3.89.10 $\int \frac{-16x^4{\log }^2(5)+(-16x^3{\log }^2(5)+(-4x^2+32x^4){\log }^2(5)\log (x))\log (\log (x))+((4x^2+48x^3){\log }^2(5)\log (x)+8x^2{\log }^2(5){\log }^2(x)){\log }^2(\log (x))+((4x+16x^2){\log }^2(5)\log (x)+(1+4x){\log }^2(5){\log }^2(x)){\log }^3(\log (x))}{3x\log (x){\log }^3(\log (x))}dx$

Optimal. Leaf size=25 $$\frac{1}{6}{\log }^2(5){(\log (x)+4x(1+\frac{x}{\log (\log (x))}))}^2$$

________________________________________________________________________________________

Rubi [F]  time = 1.27, 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 = {} $$\begin{array}{c}\int \frac{-16x^4{\log }^2(5)+(-16x^3{\log }^2(5)+(-4x^2+32x^4){\log }^2(5)\log (x))\log (\log (x))+((4x^2+48x^3){\log }^2(5)\log (x)+8x^2{\log }^2(5){\log }^2(x)){\log }^2(\log (x))+((4x+16x^2){\log }^2(5)\log (x)+(1+4x){\log }^2(5){\log }^2(x)){\log }^3(\log (x))}{3x\log (x){\log }^3(\log (x))}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(-16*x^4*Log[5]^2 + (-16*x^3*Log[5]^2 + (-4*x^2 + 32*x^4)*Log[5]^2*Log[x])*Log[Log[x]] + ((4*x^2 + 48*x^3)
*Log[5]^2*Log[x] + 8*x^2*Log[5]^2*Log[x]^2)*Log[Log[x]]^2 + ((4*x + 16*x^2)*Log[5]^2*Log[x] + (1 + 4*x)*Log[5]
^2*Log[x]^2)*Log[Log[x]]^3)/(3*x*Log[x]*Log[Log[x]]^3),x]

[Out]

(Log[5]^2*(4*x + Log[x])^2)/6 - (16*Log[5]^2*Defer[Int][x^3/(Log[x]*Log[Log[x]]^3), x])/3 - (4*Log[5]^2*Defer[
Int][x/Log[Log[x]]^2, x])/3 + (32*Log[5]^2*Defer[Int][x^3/Log[Log[x]]^2, x])/3 - (16*Log[5]^2*Defer[Int][x^2/(
Log[x]*Log[Log[x]]^2), x])/3 + (4*Log[5]^2*Defer[Int][x/Log[Log[x]], x])/3 + 16*Log[5]^2*Defer[Int][x^2/Log[Lo
g[x]], x] + (8*Log[5]^2*Defer[Int][(x*Log[x])/Log[Log[x]], x])/3

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\frac{1}{3}\int \frac{-16x^4{\log }^2(5)+(-16x^3{\log }^2(5)+(-4x^2+32x^4){\log }^2(5)\log (x))\log (\log (x))+((4x^2+48x^3){\log }^2(5)\log (x)+8x^2{\log }^2(5){\log }^2(x)){\log }^2(\log (x))+((4x+16x^2){\log }^2(5)\log (x)+(1+4x){\log }^2(5){\log }^2(x)){\log }^3(\log (x))}{x\log (x){\log }^3(\log (x))}dx\\ & =\frac{1}{3}\int \frac{{\log }^2(5)(-16x^4+4x^2(-4x+(-1+8x^2)\log (x))\log (\log (x))+4x^2\log (x)(1+12x+2\log (x)){\log }^2(\log (x))+(1+4x)\log (x)(4x+\log (x)){\log }^3(\log (x)))}{x\log (x){\log }^3(\log (x))}dx\\ & =\frac{1}{3}{\log }^2(5)\int \frac{-16x^4+4x^2(-4x+(-1+8x^2)\log (x))\log (\log (x))+4x^2\log (x)(1+12x+2\log (x)){\log }^2(\log (x))+(1+4x)\log (x)(4x+\log (x)){\log }^3(\log (x))}{x\log (x){\log }^3(\log (x))}dx\\ & =\frac{1}{3}{\log }^2(5)\int (\frac{(1+4x)(4x+\log (x))}{x}-\frac{16x^3}{\log (x){\log }^3(\log (x))}+\frac{4x(-4x-\log (x)+8x^2\log (x))}{\log (x){\log }^2(\log (x))}+\frac{4x(1+12x+2\log (x))}{\log (\log (x))})dx\\ & =\frac{1}{3}{\log }^2(5)\int \frac{(1+4x)(4x+\log (x))}{x}dx+\frac{1}{3}(4{\log }^2(5))\int \frac{x(-4x-\log (x)+8x^2\log (x))}{\log (x){\log }^2(\log (x))}dx+\frac{1}{3}(4{\log }^2(5))\int \frac{x(1+12x+2\log (x))}{\log (\log (x))}dx-\frac{1}{3}(16{\log }^2(5))\int \frac{x^3}{\log (x){\log }^3(\log (x))}dx\\ & =\frac{1}{6}{\log }^2(5)(4x+\log (x){)}^2+\frac{1}{3}(4{\log }^2(5))\int (-\frac{x}{{\log }^2(\log (x))}+\frac{8x^3}{{\log }^2(\log (x))}-\frac{4x^2}{\log (x){\log }^2(\log (x))})dx+\frac{1}{3}(4{\log }^2(5))\int (\frac{x}{\log (\log (x))}+\frac{12x^2}{\log (\log (x))}+\frac{2x\log (x)}{\log (\log (x))})dx-\frac{1}{3}(16{\log }^2(5))\int \frac{x^3}{\log (x){\log }^3(\log (x))}dx\\ & =\frac{1}{6}{\log }^2(5)(4x+\log (x){)}^2-\frac{1}{3}(4{\log }^2(5))\int \frac{x}{{\log }^2(\log (x))}dx+\frac{1}{3}(4{\log }^2(5))\int \frac{x}{\log (\log (x))}dx+\frac{1}{3}(8{\log }^2(5))\int \frac{x\log (x)}{\log (\log (x))}dx-\frac{1}{3}(16{\log }^2(5))\int \frac{x^3}{\log (x){\log }^3(\log (x))}dx-\frac{1}{3}(16{\log }^2(5))\int \frac{x^2}{\log (x){\log }^2(\log (x))}dx+\frac{1}{3}(32{\log }^2(5))\int \frac{x^3}{{\log }^2(\log (x))}dx+(16{\log }^2(5))\int \frac{x^2}{\log (\log (x))}dx\end{array}\end{array}$$

________________________________________________________________________________________

Mathematica [A]  time = 0.08, size = 31, normalized size = 1.24 $$\begin{array}{c}\frac{{\log }^2(5){(4x^2+(4x+\log (x))\log (\log (x)))}^2}{6{\log }^2(\log (x))}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-16*x^4*Log[5]^2 + (-16*x^3*Log[5]^2 + (-4*x^2 + 32*x^4)*Log[5]^2*Log[x])*Log[Log[x]] + ((4*x^2 + 4
8*x^3)*Log[5]^2*Log[x] + 8*x^2*Log[5]^2*Log[x]^2)*Log[Log[x]]^2 + ((4*x + 16*x^2)*Log[5]^2*Log[x] + (1 + 4*x)*
Log[5]^2*Log[x]^2)*Log[Log[x]]^3)/(3*x*Log[x]*Log[Log[x]]^3),x]

[Out]

(Log[5]^2*(4*x^2 + (4*x + Log[x])*Log[Log[x]])^2)/(6*Log[Log[x]]^2)

________________________________________________________________________________________

fricas [B]  time = 0.69, size = 76, normalized size = 3.04 $$\begin{array}{c}\frac{16x^4\log (5{)}^2+(16x^2\log (5{)}^2+8x\log (5{)}^2\log (x)+\log (5{)}^2\log (x{)}^2)\log {(\log (x))}^2+8(4x^3\log (5{)}^2+x^2\log (5{)}^2\log (x))\log (\log (x))}{6\log {(\log (x))}^2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(((4*x+1)*log(5)^2*log(x)^2+(16*x^2+4*x)*log(5)^2*log(x))*log(log(x))^3+(8*x^2*log(5)^2*log(x)^2
+(48*x^3+4*x^2)*log(5)^2*log(x))*log(log(x))^2+((32*x^4-4*x^2)*log(5)^2*log(x)-16*x^3*log(5)^2)*log(log(x))-16
*x^4*log(5)^2)/x/log(x)/log(log(x))^3,x, algorithm="fricas")

[Out]

1/6*(16*x^4*log(5)^2 + (16*x^2*log(5)^2 + 8*x*log(5)^2*log(x) + log(5)^2*log(x)^2)*log(log(x))^2 + 8*(4*x^3*lo
g(5)^2 + x^2*log(5)^2*log(x))*log(log(x)))/log(log(x))^2

________________________________________________________________________________________

giac [B]  time = 0.26, size = 71, normalized size = 2.84 $$\begin{array}{c}\frac{8}{3}{x}^2\log (5{)}^2+\frac{4}{3}x\log (5{)}^2\log (x)+\frac{1}{6}\log (5{)}^2\log (x{)}^2+\frac{4(2x^4\log (5{)}^2+4x^3\log (5{)}^2\log (\log (x))+x^2\log (5{)}^2\log (x)\log (\log (x)))}{3\log {(\log (x))}^2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(((4*x+1)*log(5)^2*log(x)^2+(16*x^2+4*x)*log(5)^2*log(x))*log(log(x))^3+(8*x^2*log(5)^2*log(x)^2
+(48*x^3+4*x^2)*log(5)^2*log(x))*log(log(x))^2+((32*x^4-4*x^2)*log(5)^2*log(x)-16*x^3*log(5)^2)*log(log(x))-16
*x^4*log(5)^2)/x/log(x)/log(log(x))^3,x, algorithm="giac")

[Out]

8/3*x^2*log(5)^2 + 4/3*x*log(5)^2*log(x) + 1/6*log(5)^2*log(x)^2 + 4/3*(2*x^4*log(5)^2 + 4*x^3*log(5)^2*log(lo
g(x)) + x^2*log(5)^2*log(x)*log(log(x)))/log(log(x))^2

________________________________________________________________________________________

maple [B]  time = 0.06, size = 62, normalized size = 2.48

method	result	size
risch	$\frac{\ln (x{)}^2\ln (5{)}^2}{6}+\frac{4x\ln (5{)}^2\ln (x)}{3}+\frac{8x^2\ln (5{)}^2}{3}+\frac{4x^2\ln (5{)}^2(2x^2+4x\ln (\ln (x))+\ln (x)\ln (\ln (x)))}{3\ln {(\ln (x))}^2}$	$62$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(((4*x+1)*ln(5)^2*ln(x)^2+(16*x^2+4*x)*ln(5)^2*ln(x))*ln(ln(x))^3+(8*x^2*ln(5)^2*ln(x)^2+(48*x^3+4*x^2
)*ln(5)^2*ln(x))*ln(ln(x))^2+((32*x^4-4*x^2)*ln(5)^2*ln(x)-16*x^3*ln(5)^2)*ln(ln(x))-16*x^4*ln(5)^2)/x/ln(x)/l
n(ln(x))^3,x,method=_RETURNVERBOSE)

[Out]

1/6*ln(x)^2*ln(5)^2+4/3*x*ln(5)^2*ln(x)+8/3*x^2*ln(5)^2+4/3*x^2*ln(5)^2*(2*x^2+4*x*ln(ln(x))+ln(x)*ln(ln(x)))/
ln(ln(x))^2

________________________________________________________________________________________

maxima [B]  time = 0.50, size = 82, normalized size = 3.28 $$\begin{array}{c}\frac{8}{3}{x}^2\log (5{)}^2+\frac{1}{6}\log (5{)}^2\log (x{)}^2+\frac{4}{3}(x\log (x)-x)\log (5{)}^2+\frac{4}{3}x\log (5{)}^2+\frac{4(2x^4\log (5{)}^2+(4x^3\log (5{)}^2+x^2\log (5{)}^2\log (x))\log (\log (x)))}{3\log {(\log (x))}^2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(((4*x+1)*log(5)^2*log(x)^2+(16*x^2+4*x)*log(5)^2*log(x))*log(log(x))^3+(8*x^2*log(5)^2*log(x)^2
+(48*x^3+4*x^2)*log(5)^2*log(x))*log(log(x))^2+((32*x^4-4*x^2)*log(5)^2*log(x)-16*x^3*log(5)^2)*log(log(x))-16
*x^4*log(5)^2)/x/log(x)/log(log(x))^3,x, algorithm="maxima")

[Out]

8/3*x^2*log(5)^2 + 1/6*log(5)^2*log(x)^2 + 4/3*(x*log(x) - x)*log(5)^2 + 4/3*x*log(5)^2 + 4/3*(2*x^4*log(5)^2
+ (4*x^3*log(5)^2 + x^2*log(5)^2*log(x))*log(log(x)))/log(log(x))^2

________________________________________________________________________________________

mupad [B]  time = 6.03, size = 73, normalized size = 2.92 $$\begin{array}{c}\frac{8x^2{\ln (5)}^2}{3}+\frac{{\ln (5)}^2{\ln (x)}^2}{6}+\frac{4x{\ln (5)}^2\ln (x)}{3}+\frac{16x^3{\ln (5)}^2}{3\ln (\ln (x))}+\frac{8x^4{\ln (5)}^2}{3{\ln (\ln (x))}^2}+\frac{4x^2{\ln (5)}^2\ln (x)}{3\ln (\ln (x))}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((16*x^4*log(5)^2)/3 - (log(log(x))^3*(log(5)^2*log(x)*(4*x + 16*x^2) + log(5)^2*log(x)^2*(4*x + 1)))/3 -
(log(log(x))^2*(8*x^2*log(5)^2*log(x)^2 + log(5)^2*log(x)*(4*x^2 + 48*x^3)))/3 + (log(log(x))*(16*x^3*log(5)^
2 + log(5)^2*log(x)*(4*x^2 - 32*x^4)))/3)/(x*log(log(x))^3*log(x)),x)

[Out]

(8*x^2*log(5)^2)/3 + (log(5)^2*log(x)^2)/6 + (4*x*log(5)^2*log(x))/3 + (16*x^3*log(5)^2)/(3*log(log(x))) + (8*
x^4*log(5)^2)/(3*log(log(x))^2) + (4*x^2*log(5)^2*log(x))/(3*log(log(x)))

________________________________________________________________________________________

sympy [B]  time = 0.34, size = 83, normalized size = 3.32 $$\begin{array}{c}\frac{8x^2\log {(5)}^2}{3}+\frac{4x\log {(5)}^2\log (x)}{3}+\frac{8x^4\log {(5)}^2+(16x^3\log {(5)}^2+4x^2\log {(5)}^2\log (x))\log (\log (x))}{3\log {(\log (x))}^2}+\frac{\log {(5)}^2\log {(x)}^2}{6}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(((4*x+1)*ln(5)**2*ln(x)**2+(16*x**2+4*x)*ln(5)**2*ln(x))*ln(ln(x))**3+(8*x**2*ln(5)**2*ln(x)**2
+(48*x**3+4*x**2)*ln(5)**2*ln(x))*ln(ln(x))**2+((32*x**4-4*x**2)*ln(5)**2*ln(x)-16*x**3*ln(5)**2)*ln(ln(x))-16
*x**4*ln(5)**2)/x/ln(x)/ln(ln(x))**3,x)

[Out]

8*x**2*log(5)**2/3 + 4*x*log(5)**2*log(x)/3 + (8*x**4*log(5)**2 + (16*x**3*log(5)**2 + 4*x**2*log(5)**2*log(x)
)*log(log(x)))/(3*log(log(x))**2) + log(5)**2*log(x)**2/6

________________________________________________________________________________________