4/x)-25*x**3)/(2*x**3*ln(4/x)**3+150*x**2*ln(4/x)**2+3750*x*ln(4/x)+31250)/ln(x)**3,x)

[Out]

4*x**4 + (-8*x**5*log(x)**2 + x**4 + (16*x**5*log(2) + 200*x**4)*log(x))/(4*x**2*log(x)**4 + (-16*x**2*log(2)
- 200*x)*log(x)**3 + (16*x**2*log(2)**2 + 400*x*log(2) + 2500)*log(x)**2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (29) = 58\).

Time = 0.40 (sec) , antiderivative size = 131, normalized size of antiderivative = 4.37 \[ \int \frac {-25 x^3-x^4 \log \left (\frac {4}{x}\right )+\left (-2450 x^3+x^4-199 x^4 \log \left (\frac {4}{x}\right )-4 x^5 \log ^2\left (\frac {4}{x}\right )\right ) \log (x)+\left (10000 x^3+100 x^4+\left (700 x^4+4 x^5\right ) \log \left (\frac {4}{x}\right )+12 x^5 \log ^2\left (\frac {4}{x}\right )\right ) \log ^2(x)+\left (500000 x^3+60000 x^4 \log \left (\frac {4}{x}\right )+2400 x^5 \log ^2\left (\frac {4}{x}\right )+32 x^6 \log ^3\left (\frac {4}{x}\right )\right ) \log ^3(x)}{\left (31250+3750 x \log \left (\frac {4}{x}\right )+150 x^2 \log ^2\left (\frac {4}{x}\right )+2 x^3 \log ^3\left (\frac {4}{x}\right )\right ) \log ^3(x)} \, dx=\frac {16 \, x^{6} \log \left (x\right )^{4} + x^{4} - 32 \, {\left (2 \, x^{6} \log \left (2\right ) + 25 \, x^{5}\right )} \log \left (x\right )^{3} + 8 \, {\left (8 \, x^{6} \log \left (2\right )^{2} + x^{5} {\left (200 \, \log \left (2\right ) - 1\right )} + 1250 \, x^{4}\right )} \log \left (x\right )^{2} + 8 \, {\left (2 \, x^{5} \log \left (2\right ) + 25 \, x^{4}\right )} \log \left (x\right )}{4 \, {\left (x^{2} \log \left (x\right )^{4} - 2 \, {\left (2 \, x^{2} \log \left (2\right ) + 25 \, x\right )} \log \left (x\right )^{3} + {\left (4 \, x^{2} \log \left (2\right )^{2} + 100 \, x \log \left (2\right ) + 625\right )} \log \left (x\right )^{2}\right )}} \]

[In]

integrate(((32*x^6*log(4/x)^3+2400*x^5*log(4/x)^2+60000*x^4*log(4/x)+500000*x^3)*log(x)^3+(12*x^5*log(4/x)^2+(
4*x^5+700*x^4)*log(4/x)+100*x^4+10000*x^3)*log(x)^2+(-4*x^5*log(4/x)^2-199*x^4*log(4/x)+x^4-2450*x^3)*log(x)-x
^4*log(4/x)-25*x^3)/(2*x^3*log(4/x)^3+150*x^2*log(4/x)^2+3750*x*log(4/x)+31250)/log(x)^3,x, algorithm="maxima"
)

[Out]

1/4*(16*x^6*log(x)^4 + x^4 - 32*(2*x^6*log(2) + 25*x^5)*log(x)^3 + 8*(8*x^6*log(2)^2 + x^5*(200*log(2) - 1) +
1250*x^4)*log(x)^2 + 8*(2*x^5*log(2) + 25*x^4)*log(x))/(x^2*log(x)^4 - 2*(2*x^2*log(2) + 25*x)*log(x)^3 + (4*x
^2*log(2)^2 + 100*x*log(2) + 625)*log(x)^2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 316 vs. \(2 (29) = 58\).

Time = 0.31 (sec) , antiderivative size = 316, normalized size of antiderivative = 10.53 \[ \int \frac {-25 x^3-x^4 \log \left (\frac {4}{x}\right )+\left (-2450 x^3+x^4-199 x^4 \log \left (\frac {4}{x}\right )-4 x^5 \log ^2\left (\frac {4}{x}\right )\right ) \log (x)+\left (10000 x^3+100 x^4+\left (700 x^4+4 x^5\right ) \log \left (\frac {4}{x}\right )+12 x^5 \log ^2\left (\frac {4}{x}\right )\right ) \log ^2(x)+\left (500000 x^3+60000 x^4 \log \left (\frac {4}{x}\right )+2400 x^5 \log ^2\left (\frac {4}{x}\right )+32 x^6 \log ^3\left (\frac {4}{x}\right )\right ) \log ^3(x)}{\left (31250+3750 x \log \left (\frac {4}{x}\right )+150 x^2 \log ^2\left (\frac {4}{x}\right )+2 x^3 \log ^3\left (\frac {4}{x}\right )\right ) \log ^3(x)} \, dx=4 \, x^{4} + \frac {64 \, x^{8} \log \left (2\right )^{3} - 32 \, x^{8} \log \left (2\right )^{2} \log \left (x\right ) + 2400 \, x^{7} \log \left (2\right )^{2} - 800 \, x^{7} \log \left (2\right ) \log \left (x\right ) + 6 \, x^{7} \log \left (2\right ) - 2 \, x^{7} \log \left (x\right ) + 30000 \, x^{6} \log \left (2\right ) - 5000 \, x^{6} \log \left (x\right ) + 75 \, x^{6} + 125000 \, x^{5}}{4 \, {\left (32 \, x^{5} \log \left (2\right )^{5} - 32 \, x^{5} \log \left (2\right )^{4} \log \left (x\right ) + 8 \, x^{5} \log \left (2\right )^{3} \log \left (x\right )^{2} + 2000 \, x^{4} \log \left (2\right )^{4} - 1600 \, x^{4} \log \left (2\right )^{3} \log \left (x\right ) + 300 \, x^{4} \log \left (2\right )^{2} \log \left (x\right )^{2} + 50000 \, x^{3} \log \left (2\right )^{3} - 30000 \, x^{3} \log \left (2\right )^{2} \log \left (x\right ) + 3750 \, x^{3} \log \left (2\right ) \log \left (x\right )^{2} + 625000 \, x^{2} \log \left (2\right )^{2} - 250000 \, x^{2} \log \left (2\right ) \log \left (x\right ) + 15625 \, x^{2} \log \left (x\right )^{2} + 3906250 \, x \log \left (2\right ) - 781250 \, x \log \left (x\right ) + 9765625\right )}} + \frac {32 \, x^{6} \log \left (2\right )^{2} \log \left (x\right ) + 800 \, x^{5} \log \left (2\right ) \log \left (x\right ) + 2 \, x^{5} \log \left (2\right ) + 2 \, x^{5} \log \left (x\right ) + 5000 \, x^{4} \log \left (x\right ) + 25 \, x^{4}}{4 \, {\left (8 \, x^{3} \log \left (2\right )^{3} \log \left (x\right )^{2} + 300 \, x^{2} \log \left (2\right )^{2} \log \left (x\right )^{2} + 3750 \, x \log \left (2\right ) \log \left (x\right )^{2} + 15625 \, \log \left (x\right )^{2}\right )}} \]

[In]

integrate(((32*x^6*log(4/x)^3+2400*x^5*log(4/x)^2+60000*x^4*log(4/x)+500000*x^3)*log(x)^3+(12*x^5*log(4/x)^2+(
4*x^5+700*x^4)*log(4/x)+100*x^4+10000*x^3)*log(x)^2+(-4*x^5*log(4/x)^2-199*x^4*log(4/x)+x^4-2450*x^3)*log(x)-x
^4*log(4/x)-25*x^3)/(2*x^3*log(4/x)^3+150*x^2*log(4/x)^2+3750*x*log(4/x)+31250)/log(x)^3,x, algorithm="giac")

[Out]

4*x^4 + 1/4*(64*x^8*log(2)^3 - 32*x^8*log(2)^2*log(x) + 2400*x^7*log(2)^2 - 800*x^7*log(2)*log(x) + 6*x^7*log(
2) - 2*x^7*log(x) + 30000*x^6*log(2) - 5000*x^6*log(x) + 75*x^6 + 125000*x^5)/(32*x^5*log(2)^5 - 32*x^5*log(2)
^4*log(x) + 8*x^5*log(2)^3*log(x)^2 + 2000*x^4*log(2)^4 - 1600*x^4*log(2)^3*log(x) + 300*x^4*log(2)^2*log(x)^2
+ 50000*x^3*log(2)^3 - 30000*x^3*log(2)^2*log(x) + 3750*x^3*log(2)*log(x)^2 + 625000*x^2*log(2)^2 - 250000*x^
2*log(2)*log(x) + 15625*x^2*log(x)^2 + 3906250*x*log(2) - 781250*x*log(x) + 9765625) + 1/4*(32*x^6*log(2)^2*lo
g(x) + 800*x^5*log(2)*log(x) + 2*x^5*log(2) + 2*x^5*log(x) + 5000*x^4*log(x) + 25*x^4)/(8*x^3*log(2)^3*log(x)^
2 + 300*x^2*log(2)^2*log(x)^2 + 3750*x*log(2)*log(x)^2 + 15625*log(x)^2)

Mupad [B] (verification not implemented)

Time = 9.57 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.33 \[ \int \frac {-25 x^3-x^4 \log \left (\frac {4}{x}\right )+\left (-2450 x^3+x^4-199 x^4 \log \left (\frac {4}{x}\right )-4 x^5 \log ^2\left (\frac {4}{x}\right )\right ) \log (x)+\left (10000 x^3+100 x^4+\left (700 x^4+4 x^5\right ) \log \left (\frac {4}{x}\right )+12 x^5 \log ^2\left (\frac {4}{x}\right )\right ) \log ^2(x)+\left (500000 x^3+60000 x^4 \log \left (\frac {4}{x}\right )+2400 x^5 \log ^2\left (\frac {4}{x}\right )+32 x^6 \log ^3\left (\frac {4}{x}\right )\right ) \log ^3(x)}{\left (31250+3750 x \log \left (\frac {4}{x}\right )+150 x^2 \log ^2\left (\frac {4}{x}\right )+2 x^3 \log ^3\left (\frac {4}{x}\right )\right ) \log ^3(x)} \, dx=\frac {x^4\,{\left (100\,\ln \left (x\right )+4\,x\,\ln \left (\frac {4}{x}\right )\,\ln \left (x\right )+1\right )}^2}{4\,{\ln \left (x\right )}^2\,{\left (x\,\ln \left (\frac {4}{x}\right )+25\right )}^2} \]

[In]

int(-(log(x)*(4*x^5*log(4/x)^2 + 2450*x^3 - x^4 + 199*x^4*log(4/x)) - log(x)^2*(12*x^5*log(4/x)^2 + log(4/x)*(
700*x^4 + 4*x^5) + 10000*x^3 + 100*x^4) - log(x)^3*(2400*x^5*log(4/x)^2 + 32*x^6*log(4/x)^3 + 500000*x^3 + 600
00*x^4*log(4/x)) + 25*x^3 + x^4*log(4/x))/(log(x)^3*(150*x^2*log(4/x)^2 + 2*x^3*log(4/x)^3 + 3750*x*log(4/x) +
31250)),x)

[Out]

(x^4*(100*log(x) + 4*x*log(4/x)*log(x) + 1)^2)/(4*log(x)^2*(x*log(4/x) + 25)^2)