\(\int \frac {(2+2 x) \log (\frac {5}{4 x^3})+(2+4 x+2 x^2) \log ^2(\frac {5}{4 x^3})+(-3-3 x+x \log (\frac {5}{4 x^3})) \log (x^2)}{x+(2 x+2 x^2) \log (\frac {5}{4 x^3})+(x+2 x^2+x^3) \log ^2(\frac {5}{4 x^3})} \, dx\) [4769]

Optimal result

Integrand size = 100, antiderivative size = 27 \[ \int \frac {(2+2 x) \log \left (\frac {5}{4 x^3}\right )+\left (2+4 x+2 x^2\right ) \log ^2\left (\frac {5}{4 x^3}\right )+\left (-3-3 x+x \log \left (\frac {5}{4 x^3}\right )\right ) \log \left (x^2\right )}{x+\left (2 x+2 x^2\right ) \log \left (\frac {5}{4 x^3}\right )+\left (x+2 x^2+x^3\right ) \log ^2\left (\frac {5}{4 x^3}\right )} \, dx=\frac {x \log \left (x^2\right )}{x+\frac {x}{(1+x) \log \left (\frac {5}{4 x^3}\right )}} \]

[Out]

x*ln(x^2)/(x/(1+x)/ln(5/4/x^3)+x)

Rubi [F]

\[ \int \frac {(2+2 x) \log \left (\frac {5}{4 x^3}\right )+\left (2+4 x+2 x^2\right ) \log ^2\left (\frac {5}{4 x^3}\right )+\left (-3-3 x+x \log \left (\frac {5}{4 x^3}\right )\right ) \log \left (x^2\right )}{x+\left (2 x+2 x^2\right ) \log \left (\frac {5}{4 x^3}\right )+\left (x+2 x^2+x^3\right ) \log ^2\left (\frac {5}{4 x^3}\right )} \, dx=\int \frac {(2+2 x) \log \left (\frac {5}{4 x^3}\right )+\left (2+4 x+2 x^2\right ) \log ^2\left (\frac {5}{4 x^3}\right )+\left (-3-3 x+x \log \left (\frac {5}{4 x^3}\right )\right ) \log \left (x^2\right )}{x+\left (2 x+2 x^2\right ) \log \left (\frac {5}{4 x^3}\right )+\left (x+2 x^2+x^3\right ) \log ^2\left (\frac {5}{4 x^3}\right )} \, dx \]

[In]

Int[((2 + 2*x)*Log[5/(4*x^3)] + (2 + 4*x + 2*x^2)*Log[5/(4*x^3)]^2 + (-3 - 3*x + x*Log[5/(4*x^3)])*Log[x^2])/(
x + (2*x + 2*x^2)*Log[5/(4*x^3)] + (x + 2*x^2 + x^3)*Log[5/(4*x^3)]^2),x]

[Out]

2*Log[x] - 2*Defer[Int][(x + x*(1 + x)*Log[5/(4*x^3)])^(-1), x] - 3*Defer[Int][Log[x^2]/(1 + (1 + x)*Log[5/(4*
x^3)])^2, x] - 3*Defer[Int][Log[x^2]/(x*(1 + (1 + x)*Log[5/(4*x^3)])^2), x] + Defer[Int][(Log[5/(4*x^3)]*Log[x
^2])/(1 + (1 + x)*Log[5/(4*x^3)])^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {2 (1+x)^2 \log ^2\left (\frac {5}{4 x^3}\right )-3 (1+x) \log \left (x^2\right )+\log \left (\frac {5}{4 x^3}\right ) \left (2+2 x+x \log \left (x^2\right )\right )}{x \left (1+(1+x) \log \left (\frac {5}{4 x^3}\right )\right )^2} \, dx \\ & = \int \left (\frac {2 (1+x) \log \left (\frac {5}{4 x^3}\right )}{x \left (1+\log \left (\frac {5}{4 x^3}\right )+x \log \left (\frac {5}{4 x^3}\right )\right )}+\frac {\left (-3-3 x+x \log \left (\frac {5}{4 x^3}\right )\right ) \log \left (x^2\right )}{x \left (1+\log \left (\frac {5}{4 x^3}\right )+x \log \left (\frac {5}{4 x^3}\right )\right )^2}\right ) \, dx \\ & = 2 \int \frac {(1+x) \log \left (\frac {5}{4 x^3}\right )}{x \left (1+\log \left (\frac {5}{4 x^3}\right )+x \log \left (\frac {5}{4 x^3}\right )\right )} \, dx+\int \frac {\left (-3-3 x+x \log \left (\frac {5}{4 x^3}\right )\right ) \log \left (x^2\right )}{x \left (1+\log \left (\frac {5}{4 x^3}\right )+x \log \left (\frac {5}{4 x^3}\right )\right )^2} \, dx \\ & = 2 \int \frac {(1+x) \log \left (\frac {5}{4 x^3}\right )}{x+x (1+x) \log \left (\frac {5}{4 x^3}\right )} \, dx+\int \left (-\frac {3 \log \left (x^2\right )}{\left (1+\log \left (\frac {5}{4 x^3}\right )+x \log \left (\frac {5}{4 x^3}\right )\right )^2}-\frac {3 \log \left (x^2\right )}{x \left (1+\log \left (\frac {5}{4 x^3}\right )+x \log \left (\frac {5}{4 x^3}\right )\right )^2}+\frac {\log \left (\frac {5}{4 x^3}\right ) \log \left (x^2\right )}{\left (1+\log \left (\frac {5}{4 x^3}\right )+x \log \left (\frac {5}{4 x^3}\right )\right )^2}\right ) \, dx \\ & = 2 \int \left (\frac {1}{x}-\frac {1}{x \left (1+\log \left (\frac {5}{4 x^3}\right )+x \log \left (\frac {5}{4 x^3}\right )\right )}\right ) \, dx-3 \int \frac {\log \left (x^2\right )}{\left (1+\log \left (\frac {5}{4 x^3}\right )+x \log \left (\frac {5}{4 x^3}\right )\right )^2} \, dx-3 \int \frac {\log \left (x^2\right )}{x \left (1+\log \left (\frac {5}{4 x^3}\right )+x \log \left (\frac {5}{4 x^3}\right )\right )^2} \, dx+\int \frac {\log \left (\frac {5}{4 x^3}\right ) \log \left (x^2\right )}{\left (1+\log \left (\frac {5}{4 x^3}\right )+x \log \left (\frac {5}{4 x^3}\right )\right )^2} \, dx \\ & = 2 \log (x)-2 \int \frac {1}{x \left (1+\log \left (\frac {5}{4 x^3}\right )+x \log \left (\frac {5}{4 x^3}\right )\right )} \, dx-3 \int \frac {\log \left (x^2\right )}{\left (1+(1+x) \log \left (\frac {5}{4 x^3}\right )\right )^2} \, dx-3 \int \frac {\log \left (x^2\right )}{x \left (1+(1+x) \log \left (\frac {5}{4 x^3}\right )\right )^2} \, dx+\int \frac {\log \left (\frac {5}{4 x^3}\right ) \log \left (x^2\right )}{\left (1+(1+x) \log \left (\frac {5}{4 x^3}\right )\right )^2} \, dx \\ & = 2 \log (x)-2 \int \frac {1}{x+x (1+x) \log \left (\frac {5}{4 x^3}\right )} \, dx-3 \int \frac {\log \left (x^2\right )}{\left (1+(1+x) \log \left (\frac {5}{4 x^3}\right )\right )^2} \, dx-3 \int \frac {\log \left (x^2\right )}{x \left (1+(1+x) \log \left (\frac {5}{4 x^3}\right )\right )^2} \, dx+\int \frac {\log \left (\frac {5}{4 x^3}\right ) \log \left (x^2\right )}{\left (1+(1+x) \log \left (\frac {5}{4 x^3}\right )\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {(2+2 x) \log \left (\frac {5}{4 x^3}\right )+\left (2+4 x+2 x^2\right ) \log ^2\left (\frac {5}{4 x^3}\right )+\left (-3-3 x+x \log \left (\frac {5}{4 x^3}\right )\right ) \log \left (x^2\right )}{x+\left (2 x+2 x^2\right ) \log \left (\frac {5}{4 x^3}\right )+\left (x+2 x^2+x^3\right ) \log ^2\left (\frac {5}{4 x^3}\right )} \, dx=2 \log (x)-\frac {\log \left (x^2\right )}{1+(1+x) \log \left (\frac {5}{4 x^3}\right )} \]

[In]

Integrate[((2 + 2*x)*Log[5/(4*x^3)] + (2 + 4*x + 2*x^2)*Log[5/(4*x^3)]^2 + (-3 - 3*x + x*Log[5/(4*x^3)])*Log[x
^2])/(x + (2*x + 2*x^2)*Log[5/(4*x^3)] + (x + 2*x^2 + x^3)*Log[5/(4*x^3)]^2),x]

[Out]

2*Log[x] - Log[x^2]/(1 + (1 + x)*Log[5/(4*x^3)])

Maple [A] (verified)

Time = 2.77 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.74

method	result	size
parallelrisch	\(\frac {8 \ln \left (x^{2}\right ) \ln \left (\frac {5}{4 x^{3}}\right )+8 x \ln \left (x^{2}\right ) \ln \left (\frac {5}{4 x^{3}}\right )}{8 x \ln \left (\frac {5}{4 x^{3}}\right )+8 \ln \left (\frac {5}{4 x^{3}}\right )+8}\)	\(47\)
risch	\(\frac {2 x \ln \left (x \right )+2 \ln \left (x \right )+\frac {2}{3}}{1+x}-\frac {4-2 i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x^{3}\right )^{2}+2 i x \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x^{3}\right )+2 i x \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+2 i x \pi \operatorname {csgn}\left (i x^{3}\right )^{3}+2 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x^{3}\right )-i x \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+4 x \ln \left (5\right )-8 x \ln \left (2\right )+4 \ln \left (5\right )-8 \ln \left (2\right )+2 i \pi \operatorname {csgn}\left (i x^{3}\right )^{3}+2 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-2 i x \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x^{3}\right )^{2}-2 i x \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{3}\right )^{2}-2 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{3}\right )^{2}-i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}-i x \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )}{3 \left (1+x \right ) \left (2-i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x^{3}\right )^{2}+i x \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x^{3}\right )-2 i x \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+i x \pi \operatorname {csgn}\left (i x^{3}\right )^{3}+i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x^{3}\right )+i x \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+i \pi \operatorname {csgn}\left (i x^{3}\right )^{3}-2 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-i x \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x^{3}\right )^{2}-i x \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{3}\right )^{2}-i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{3}\right )^{2}+2 x \ln \left (5\right )-4 x \ln \left (2\right )-6 x \ln \left (x \right )+2 \ln \left (5\right )-4 \ln \left (2\right )-6 \ln \left (x \right )+i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+i x \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )\right )}\)	\(584\)

[In]

int(((x*ln(5/4/x^3)-3*x-3)*ln(x^2)+(2*x^2+4*x+2)*ln(5/4/x^3)^2+(2+2*x)*ln(5/4/x^3))/((x^3+2*x^2+x)*ln(5/4/x^3)
^2+(2*x^2+2*x)*ln(5/4/x^3)+x),x,method=_RETURNVERBOSE)

[Out]

1/8*(8*ln(x^2)*ln(5/4/x^3)+8*x*ln(x^2)*ln(5/4/x^3))/(x*ln(5/4/x^3)+ln(5/4/x^3)+1)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {(2+2 x) \log \left (\frac {5}{4 x^3}\right )+\left (2+4 x+2 x^2\right ) \log ^2\left (\frac {5}{4 x^3}\right )+\left (-3-3 x+x \log \left (\frac {5}{4 x^3}\right )\right ) \log \left (x^2\right )}{x+\left (2 x+2 x^2\right ) \log \left (\frac {5}{4 x^3}\right )+\left (x+2 x^2+x^3\right ) \log ^2\left (\frac {5}{4 x^3}\right )} \, dx=-\frac {2 \, {\left (x + 1\right )} \log \left (\frac {5}{4 \, x^{3}}\right )^{2} + \log \left (\frac {25}{16}\right )}{3 \, {\left ({\left (x + 1\right )} \log \left (\frac {5}{4 \, x^{3}}\right ) + 1\right )}} \]

[In]

integrate(((x*log(5/4/x^3)-3*x-3)*log(x^2)+(2*x^2+4*x+2)*log(5/4/x^3)^2+(2+2*x)*log(5/4/x^3))/((x^3+2*x^2+x)*l
og(5/4/x^3)^2+(2*x^2+2*x)*log(5/4/x^3)+x),x, algorithm="fricas")

[Out]

-1/3*(2*(x + 1)*log(5/4/x^3)^2 + log(25/16))/((x + 1)*log(5/4/x^3) + 1)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (20) = 40\).

Time = 0.32 (sec) , antiderivative size = 97, normalized size of antiderivative = 3.59 \[ \int \frac {(2+2 x) \log \left (\frac {5}{4 x^3}\right )+\left (2+4 x+2 x^2\right ) \log ^2\left (\frac {5}{4 x^3}\right )+\left (-3-3 x+x \log \left (\frac {5}{4 x^3}\right )\right ) \log \left (x^2\right )}{x+\left (2 x+2 x^2\right ) \log \left (\frac {5}{4 x^3}\right )+\left (x+2 x^2+x^3\right ) \log ^2\left (\frac {5}{4 x^3}\right )} \, dx=\frac {- 8 x \log {\left (2 \right )} + 4 x \log {\left (5 \right )} - 8 \log {\left (2 \right )} + 4 + 4 \log {\left (5 \right )}}{- 6 x^{2} \log {\left (5 \right )} + 12 x^{2} \log {\left (2 \right )} - 12 x \log {\left (5 \right )} - 6 x + 24 x \log {\left (2 \right )} + \left (9 x^{2} + 18 x + 9\right ) \log {\left (x^{2} \right )} - 6 \log {\left (5 \right )} - 6 + 12 \log {\left (2 \right )}} + 2 \log {\left (x \right )} + \frac {2}{3 x + 3} \]

[In]

integrate(((x*ln(5/4/x**3)-3*x-3)*ln(x**2)+(2*x**2+4*x+2)*ln(5/4/x**3)**2+(2+2*x)*ln(5/4/x**3))/((x**3+2*x**2+
x)*ln(5/4/x**3)**2+(2*x**2+2*x)*ln(5/4/x**3)+x),x)

[Out]

(-8*x*log(2) + 4*x*log(5) - 8*log(2) + 4 + 4*log(5))/(-6*x**2*log(5) + 12*x**2*log(2) - 12*x*log(5) - 6*x + 24
*x*log(2) + (9*x**2 + 18*x + 9)*log(x**2) - 6*log(5) - 6 + 12*log(2)) + 2*log(x) + 2/(3*x + 3)

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.30 \[ \int \frac {(2+2 x) \log \left (\frac {5}{4 x^3}\right )+\left (2+4 x+2 x^2\right ) \log ^2\left (\frac {5}{4 x^3}\right )+\left (-3-3 x+x \log \left (\frac {5}{4 x^3}\right )\right ) \log \left (x^2\right )}{x+\left (2 x+2 x^2\right ) \log \left (\frac {5}{4 x^3}\right )+\left (x+2 x^2+x^3\right ) \log ^2\left (\frac {5}{4 x^3}\right )} \, dx=-\frac {2 \, \log \left (x\right )}{x {\left (\log \left (5\right ) - 2 \, \log \left (2\right )\right )} - 3 \, {\left (x + 1\right )} \log \left (x\right ) + \log \left (5\right ) - 2 \, \log \left (2\right ) + 1} + 2 \, \log \left (x\right ) \]

[In]

integrate(((x*log(5/4/x^3)-3*x-3)*log(x^2)+(2*x^2+4*x+2)*log(5/4/x^3)^2+(2+2*x)*log(5/4/x^3))/((x^3+2*x^2+x)*l
og(5/4/x^3)^2+(2*x^2+2*x)*log(5/4/x^3)+x),x, algorithm="maxima")

[Out]

-2*log(x)/(x*(log(5) - 2*log(2)) - 3*(x + 1)*log(x) + log(5) - 2*log(2) + 1) + 2*log(x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (25) = 50\).

Time = 0.32 (sec) , antiderivative size = 81, normalized size of antiderivative = 3.00 \[ \int \frac {(2+2 x) \log \left (\frac {5}{4 x^3}\right )+\left (2+4 x+2 x^2\right ) \log ^2\left (\frac {5}{4 x^3}\right )+\left (-3-3 x+x \log \left (\frac {5}{4 x^3}\right )\right ) \log \left (x^2\right )}{x+\left (2 x+2 x^2\right ) \log \left (\frac {5}{4 x^3}\right )+\left (x+2 x^2+x^3\right ) \log ^2\left (\frac {5}{4 x^3}\right )} \, dx=-\frac {2 \, {\left (x \log \left (5\right ) - 2 \, x \log \left (2\right ) + \log \left (5\right ) - 2 \, \log \left (2\right ) + 1\right )}}{3 \, {\left (x^{2} \log \left (5\right ) - 2 \, x^{2} \log \left (2\right ) - 3 \, x^{2} \log \left (x\right ) + 2 \, x \log \left (5\right ) - 4 \, x \log \left (2\right ) - 6 \, x \log \left (x\right ) + x + \log \left (5\right ) - 2 \, \log \left (2\right ) - 3 \, \log \left (x\right ) + 1\right )}} + \frac {2}{3 \, {\left (x + 1\right )}} + 2 \, \log \left (x\right ) \]

[In]

integrate(((x*log(5/4/x^3)-3*x-3)*log(x^2)+(2*x^2+4*x+2)*log(5/4/x^3)^2+(2+2*x)*log(5/4/x^3))/((x^3+2*x^2+x)*l
og(5/4/x^3)^2+(2*x^2+2*x)*log(5/4/x^3)+x),x, algorithm="giac")

[Out]

-2/3*(x*log(5) - 2*x*log(2) + log(5) - 2*log(2) + 1)/(x^2*log(5) - 2*x^2*log(2) - 3*x^2*log(x) + 2*x*log(5) -
4*x*log(2) - 6*x*log(x) + x + log(5) - 2*log(2) - 3*log(x) + 1) + 2/3/(x + 1) + 2*log(x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(2+2 x) \log \left (\frac {5}{4 x^3}\right )+\left (2+4 x+2 x^2\right ) \log ^2\left (\frac {5}{4 x^3}\right )+\left (-3-3 x+x \log \left (\frac {5}{4 x^3}\right )\right ) \log \left (x^2\right )}{x+\left (2 x+2 x^2\right ) \log \left (\frac {5}{4 x^3}\right )+\left (x+2 x^2+x^3\right ) \log ^2\left (\frac {5}{4 x^3}\right )} \, dx=\int \frac {\ln \left (\frac {5}{4\,x^3}\right )\,\left (2\,x+2\right )-\ln \left (x^2\right )\,\left (3\,x-x\,\ln \left (\frac {5}{4\,x^3}\right )+3\right )+{\ln \left (\frac {5}{4\,x^3}\right )}^2\,\left (2\,x^2+4\,x+2\right )}{\left (x^3+2\,x^2+x\right )\,{\ln \left (\frac {5}{4\,x^3}\right )}^2+\left (2\,x^2+2\,x\right )\,\ln \left (\frac {5}{4\,x^3}\right )+x} \,d x \]

[In]

int((log(5/(4*x^3))*(2*x + 2) - log(x^2)*(3*x - x*log(5/(4*x^3)) + 3) + log(5/(4*x^3))^2*(4*x + 2*x^2 + 2))/(x
+ log(5/(4*x^3))^2*(x + 2*x^2 + x^3) + log(5/(4*x^3))*(2*x + 2*x^2)),x)

[Out]

int((log(5/(4*x^3))*(2*x + 2) - log(x^2)*(3*x - x*log(5/(4*x^3)) + 3) + log(5/(4*x^3))^2*(4*x + 2*x^2 + 2))/(x
+ log(5/(4*x^3))^2*(x + 2*x^2 + x^3) + log(5/(4*x^3))*(2*x + 2*x^2)), x)