3.21.80 \(\int \frac {4+(320+60 x-5 x^2) \log ^2(\frac {4+x}{x})+\log ^2(2 x) (4+(-20 x-5 x^2) \log ^2(\frac {4+x}{x}))+\log (2 x) (-8+(40 x+10 x^2) \log ^2(\frac {4+x}{x}))}{(20 x+5 x^2) \log ^2(\frac {4+x}{x})+(-40 x-10 x^2) \log (2 x) \log ^2(\frac {4+x}{x})+(20 x+5 x^2) \log ^2(2 x) \log ^2(\frac {4+x}{x})} \, dx\)

Optimal. Leaf size=31 \[ 5-x+\frac {16}{1-\log (2 x)}+\frac {1}{5 \log \left (\frac {4+x}{x}\right )} \]

________________________________________________________________________________________

Rubi [F]  time = 5.62, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {4+\left (320+60 x-5 x^2\right ) \log ^2\left (\frac {4+x}{x}\right )+\log ^2(2 x) \left (4+\left (-20 x-5 x^2\right ) \log ^2\left (\frac {4+x}{x}\right )\right )+\log (2 x) \left (-8+\left (40 x+10 x^2\right ) \log ^2\left (\frac {4+x}{x}\right )\right )}{\left (20 x+5 x^2\right ) \log ^2\left (\frac {4+x}{x}\right )+\left (-40 x-10 x^2\right ) \log (2 x) \log ^2\left (\frac {4+x}{x}\right )+\left (20 x+5 x^2\right ) \log ^2(2 x) \log ^2\left (\frac {4+x}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4 + (320 + 60*x - 5*x^2)*Log[(4 + x)/x]^2 + Log[2*x]^2*(4 + (-20*x - 5*x^2)*Log[(4 + x)/x]^2) + Log[2*x]*
(-8 + (40*x + 10*x^2)*Log[(4 + x)/x]^2))/((20*x + 5*x^2)*Log[(4 + x)/x]^2 + (-40*x - 10*x^2)*Log[2*x]*Log[(4 +
x)/x]^2 + (20*x + 5*x^2)*Log[2*x]^2*Log[(4 + x)/x]^2),x]

[Out]

-x + 16/(1 - Log[2*x]) + Defer[Int][1/(x*Log[1 + 4/x]^2), x]/5 - Defer[Int][1/((4 + x)*Log[1 + 4/x]^2), x]/5 +
(2*Defer[Int][1/((-4 - x)*Log[1 + 4/x]^2*(1 - Log[2*x])^2), x])/5 + (2*Defer[Int][1/((4 + x)*Log[1 + 4/x]^2*(
1 - Log[2*x])^2), x])/5 + (2*Defer[Int][1/((-4 - x)*Log[1 + 4/x]^2*(1 - Log[2*x])), x])/5 + (2*Defer[Int][1/(x
*Log[1 + 4/x]^2*(1 - Log[2*x])), x])/5 + (2*Defer[Int][1/((4 + x)*Log[1 + 4/x]^2*(1 - Log[2*x])), x])/5 + (2*D
efer[Int][1/(x*Log[1 + 4/x]^2*(-1 + Log[2*x])), x])/5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4-5 \left (-64-12 x+x^2\right ) \log ^2\left (\frac {4+x}{x}\right )+\log ^2(2 x) \left (4-5 x (4+x) \log ^2\left (\frac {4+x}{x}\right )\right )+2 \log (2 x) \left (-4+5 x (4+x) \log ^2\left (\frac {4+x}{x}\right )\right )}{5 x (4+x) \log ^2\left (1+\frac {4}{x}\right ) (1-\log (2 x))^2} \, dx\\ &=\frac {1}{5} \int \frac {4-5 \left (-64-12 x+x^2\right ) \log ^2\left (\frac {4+x}{x}\right )+\log ^2(2 x) \left (4-5 x (4+x) \log ^2\left (\frac {4+x}{x}\right )\right )+2 \log (2 x) \left (-4+5 x (4+x) \log ^2\left (\frac {4+x}{x}\right )\right )}{x (4+x) \log ^2\left (1+\frac {4}{x}\right ) (1-\log (2 x))^2} \, dx\\ &=\frac {1}{5} \int \left (\frac {4}{x (4+x) \log ^2\left (1+\frac {4}{x}\right ) (1-\log (2 x))^2}+\frac {8 \log (2 x)}{(-4-x) x \log ^2\left (1+\frac {4}{x}\right ) (1-\log (2 x))^2}+\frac {4 \log ^2(2 x)}{x (4+x) \log ^2\left (1+\frac {4}{x}\right ) (1-\log (2 x))^2}-\frac {5 \left (-16+x-2 x \log (2 x)+x \log ^2(2 x)\right )}{x (-1+\log (2 x))^2}\right ) \, dx\\ &=\frac {4}{5} \int \frac {1}{x (4+x) \log ^2\left (1+\frac {4}{x}\right ) (1-\log (2 x))^2} \, dx+\frac {4}{5} \int \frac {\log ^2(2 x)}{x (4+x) \log ^2\left (1+\frac {4}{x}\right ) (1-\log (2 x))^2} \, dx+\frac {8}{5} \int \frac {\log (2 x)}{(-4-x) x \log ^2\left (1+\frac {4}{x}\right ) (1-\log (2 x))^2} \, dx-\int \frac {-16+x-2 x \log (2 x)+x \log ^2(2 x)}{x (-1+\log (2 x))^2} \, dx\\ &=\frac {4}{5} \int \left (\frac {1}{4 (-4-x) \log ^2\left (1+\frac {4}{x}\right ) (1-\log (2 x))^2}+\frac {1}{4 x \log ^2\left (1+\frac {4}{x}\right ) (1-\log (2 x))^2}\right ) \, dx+\frac {4}{5} \int \left (\frac {1}{x (4+x) \log ^2\left (1+\frac {4}{x}\right )}+\frac {1}{x (4+x) \log ^2\left (1+\frac {4}{x}\right ) (1-\log (2 x))^2}+\frac {2}{(-4-x) x \log ^2\left (1+\frac {4}{x}\right ) (1-\log (2 x))}\right ) \, dx+\frac {8}{5} \int \left (\frac {1}{(-4-x) x \log ^2\left (1+\frac {4}{x}\right ) (1-\log (2 x))^2}+\frac {1}{x (4+x) \log ^2\left (1+\frac {4}{x}\right ) (1-\log (2 x))}\right ) \, dx-\int \left (1-\frac {16}{x (-1+\log (2 x))^2}\right ) \, dx\\ &=-x+\frac {1}{5} \int \frac {1}{(-4-x) \log ^2\left (1+\frac {4}{x}\right ) (1-\log (2 x))^2} \, dx+\frac {1}{5} \int \frac {1}{x \log ^2\left (1+\frac {4}{x}\right ) (1-\log (2 x))^2} \, dx+\frac {4}{5} \int \frac {1}{x (4+x) \log ^2\left (1+\frac {4}{x}\right )} \, dx+\frac {4}{5} \int \frac {1}{x (4+x) \log ^2\left (1+\frac {4}{x}\right ) (1-\log (2 x))^2} \, dx+\frac {8}{5} \int \frac {1}{(-4-x) x \log ^2\left (1+\frac {4}{x}\right ) (1-\log (2 x))^2} \, dx+\frac {8}{5} \int \frac {1}{(-4-x) x \log ^2\left (1+\frac {4}{x}\right ) (1-\log (2 x))} \, dx+\frac {8}{5} \int \frac {1}{x (4+x) \log ^2\left (1+\frac {4}{x}\right ) (1-\log (2 x))} \, dx+16 \int \frac {1}{x (-1+\log (2 x))^2} \, dx\\ &=-x+\frac {1}{5} \int \frac {1}{(-4-x) \log ^2\left (1+\frac {4}{x}\right ) (1-\log (2 x))^2} \, dx+\frac {1}{5} \int \frac {1}{x \log ^2\left (1+\frac {4}{x}\right ) (1-\log (2 x))^2} \, dx+\frac {4}{5} \int \left (\frac {1}{4 x \log ^2\left (1+\frac {4}{x}\right )}-\frac {1}{4 (4+x) \log ^2\left (1+\frac {4}{x}\right )}\right ) \, dx+\frac {4}{5} \int \left (\frac {1}{4 (-4-x) \log ^2\left (1+\frac {4}{x}\right ) (1-\log (2 x))^2}+\frac {1}{4 x \log ^2\left (1+\frac {4}{x}\right ) (1-\log (2 x))^2}\right ) \, dx+\frac {8}{5} \int \left (-\frac {1}{4 x \log ^2\left (1+\frac {4}{x}\right ) (1-\log (2 x))^2}+\frac {1}{4 (4+x) \log ^2\left (1+\frac {4}{x}\right ) (1-\log (2 x))^2}\right ) \, dx+\frac {8}{5} \int \left (\frac {1}{4 (-4-x) \log ^2\left (1+\frac {4}{x}\right ) (1-\log (2 x))}+\frac {1}{4 x \log ^2\left (1+\frac {4}{x}\right ) (1-\log (2 x))}\right ) \, dx+\frac {8}{5} \int \left (\frac {1}{4 (4+x) \log ^2\left (1+\frac {4}{x}\right ) (1-\log (2 x))}+\frac {1}{4 x \log ^2\left (1+\frac {4}{x}\right ) (-1+\log (2 x))}\right ) \, dx+16 \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,-1+\log (2 x)\right )\\ &=-x+\frac {16}{1-\log (2 x)}+\frac {1}{5} \int \frac {1}{x \log ^2\left (1+\frac {4}{x}\right )} \, dx-\frac {1}{5} \int \frac {1}{(4+x) \log ^2\left (1+\frac {4}{x}\right )} \, dx+2 \left (\frac {1}{5} \int \frac {1}{(-4-x) \log ^2\left (1+\frac {4}{x}\right ) (1-\log (2 x))^2} \, dx\right )+2 \left (\frac {1}{5} \int \frac {1}{x \log ^2\left (1+\frac {4}{x}\right ) (1-\log (2 x))^2} \, dx\right )-\frac {2}{5} \int \frac {1}{x \log ^2\left (1+\frac {4}{x}\right ) (1-\log (2 x))^2} \, dx+\frac {2}{5} \int \frac {1}{(4+x) \log ^2\left (1+\frac {4}{x}\right ) (1-\log (2 x))^2} \, dx+\frac {2}{5} \int \frac {1}{(-4-x) \log ^2\left (1+\frac {4}{x}\right ) (1-\log (2 x))} \, dx+\frac {2}{5} \int \frac {1}{x \log ^2\left (1+\frac {4}{x}\right ) (1-\log (2 x))} \, dx+\frac {2}{5} \int \frac {1}{(4+x) \log ^2\left (1+\frac {4}{x}\right ) (1-\log (2 x))} \, dx+\frac {2}{5} \int \frac {1}{x \log ^2\left (1+\frac {4}{x}\right ) (-1+\log (2 x))} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.08, size = 28, normalized size = 0.90 \begin {gather*} \frac {1}{5} \left (-5 x-\frac {80}{-1+\log (2 x)}+\frac {1}{\log \left (\frac {4+x}{x}\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 + (320 + 60*x - 5*x^2)*Log[(4 + x)/x]^2 + Log[2*x]^2*(4 + (-20*x - 5*x^2)*Log[(4 + x)/x]^2) + Log
[2*x]*(-8 + (40*x + 10*x^2)*Log[(4 + x)/x]^2))/((20*x + 5*x^2)*Log[(4 + x)/x]^2 + (-40*x - 10*x^2)*Log[2*x]*Lo
g[(4 + x)/x]^2 + (20*x + 5*x^2)*Log[2*x]^2*Log[(4 + x)/x]^2),x]

[Out]

(-5*x - 80/(-1 + Log[2*x]) + Log[(4 + x)/x]^(-1))/5

________________________________________________________________________________________

fricas [B]  time = 0.94, size = 61, normalized size = 1.97 \begin {gather*} -\frac {{\left (5 \, x \log \left (\frac {x + 4}{x}\right ) - 1\right )} \log \left (2 \, x\right ) - 5 \, {\left (x - 16\right )} \log \left (\frac {x + 4}{x}\right ) + 1}{5 \, {\left (\log \left (2 \, x\right ) \log \left (\frac {x + 4}{x}\right ) - \log \left (\frac {x + 4}{x}\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-5*x^2-20*x)*log((4+x)/x)^2+4)*log(2*x)^2+((10*x^2+40*x)*log((4+x)/x)^2-8)*log(2*x)+(-5*x^2+60*x+
320)*log((4+x)/x)^2+4)/((5*x^2+20*x)*log((4+x)/x)^2*log(2*x)^2+(-10*x^2-40*x)*log((4+x)/x)^2*log(2*x)+(5*x^2+2
0*x)*log((4+x)/x)^2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.53, size = 27, normalized size = 0.87 \begin {gather*} -x - \frac {16}{\log \relax (2) + \log \relax (x) - 1} + \frac {1}{5 \, {\left (\log \left (x + 4\right ) - \log \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-5*x^2-20*x)*log((4+x)/x)^2+4)*log(2*x)^2+((10*x^2+40*x)*log((4+x)/x)^2-8)*log(2*x)+(-5*x^2+60*x+
320)*log((4+x)/x)^2+4)/((5*x^2+20*x)*log((4+x)/x)^2*log(2*x)^2+(-10*x^2-40*x)*log((4+x)/x)^2*log(2*x)+(5*x^2+2
0*x)*log((4+x)/x)^2),x, algorithm="giac")

[Out]

-x - 16/(log(2) + log(x) - 1) + 1/5/(log(x + 4) - log(x))

________________________________________________________________________________________

maple [A]  time = 0.67, size = 38, normalized size = 1.23

method	result	size
default	\(\frac {1}{5 \ln \left (\frac {4}{x}+1\right )}+\frac {x \left (1-\ln \relax (2)\right )-x \ln \relax (x )-16}{\ln \relax (x )+\ln \relax (2)-1}\)	\(38\)
risch	\(-\frac {2 i x \ln \relax (2)+2 i x \ln \relax (x )-2 i x +32 i}{2 i \ln \relax (2)+2 i \ln \relax (x )-2 i}+\frac {2 i}{5 \left (\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (4+x \right )\right ) \mathrm {csgn}\left (\frac {i \left (4+x \right )}{x}\right )-\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (4+x \right )}{x}\right )^{2}-\pi \,\mathrm {csgn}\left (i \left (4+x \right )\right ) \mathrm {csgn}\left (\frac {i \left (4+x \right )}{x}\right )^{2}+\pi \mathrm {csgn}\left (\frac {i \left (4+x \right )}{x}\right )^{3}-2 i \ln \relax (x )+2 i \ln \left (4+x \right )\right )}\)	\(140\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-5*x^2-20*x)*ln((4+x)/x)^2+4)*ln(2*x)^2+((10*x^2+40*x)*ln((4+x)/x)^2-8)*ln(2*x)+(-5*x^2+60*x+320)*ln((4
+x)/x)^2+4)/((5*x^2+20*x)*ln((4+x)/x)^2*ln(2*x)^2+(-10*x^2-40*x)*ln((4+x)/x)^2*ln(2*x)+(5*x^2+20*x)*ln((4+x)/x
)^2),x,method=_RETURNVERBOSE)

[Out]

1/5/ln(4/x+1)+(x*(1-ln(2))-x*ln(x)-16)/(ln(x)+ln(2)-1)

________________________________________________________________________________________

maxima [B]  time = 0.88, size = 71, normalized size = 2.29 \begin {gather*} \frac {5 \, x \log \relax (x)^{2} - 5 \, {\left (x {\left (\log \relax (2) - 1\right )} + x \log \relax (x) + 16\right )} \log \left (x + 4\right ) + {\left (5 \, x {\left (\log \relax (2) - 1\right )} + 81\right )} \log \relax (x) + \log \relax (2) - 1}{5 \, {\left ({\left (\log \relax (2) + \log \relax (x) - 1\right )} \log \left (x + 4\right ) - {\left (\log \relax (2) - 1\right )} \log \relax (x) - \log \relax (x)^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-5*x^2-20*x)*log((4+x)/x)^2+4)*log(2*x)^2+((10*x^2+40*x)*log((4+x)/x)^2-8)*log(2*x)+(-5*x^2+60*x+
320)*log((4+x)/x)^2+4)/((5*x^2+20*x)*log((4+x)/x)^2*log(2*x)^2+(-10*x^2-40*x)*log((4+x)/x)^2*log(2*x)+(5*x^2+2
0*x)*log((4+x)/x)^2),x, algorithm="maxima")

[Out]

1/5*(5*x*log(x)^2 - 5*(x*(log(2) - 1) + x*log(x) + 16)*log(x + 4) + (5*x*(log(2) - 1) + 81)*log(x) + log(2) -
1)/((log(2) + log(x) - 1)*log(x + 4) - (log(2) - 1)*log(x) - log(x)^2)

________________________________________________________________________________________

mupad [B]  time = 1.48, size = 26, normalized size = 0.84 \begin {gather*} \frac {1}{5\,\ln \left (\frac {x+4}{x}\right )}-\frac {16}{\ln \left (2\,x\right )-1}-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log((x + 4)/x)^2*(60*x - 5*x^2 + 320) - log(2*x)^2*(log((x + 4)/x)^2*(20*x + 5*x^2) - 4) + log(2*x)*(log(
(x + 4)/x)^2*(40*x + 10*x^2) - 8) + 4)/(log((x + 4)/x)^2*(20*x + 5*x^2) + log(2*x)^2*log((x + 4)/x)^2*(20*x +
5*x^2) - log(2*x)*log((x + 4)/x)^2*(40*x + 10*x^2)),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.37, size = 19, normalized size = 0.61 \begin {gather*} - x + \frac {1}{5 \log {\left (\frac {x + 4}{x} \right )}} - \frac {16}{\log {\left (2 x \right )} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-5*x**2-20*x)*ln((4+x)/x)**2+4)*ln(2*x)**2+((10*x**2+40*x)*ln((4+x)/x)**2-8)*ln(2*x)+(-5*x**2+60*
x+320)*ln((4+x)/x)**2+4)/((5*x**2+20*x)*ln((4+x)/x)**2*ln(2*x)**2+(-10*x**2-40*x)*ln((4+x)/x)**2*ln(2*x)+(5*x*
*2+20*x)*ln((4+x)/x)**2),x)

[Out]

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

________________________________________________________________________________________