3.10.84 \(\int \frac {-10 x-5 x^2+(40 x+40 x^2) \log (x)+(70+35 x) \log ^2(x)+20 x \log ^3(x)+((-10 x-10 x^2) \log (x)+(-20-10 x) \log ^2(x)-5 x \log ^3(x)) \log (\log (x))}{(64 x^3+64 x^4+16 x^5) \log (x)+(128 x^2+128 x^3+32 x^4) \log ^3(x)+(64 x+64 x^2+16 x^3) \log ^5(x)+((-32 x^3-32 x^4-8 x^5) \log (x)+(-64 x^2-64 x^3-16 x^4) \log ^3(x)+(-32 x-32 x^2-8 x^3) \log ^5(x)) \log (\log (x))+((4 x^3+4 x^4+x^5) \log (x)+(8 x^2+8 x^3+2 x^4) \log ^3(x)+(4 x+4 x^2+x^3) \log ^5(x)) \log ^2(\log (x))} \, dx\)

Optimal. Leaf size=28 \[ \frac {5}{2 \left (1+\frac {x}{2}\right ) \left (x+\log ^2(x)\right ) (-4+\log (\log (x)))} \]

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Rubi [F]  time = 13.67, 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 {-10 x-5 x^2+\left (40 x+40 x^2\right ) \log (x)+(70+35 x) \log ^2(x)+20 x \log ^3(x)+\left (\left (-10 x-10 x^2\right ) \log (x)+(-20-10 x) \log ^2(x)-5 x \log ^3(x)\right ) \log (\log (x))}{\left (64 x^3+64 x^4+16 x^5\right ) \log (x)+\left (128 x^2+128 x^3+32 x^4\right ) \log ^3(x)+\left (64 x+64 x^2+16 x^3\right ) \log ^5(x)+\left (\left (-32 x^3-32 x^4-8 x^5\right ) \log (x)+\left (-64 x^2-64 x^3-16 x^4\right ) \log ^3(x)+\left (-32 x-32 x^2-8 x^3\right ) \log ^5(x)\right ) \log (\log (x))+\left (\left (4 x^3+4 x^4+x^5\right ) \log (x)+\left (8 x^2+8 x^3+2 x^4\right ) \log ^3(x)+\left (4 x+4 x^2+x^3\right ) \log ^5(x)\right ) \log ^2(\log (x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-10*x - 5*x^2 + (40*x + 40*x^2)*Log[x] + (70 + 35*x)*Log[x]^2 + 20*x*Log[x]^3 + ((-10*x - 10*x^2)*Log[x]
+ (-20 - 10*x)*Log[x]^2 - 5*x*Log[x]^3)*Log[Log[x]])/((64*x^3 + 64*x^4 + 16*x^5)*Log[x] + (128*x^2 + 128*x^3 +
32*x^4)*Log[x]^3 + (64*x + 64*x^2 + 16*x^3)*Log[x]^5 + ((-32*x^3 - 32*x^4 - 8*x^5)*Log[x] + (-64*x^2 - 64*x^3
- 16*x^4)*Log[x]^3 + (-32*x - 32*x^2 - 8*x^3)*Log[x]^5)*Log[Log[x]] + ((4*x^3 + 4*x^4 + x^5)*Log[x] + (8*x^2
+ 8*x^3 + 2*x^4)*Log[x]^3 + (4*x + 4*x^2 + x^3)*Log[x]^5)*Log[Log[x]]^2),x]

[Out]

(-5*Defer[Int][1/(x*Log[x]*(x + Log[x]^2)*(-4 + Log[Log[x]])^2), x])/2 + (5*Defer[Int][1/((2 + x)*Log[x]*(x +
Log[x]^2)*(-4 + Log[Log[x]])^2), x])/2 + 10*Defer[Int][1/((2 + x)^2*(x + Log[x]^2)^2*(-4 + Log[Log[x]])), x] -
10*Defer[Int][1/((2 + x)*(x + Log[x]^2)^2*(-4 + Log[Log[x]])), x] - 5*Defer[Int][Log[x]/(x*(x + Log[x]^2)^2*(
-4 + Log[Log[x]])), x] + 5*Defer[Int][Log[x]/((2 + x)*(x + Log[x]^2)^2*(-4 + Log[Log[x]])), x] - 5*Defer[Int][
Log[x]^2/((2 + x)^2*(x + Log[x]^2)^2*(-4 + Log[Log[x]])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 \left (-x (2+x)-2 x (1+x) \log (x) (-4+\log (\log (x)))-x \log ^3(x) (-4+\log (\log (x)))-(2+x) \log ^2(x) (-7+2 \log (\log (x)))\right )}{x (2+x)^2 \log (x) \left (x+\log ^2(x)\right )^2 (4-\log (\log (x)))^2} \, dx\\ &=5 \int \frac {-x (2+x)-2 x (1+x) \log (x) (-4+\log (\log (x)))-x \log ^3(x) (-4+\log (\log (x)))-(2+x) \log ^2(x) (-7+2 \log (\log (x)))}{x (2+x)^2 \log (x) \left (x+\log ^2(x)\right )^2 (4-\log (\log (x)))^2} \, dx\\ &=5 \int \left (-\frac {1}{x (2+x) \log (x) \left (x+\log ^2(x)\right ) (-4+\log (\log (x)))^2}+\frac {-2 x-2 x^2-4 \log (x)-2 x \log (x)-x \log ^2(x)}{x (2+x)^2 \left (x+\log ^2(x)\right )^2 (-4+\log (\log (x)))}\right ) \, dx\\ &=-\left (5 \int \frac {1}{x (2+x) \log (x) \left (x+\log ^2(x)\right ) (-4+\log (\log (x)))^2} \, dx\right )+5 \int \frac {-2 x-2 x^2-4 \log (x)-2 x \log (x)-x \log ^2(x)}{x (2+x)^2 \left (x+\log ^2(x)\right )^2 (-4+\log (\log (x)))} \, dx\\ &=-\left (5 \int \left (\frac {1}{2 x \log (x) \left (x+\log ^2(x)\right ) (-4+\log (\log (x)))^2}-\frac {1}{2 (2+x) \log (x) \left (x+\log ^2(x)\right ) (-4+\log (\log (x)))^2}\right ) \, dx\right )+5 \int \left (\frac {-2 x-2 x^2-4 \log (x)-2 x \log (x)-x \log ^2(x)}{4 x \left (x+\log ^2(x)\right )^2 (-4+\log (\log (x)))}+\frac {2 x+2 x^2+4 \log (x)+2 x \log (x)+x \log ^2(x)}{2 (2+x)^2 \left (x+\log ^2(x)\right )^2 (-4+\log (\log (x)))}+\frac {2 x+2 x^2+4 \log (x)+2 x \log (x)+x \log ^2(x)}{4 (2+x) \left (x+\log ^2(x)\right )^2 (-4+\log (\log (x)))}\right ) \, dx\\ &=\frac {5}{4} \int \frac {-2 x-2 x^2-4 \log (x)-2 x \log (x)-x \log ^2(x)}{x \left (x+\log ^2(x)\right )^2 (-4+\log (\log (x)))} \, dx+\frac {5}{4} \int \frac {2 x+2 x^2+4 \log (x)+2 x \log (x)+x \log ^2(x)}{(2+x) \left (x+\log ^2(x)\right )^2 (-4+\log (\log (x)))} \, dx-\frac {5}{2} \int \frac {1}{x \log (x) \left (x+\log ^2(x)\right ) (-4+\log (\log (x)))^2} \, dx+\frac {5}{2} \int \frac {1}{(2+x) \log (x) \left (x+\log ^2(x)\right ) (-4+\log (\log (x)))^2} \, dx+\frac {5}{2} \int \frac {2 x+2 x^2+4 \log (x)+2 x \log (x)+x \log ^2(x)}{(2+x)^2 \left (x+\log ^2(x)\right )^2 (-4+\log (\log (x)))} \, dx\\ &=\frac {5}{4} \int \left (-\frac {2}{\left (x+\log ^2(x)\right )^2 (-4+\log (\log (x)))}-\frac {2 x}{\left (x+\log ^2(x)\right )^2 (-4+\log (\log (x)))}-\frac {2 \log (x)}{\left (x+\log ^2(x)\right )^2 (-4+\log (\log (x)))}-\frac {4 \log (x)}{x \left (x+\log ^2(x)\right )^2 (-4+\log (\log (x)))}-\frac {\log ^2(x)}{\left (x+\log ^2(x)\right )^2 (-4+\log (\log (x)))}\right ) \, dx+\frac {5}{4} \int \left (\frac {2 x}{(2+x) \left (x+\log ^2(x)\right )^2 (-4+\log (\log (x)))}+\frac {2 x^2}{(2+x) \left (x+\log ^2(x)\right )^2 (-4+\log (\log (x)))}+\frac {4 \log (x)}{(2+x) \left (x+\log ^2(x)\right )^2 (-4+\log (\log (x)))}+\frac {2 x \log (x)}{(2+x) \left (x+\log ^2(x)\right )^2 (-4+\log (\log (x)))}+\frac {x \log ^2(x)}{(2+x) \left (x+\log ^2(x)\right )^2 (-4+\log (\log (x)))}\right ) \, dx+\frac {5}{2} \int \left (\frac {2 x}{(2+x)^2 \left (x+\log ^2(x)\right )^2 (-4+\log (\log (x)))}+\frac {2 x^2}{(2+x)^2 \left (x+\log ^2(x)\right )^2 (-4+\log (\log (x)))}+\frac {4 \log (x)}{(2+x)^2 \left (x+\log ^2(x)\right )^2 (-4+\log (\log (x)))}+\frac {2 x \log (x)}{(2+x)^2 \left (x+\log ^2(x)\right )^2 (-4+\log (\log (x)))}+\frac {x \log ^2(x)}{(2+x)^2 \left (x+\log ^2(x)\right )^2 (-4+\log (\log (x)))}\right ) \, dx-\frac {5}{2} \int \frac {1}{x \log (x) \left (x+\log ^2(x)\right ) (-4+\log (\log (x)))^2} \, dx+\frac {5}{2} \int \frac {1}{(2+x) \log (x) \left (x+\log ^2(x)\right ) (-4+\log (\log (x)))^2} \, dx\\ &=-\left (\frac {5}{4} \int \frac {\log ^2(x)}{\left (x+\log ^2(x)\right )^2 (-4+\log (\log (x)))} \, dx\right )+\frac {5}{4} \int \frac {x \log ^2(x)}{(2+x) \left (x+\log ^2(x)\right )^2 (-4+\log (\log (x)))} \, dx-\frac {5}{2} \int \frac {1}{x \log (x) \left (x+\log ^2(x)\right ) (-4+\log (\log (x)))^2} \, dx+\frac {5}{2} \int \frac {1}{(2+x) \log (x) \left (x+\log ^2(x)\right ) (-4+\log (\log (x)))^2} \, dx-\frac {5}{2} \int \frac {1}{\left (x+\log ^2(x)\right )^2 (-4+\log (\log (x)))} \, dx-\frac {5}{2} \int \frac {x}{\left (x+\log ^2(x)\right )^2 (-4+\log (\log (x)))} \, dx+\frac {5}{2} \int \frac {x}{(2+x) \left (x+\log ^2(x)\right )^2 (-4+\log (\log (x)))} \, dx+\frac {5}{2} \int \frac {x^2}{(2+x) \left (x+\log ^2(x)\right )^2 (-4+\log (\log (x)))} \, dx-\frac {5}{2} \int \frac {\log (x)}{\left (x+\log ^2(x)\right )^2 (-4+\log (\log (x)))} \, dx+\frac {5}{2} \int \frac {x \log (x)}{(2+x) \left (x+\log ^2(x)\right )^2 (-4+\log (\log (x)))} \, dx+\frac {5}{2} \int \frac {x \log ^2(x)}{(2+x)^2 \left (x+\log ^2(x)\right )^2 (-4+\log (\log (x)))} \, dx+5 \int \frac {x}{(2+x)^2 \left (x+\log ^2(x)\right )^2 (-4+\log (\log (x)))} \, dx+5 \int \frac {x^2}{(2+x)^2 \left (x+\log ^2(x)\right )^2 (-4+\log (\log (x)))} \, dx-5 \int \frac {\log (x)}{x \left (x+\log ^2(x)\right )^2 (-4+\log (\log (x)))} \, dx+5 \int \frac {x \log (x)}{(2+x)^2 \left (x+\log ^2(x)\right )^2 (-4+\log (\log (x)))} \, dx+5 \int \frac {\log (x)}{(2+x) \left (x+\log ^2(x)\right )^2 (-4+\log (\log (x)))} \, dx+10 \int \frac {\log (x)}{(2+x)^2 \left (x+\log ^2(x)\right )^2 (-4+\log (\log (x)))} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.62, size = 22, normalized size = 0.79 \begin {gather*} \frac {5}{(2+x) \left (x+\log ^2(x)\right ) (-4+\log (\log (x)))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-10*x - 5*x^2 + (40*x + 40*x^2)*Log[x] + (70 + 35*x)*Log[x]^2 + 20*x*Log[x]^3 + ((-10*x - 10*x^2)*L
og[x] + (-20 - 10*x)*Log[x]^2 - 5*x*Log[x]^3)*Log[Log[x]])/((64*x^3 + 64*x^4 + 16*x^5)*Log[x] + (128*x^2 + 128
*x^3 + 32*x^4)*Log[x]^3 + (64*x + 64*x^2 + 16*x^3)*Log[x]^5 + ((-32*x^3 - 32*x^4 - 8*x^5)*Log[x] + (-64*x^2 -
64*x^3 - 16*x^4)*Log[x]^3 + (-32*x - 32*x^2 - 8*x^3)*Log[x]^5)*Log[Log[x]] + ((4*x^3 + 4*x^4 + x^5)*Log[x] + (
8*x^2 + 8*x^3 + 2*x^4)*Log[x]^3 + (4*x + 4*x^2 + x^3)*Log[x]^5)*Log[Log[x]]^2),x]

[Out]

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

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fricas [A]  time = 0.47, size = 42, normalized size = 1.50 \begin {gather*} -\frac {5}{4 \, {\left (x + 2\right )} \log \relax (x)^{2} + 4 \, x^{2} - {\left ({\left (x + 2\right )} \log \relax (x)^{2} + x^{2} + 2 \, x\right )} \log \left (\log \relax (x)\right ) + 8 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x*log(x)^3+(-10*x-20)*log(x)^2+(-10*x^2-10*x)*log(x))*log(log(x))+20*x*log(x)^3+(35*x+70)*log(x
)^2+(40*x^2+40*x)*log(x)-5*x^2-10*x)/(((x^3+4*x^2+4*x)*log(x)^5+(2*x^4+8*x^3+8*x^2)*log(x)^3+(x^5+4*x^4+4*x^3)
*log(x))*log(log(x))^2+((-8*x^3-32*x^2-32*x)*log(x)^5+(-16*x^4-64*x^3-64*x^2)*log(x)^3+(-8*x^5-32*x^4-32*x^3)*
log(x))*log(log(x))+(16*x^3+64*x^2+64*x)*log(x)^5+(32*x^4+128*x^3+128*x^2)*log(x)^3+(16*x^5+64*x^4+64*x^3)*log
(x)),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.80, size = 57, normalized size = 2.04 \begin {gather*} \frac {5}{x \log \relax (x)^{2} \log \left (\log \relax (x)\right ) - 4 \, x \log \relax (x)^{2} + x^{2} \log \left (\log \relax (x)\right ) + 2 \, \log \relax (x)^{2} \log \left (\log \relax (x)\right ) - 4 \, x^{2} - 8 \, \log \relax (x)^{2} + 2 \, x \log \left (\log \relax (x)\right ) - 8 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x*log(x)^3+(-10*x-20)*log(x)^2+(-10*x^2-10*x)*log(x))*log(log(x))+20*x*log(x)^3+(35*x+70)*log(x
)^2+(40*x^2+40*x)*log(x)-5*x^2-10*x)/(((x^3+4*x^2+4*x)*log(x)^5+(2*x^4+8*x^3+8*x^2)*log(x)^3+(x^5+4*x^4+4*x^3)
*log(x))*log(log(x))^2+((-8*x^3-32*x^2-32*x)*log(x)^5+(-16*x^4-64*x^3-64*x^2)*log(x)^3+(-8*x^5-32*x^4-32*x^3)*
log(x))*log(log(x))+(16*x^3+64*x^2+64*x)*log(x)^5+(32*x^4+128*x^3+128*x^2)*log(x)^3+(16*x^5+64*x^4+64*x^3)*log
(x)),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.03, size = 31, normalized size = 1.11

method	result	size
risch	\(\frac {5}{\left (x \ln \relax (x )^{2}+x^{2}+2 \ln \relax (x )^{2}+2 x \right ) \left (\ln \left (\ln \relax (x )\right )-4\right )}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-5*x*ln(x)^3+(-10*x-20)*ln(x)^2+(-10*x^2-10*x)*ln(x))*ln(ln(x))+20*x*ln(x)^3+(35*x+70)*ln(x)^2+(40*x^2+4
0*x)*ln(x)-5*x^2-10*x)/(((x^3+4*x^2+4*x)*ln(x)^5+(2*x^4+8*x^3+8*x^2)*ln(x)^3+(x^5+4*x^4+4*x^3)*ln(x))*ln(ln(x)
)^2+((-8*x^3-32*x^2-32*x)*ln(x)^5+(-16*x^4-64*x^3-64*x^2)*ln(x)^3+(-8*x^5-32*x^4-32*x^3)*ln(x))*ln(ln(x))+(16*
x^3+64*x^2+64*x)*ln(x)^5+(32*x^4+128*x^3+128*x^2)*ln(x)^3+(16*x^5+64*x^4+64*x^3)*ln(x)),x,method=_RETURNVERBOS
E)

[Out]

5/(x*ln(x)^2+x^2+2*ln(x)^2+2*x)/(ln(ln(x))-4)

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maxima [A]  time = 0.52, size = 42, normalized size = 1.50 \begin {gather*} -\frac {5}{4 \, {\left (x + 2\right )} \log \relax (x)^{2} + 4 \, x^{2} - {\left ({\left (x + 2\right )} \log \relax (x)^{2} + x^{2} + 2 \, x\right )} \log \left (\log \relax (x)\right ) + 8 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x*log(x)^3+(-10*x-20)*log(x)^2+(-10*x^2-10*x)*log(x))*log(log(x))+20*x*log(x)^3+(35*x+70)*log(x
)^2+(40*x^2+40*x)*log(x)-5*x^2-10*x)/(((x^3+4*x^2+4*x)*log(x)^5+(2*x^4+8*x^3+8*x^2)*log(x)^3+(x^5+4*x^4+4*x^3)
*log(x))*log(log(x))^2+((-8*x^3-32*x^2-32*x)*log(x)^5+(-16*x^4-64*x^3-64*x^2)*log(x)^3+(-8*x^5-32*x^4-32*x^3)*
log(x))*log(log(x))+(16*x^3+64*x^2+64*x)*log(x)^5+(32*x^4+128*x^3+128*x^2)*log(x)^3+(16*x^5+64*x^4+64*x^3)*log
(x)),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.50, size = 22, normalized size = 0.79 \begin {gather*} \frac {5}{\left ({\ln \relax (x)}^2+x\right )\,\left (\ln \left (\ln \relax (x)\right )-4\right )\,\left (x+2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(10*x - 20*x*log(x)^3 - log(x)*(40*x + 40*x^2) + log(log(x))*(5*x*log(x)^3 + log(x)*(10*x + 10*x^2) + log
(x)^2*(10*x + 20)) + 5*x^2 - log(x)^2*(35*x + 70))/(log(x)^5*(64*x + 64*x^2 + 16*x^3) - log(log(x))*(log(x)^5*
(32*x + 32*x^2 + 8*x^3) + log(x)*(32*x^3 + 32*x^4 + 8*x^5) + log(x)^3*(64*x^2 + 64*x^3 + 16*x^4)) + log(x)*(64
*x^3 + 64*x^4 + 16*x^5) + log(x)^3*(128*x^2 + 128*x^3 + 32*x^4) + log(log(x))^2*(log(x)^3*(8*x^2 + 8*x^3 + 2*x
^4) + log(x)^5*(4*x + 4*x^2 + x^3) + log(x)*(4*x^3 + 4*x^4 + x^5))),x)

[Out]

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

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sympy [B]  time = 0.41, size = 49, normalized size = 1.75 \begin {gather*} \frac {5}{- 4 x^{2} - 4 x \log {\relax (x )}^{2} - 8 x + \left (x^{2} + x \log {\relax (x )}^{2} + 2 x + 2 \log {\relax (x )}^{2}\right ) \log {\left (\log {\relax (x )} \right )} - 8 \log {\relax (x )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x*ln(x)**3+(-10*x-20)*ln(x)**2+(-10*x**2-10*x)*ln(x))*ln(ln(x))+20*x*ln(x)**3+(35*x+70)*ln(x)**
2+(40*x**2+40*x)*ln(x)-5*x**2-10*x)/(((x**3+4*x**2+4*x)*ln(x)**5+(2*x**4+8*x**3+8*x**2)*ln(x)**3+(x**5+4*x**4+
4*x**3)*ln(x))*ln(ln(x))**2+((-8*x**3-32*x**2-32*x)*ln(x)**5+(-16*x**4-64*x**3-64*x**2)*ln(x)**3+(-8*x**5-32*x
**4-32*x**3)*ln(x))*ln(ln(x))+(16*x**3+64*x**2+64*x)*ln(x)**5+(32*x**4+128*x**3+128*x**2)*ln(x)**3+(16*x**5+64
*x**4+64*x**3)*ln(x)),x)

[Out]

5/(-4*x**2 - 4*x*log(x)**2 - 8*x + (x**2 + x*log(x)**2 + 2*x + 2*log(x)**2)*log(log(x)) - 8*log(x)**2)

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