3.48.87 \(\int \frac {-16 x+4 x^3+(-80+20 x^2) \log (3)+(-8 x-12 x^2+2 x^3+x^4+(-40-80 x+10 x^2) \log (3)+(-100-25 x^2) \log ^2(3)) \log (x)+(16-4 x^2+(8+16 x-2 x^2+(40+10 x^2) \log (3)) \log (x)) \log (\frac {5}{2 x \log ^2(x)})+(-4-x^2) \log (x) \log ^2(\frac {5}{2 x \log ^2(x)})}{(16-8 x^2+x^4) \log ^2(3) \log (x)} \, dx\)

Optimal. Leaf size=35 \[ \frac {\left (5+\frac {x-\log \left (\frac {5}{2 x \log ^2(x)}\right )}{\log (3)}\right )^2}{-\frac {4}{x}+x} \]

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Rubi [F]  time = 5.53, 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 {-16 x+4 x^3+\left (-80+20 x^2\right ) \log (3)+\left (-8 x-12 x^2+2 x^3+x^4+\left (-40-80 x+10 x^2\right ) \log (3)+\left (-100-25 x^2\right ) \log ^2(3)\right ) \log (x)+\left (16-4 x^2+\left (8+16 x-2 x^2+\left (40+10 x^2\right ) \log (3)\right ) \log (x)\right ) \log \left (\frac {5}{2 x \log ^2(x)}\right )+\left (-4-x^2\right ) \log (x) \log ^2\left (\frac {5}{2 x \log ^2(x)}\right )}{\left (16-8 x^2+x^4\right ) \log ^2(3) \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-16*x + 4*x^3 + (-80 + 20*x^2)*Log[3] + (-8*x - 12*x^2 + 2*x^3 + x^4 + (-40 - 80*x + 10*x^2)*Log[3] + (-1
00 - 25*x^2)*Log[3]^2)*Log[x] + (16 - 4*x^2 + (8 + 16*x - 2*x^2 + (40 + 10*x^2)*Log[3])*Log[x])*Log[5/(2*x*Log
[x]^2)] + (-4 - x^2)*Log[x]*Log[5/(2*x*Log[x]^2)]^2)/((16 - 8*x^2 + x^4)*Log[3]^2*Log[x]),x]

[Out]

(3*x)/(2*Log[3]^2) - (6*x)/((4 - x^2)*Log[3]^2) + x^3/(2*(4 - x^2)*Log[3]^2) + (2*(2 - Log[243]))/((4 - x^2)*L
og[3]^2) - (2*(2 + Log[243]))/((4 - x^2)*Log[3]^2) - Log[2 - x]/(4*Log[3]^2) + ((2 - Log[243])*Log[2 - x])/(8*
Log[3]^2) - ((2 + Log[243])*Log[2 - x])/(8*Log[3]^2) + (2*Log[x])/Log[3]^2 - Log[2 + x]/(4*Log[3]^2) - ((2 - L
og[243])*Log[2 + x])/(8*Log[3]^2) + ((2 + Log[243])*Log[2 + x])/(8*Log[3]^2) - (3*Log[4 - x^2])/(4*Log[3]^2) +
((2 - Log[243])*Log[4 - x^2])/(2*Log[3]^2) - Log[5/(486*x*Log[x]^2)]/(2*(2 - x)*Log[3]^2) - Log[5/(486*x*Log[
x]^2)]/(2*(2 + x)*Log[3]^2) + (6*Log[5/(486*x*Log[x]^2)])/((4 - x^2)*Log[3]^2) - ((2 - Log[243])*Log[5/(486*x*
Log[x]^2)])/(4*(2 - x)*Log[3]^2) + ((2 - Log[243])*Log[5/(486*x*Log[x]^2)])/(4*(2 + x)*Log[3]^2) + ((2 + Log[2
43])*Log[5/(486*x*Log[x]^2)])/(4*(2 - x)*Log[3]^2) - ((2 + Log[243])*Log[5/(486*x*Log[x]^2)])/(4*(2 + x)*Log[3
]^2) + Log[5/(2*x*Log[x]^2)]/((2 - x)*Log[3]^2) + Log[5/(2*x*Log[x]^2)]/((2 + x)*Log[3]^2) - Defer[Int][1/((-2
+ x)*x*Log[x]), x]/Log[3]^2 + ((2 - Log[243])*Defer[Int][1/((-2 + x)*x*Log[x]), x])/(2*Log[3]^2) - ((2 + Log[
243])*Defer[Int][1/((-2 + x)*x*Log[x]), x])/(2*Log[3]^2) + Defer[Int][1/(x*(2 + x)*Log[x]), x]/Log[3]^2 + ((2
- Log[243])*Defer[Int][1/(x*(2 + x)*Log[x]), x])/(2*Log[3]^2) - ((2 + Log[243])*Defer[Int][1/(x*(2 + x)*Log[x]
), x])/(2*Log[3]^2) - (12*Defer[Int][1/(x*(-4 + x^2)*Log[x]), x])/Log[3]^2 + (4*Defer[Int][x/((-4 + x^2)*Log[x
]), x])/Log[3]^2 - Defer[Int][Log[5/(486*x*Log[x]^2)]/(-2 + x), x]/(2*Log[3]^2) - ((2 - Log[243])*Defer[Int][L
og[5/(486*x*Log[x]^2)]/(-2 + x), x])/(8*Log[3]^2) - ((2 + Log[243])*Defer[Int][Log[5/(486*x*Log[x]^2)]/(-2 + x
), x])/(8*Log[3]^2) - Defer[Int][Log[5/(486*x*Log[x]^2)]/(2 + x), x]/(2*Log[3]^2) + ((2 - Log[243])*Defer[Int]
[Log[5/(486*x*Log[x]^2)]/(2 + x), x])/(8*Log[3]^2) + ((2 + Log[243])*Defer[Int][Log[5/(486*x*Log[x]^2)]/(2 + x
), x])/(8*Log[3]^2) - Defer[Int][Log[5/(486*x*Log[x]^2)]/((-2 + x)*Log[x]), x]/Log[3]^2 + Defer[Int][Log[5/(48
6*x*Log[x]^2)]/((2 + x)*Log[x]), x]/Log[3]^2 + Defer[Int][Log[5/(2*x*Log[x]^2)]/(-2 + x), x]/(2*Log[3]^2) + De
fer[Int][Log[5/(2*x*Log[x]^2)]/(2 + x), x]/(2*Log[3]^2) - Defer[Int][(Log[5/(486*x*Log[x]^2)]*Log[5/(2*x*Log[x
]^2)])/(-2 + x)^2, x]/(2*Log[3]^2) - Defer[Int][(Log[5/(486*x*Log[x]^2)]*Log[5/(2*x*Log[x]^2)])/(2 + x)^2, x]/
(2*Log[3]^2)

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-16 x+4 x^3+\left (-80+20 x^2\right ) \log (3)+\left (-8 x-12 x^2+2 x^3+x^4+\left (-40-80 x+10 x^2\right ) \log (3)+\left (-100-25 x^2\right ) \log ^2(3)\right ) \log (x)+\left (16-4 x^2+\left (8+16 x-2 x^2+\left (40+10 x^2\right ) \log (3)\right ) \log (x)\right ) \log \left (\frac {5}{2 x \log ^2(x)}\right )+\left (-4-x^2\right ) \log (x) \log ^2\left (\frac {5}{2 x \log ^2(x)}\right )}{\left (16-8 x^2+x^4\right ) \log (x)} \, dx}{\log ^2(3)}\\ &=\frac {\int \frac {-16 x+4 x^3+\left (-80+20 x^2\right ) \log (3)+\left (-8 x-12 x^2+2 x^3+x^4+\left (-40-80 x+10 x^2\right ) \log (3)+\left (-100-25 x^2\right ) \log ^2(3)\right ) \log (x)+\left (16-4 x^2+\left (8+16 x-2 x^2+\left (40+10 x^2\right ) \log (3)\right ) \log (x)\right ) \log \left (\frac {5}{2 x \log ^2(x)}\right )+\left (-4-x^2\right ) \log (x) \log ^2\left (\frac {5}{2 x \log ^2(x)}\right )}{\left (-4+x^2\right )^2 \log (x)} \, dx}{\log ^2(3)}\\ &=\frac {\int \frac {\left (x-\log \left (\frac {5}{486 x \log ^2(x)}\right )\right ) \left (4 \left (-4+x^2\right )+\log (x) \left (-12 x+x^3+x^2 (2-5 \log (3))-4 (2+\log (243))+\left (4+x^2\right ) \log \left (\frac {5}{2 x \log ^2(x)}\right )\right )\right )}{\left (4-x^2\right )^2 \log (x)} \, dx}{\log ^2(3)}\\ &=\frac {\int \left (-\frac {12 x \left (x-\log \left (\frac {5}{486 x \log ^2(x)}\right )\right )}{\left (-4+x^2\right )^2}+\frac {x^3 \left (x-\log \left (\frac {5}{486 x \log ^2(x)}\right )\right )}{\left (-4+x^2\right )^2}-\frac {x^2 (-2+\log (243)) \left (x-\log \left (\frac {5}{486 x \log ^2(x)}\right )\right )}{\left (-4+x^2\right )^2}-\frac {4 (2+\log (243)) \left (x-\log \left (\frac {5}{486 x \log ^2(x)}\right )\right )}{\left (-4+x^2\right )^2}+\frac {4 \left (x-\log \left (\frac {5}{486 x \log ^2(x)}\right )\right )}{(-2+x) (2+x) \log (x)}+\frac {\left (4+x^2\right ) \left (x-\log \left (\frac {5}{486 x \log ^2(x)}\right )\right ) \log \left (\frac {5}{2 x \log ^2(x)}\right )}{\left (-4+x^2\right )^2}\right ) \, dx}{\log ^2(3)}\\ &=\frac {\int \frac {x^3 \left (x-\log \left (\frac {5}{486 x \log ^2(x)}\right )\right )}{\left (-4+x^2\right )^2} \, dx}{\log ^2(3)}+\frac {\int \frac {\left (4+x^2\right ) \left (x-\log \left (\frac {5}{486 x \log ^2(x)}\right )\right ) \log \left (\frac {5}{2 x \log ^2(x)}\right )}{\left (-4+x^2\right )^2} \, dx}{\log ^2(3)}+\frac {4 \int \frac {x-\log \left (\frac {5}{486 x \log ^2(x)}\right )}{(-2+x) (2+x) \log (x)} \, dx}{\log ^2(3)}-\frac {12 \int \frac {x \left (x-\log \left (\frac {5}{486 x \log ^2(x)}\right )\right )}{\left (-4+x^2\right )^2} \, dx}{\log ^2(3)}+\frac {(2-\log (243)) \int \frac {x^2 \left (x-\log \left (\frac {5}{486 x \log ^2(x)}\right )\right )}{\left (-4+x^2\right )^2} \, dx}{\log ^2(3)}-\frac {(4 (2+\log (243))) \int \frac {x-\log \left (\frac {5}{486 x \log ^2(x)}\right )}{\left (-4+x^2\right )^2} \, dx}{\log ^2(3)}\\ &=\frac {\int \left (\frac {x^4}{\left (-4+x^2\right )^2}-\frac {x^3 \log \left (\frac {5}{486 x \log ^2(x)}\right )}{\left (-4+x^2\right )^2}\right ) \, dx}{\log ^2(3)}+\frac {\int \left (\frac {\left (x-\log \left (\frac {5}{486 x \log ^2(x)}\right )\right ) \log \left (\frac {5}{2 x \log ^2(x)}\right )}{2 (-2+x)^2}+\frac {\left (x-\log \left (\frac {5}{486 x \log ^2(x)}\right )\right ) \log \left (\frac {5}{2 x \log ^2(x)}\right )}{2 (2+x)^2}\right ) \, dx}{\log ^2(3)}+\frac {4 \int \left (\frac {x}{\left (-4+x^2\right ) \log (x)}-\frac {\log \left (\frac {5}{486 x \log ^2(x)}\right )}{\left (-4+x^2\right ) \log (x)}\right ) \, dx}{\log ^2(3)}-\frac {12 \int \left (\frac {x^2}{\left (-4+x^2\right )^2}-\frac {x \log \left (\frac {5}{486 x \log ^2(x)}\right )}{\left (-4+x^2\right )^2}\right ) \, dx}{\log ^2(3)}+\frac {(2-\log (243)) \int \left (\frac {x^3}{\left (-4+x^2\right )^2}-\frac {x^2 \log \left (\frac {5}{486 x \log ^2(x)}\right )}{\left (-4+x^2\right )^2}\right ) \, dx}{\log ^2(3)}-\frac {(4 (2+\log (243))) \int \left (\frac {x}{\left (-4+x^2\right )^2}-\frac {\log \left (\frac {5}{486 x \log ^2(x)}\right )}{\left (-4+x^2\right )^2}\right ) \, dx}{\log ^2(3)}\\ &=\frac {\int \frac {\left (x-\log \left (\frac {5}{486 x \log ^2(x)}\right )\right ) \log \left (\frac {5}{2 x \log ^2(x)}\right )}{(-2+x)^2} \, dx}{2 \log ^2(3)}+\frac {\int \frac {\left (x-\log \left (\frac {5}{486 x \log ^2(x)}\right )\right ) \log \left (\frac {5}{2 x \log ^2(x)}\right )}{(2+x)^2} \, dx}{2 \log ^2(3)}+\frac {\int \frac {x^4}{\left (-4+x^2\right )^2} \, dx}{\log ^2(3)}-\frac {\int \frac {x^3 \log \left (\frac {5}{486 x \log ^2(x)}\right )}{\left (-4+x^2\right )^2} \, dx}{\log ^2(3)}+\frac {4 \int \frac {x}{\left (-4+x^2\right ) \log (x)} \, dx}{\log ^2(3)}-\frac {4 \int \frac {\log \left (\frac {5}{486 x \log ^2(x)}\right )}{\left (-4+x^2\right ) \log (x)} \, dx}{\log ^2(3)}-\frac {12 \int \frac {x^2}{\left (-4+x^2\right )^2} \, dx}{\log ^2(3)}+\frac {12 \int \frac {x \log \left (\frac {5}{486 x \log ^2(x)}\right )}{\left (-4+x^2\right )^2} \, dx}{\log ^2(3)}+\frac {(2-\log (243)) \int \frac {x^3}{\left (-4+x^2\right )^2} \, dx}{\log ^2(3)}-\frac {(2-\log (243)) \int \frac {x^2 \log \left (\frac {5}{486 x \log ^2(x)}\right )}{\left (-4+x^2\right )^2} \, dx}{\log ^2(3)}-\frac {(4 (2+\log (243))) \int \frac {x}{\left (-4+x^2\right )^2} \, dx}{\log ^2(3)}+\frac {(4 (2+\log (243))) \int \frac {\log \left (\frac {5}{486 x \log ^2(x)}\right )}{\left (-4+x^2\right )^2} \, dx}{\log ^2(3)}\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [F]  time = 1.56, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-16 x+4 x^3+\left (-80+20 x^2\right ) \log (3)+\left (-8 x-12 x^2+2 x^3+x^4+\left (-40-80 x+10 x^2\right ) \log (3)+\left (-100-25 x^2\right ) \log ^2(3)\right ) \log (x)+\left (16-4 x^2+\left (8+16 x-2 x^2+\left (40+10 x^2\right ) \log (3)\right ) \log (x)\right ) \log \left (\frac {5}{2 x \log ^2(x)}\right )+\left (-4-x^2\right ) \log (x) \log ^2\left (\frac {5}{2 x \log ^2(x)}\right )}{\left (16-8 x^2+x^4\right ) \log ^2(3) \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(-16*x + 4*x^3 + (-80 + 20*x^2)*Log[3] + (-8*x - 12*x^2 + 2*x^3 + x^4 + (-40 - 80*x + 10*x^2)*Log[3]
+ (-100 - 25*x^2)*Log[3]^2)*Log[x] + (16 - 4*x^2 + (8 + 16*x - 2*x^2 + (40 + 10*x^2)*Log[3])*Log[x])*Log[5/(2
*x*Log[x]^2)] + (-4 - x^2)*Log[x]*Log[5/(2*x*Log[x]^2)]^2)/((16 - 8*x^2 + x^4)*Log[3]^2*Log[x]),x]

[Out]

Integrate[(-16*x + 4*x^3 + (-80 + 20*x^2)*Log[3] + (-8*x - 12*x^2 + 2*x^3 + x^4 + (-40 - 80*x + 10*x^2)*Log[3]
+ (-100 - 25*x^2)*Log[3]^2)*Log[x] + (16 - 4*x^2 + (8 + 16*x - 2*x^2 + (40 + 10*x^2)*Log[3])*Log[x])*Log[5/(2
*x*Log[x]^2)] + (-4 - x^2)*Log[x]*Log[5/(2*x*Log[x]^2)]^2)/((16 - 8*x^2 + x^4)*Log[3]^2*Log[x]), x]

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fricas [A]  time = 1.07, size = 62, normalized size = 1.77 \begin {gather*} \frac {x^{3} + 25 \, x \log \relax (3)^{2} + x \log \left (\frac {5}{2 \, x \log \relax (x)^{2}}\right )^{2} - 2 \, {\left (x^{2} + 5 \, x \log \relax (3)\right )} \log \left (\frac {5}{2 \, x \log \relax (x)^{2}}\right ) + 40 \, \log \relax (3)}{{\left (x^{2} - 4\right )} \log \relax (3)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2-4)*log(x)*log(5/2/x/log(x)^2)^2+(((10*x^2+40)*log(3)-2*x^2+16*x+8)*log(x)-4*x^2+16)*log(5/2/x
/log(x)^2)+((-25*x^2-100)*log(3)^2+(10*x^2-80*x-40)*log(3)+x^4+2*x^3-12*x^2-8*x)*log(x)+(20*x^2-80)*log(3)+4*x
^3-16*x)/(x^4-8*x^2+16)/log(3)^2/log(x),x, algorithm="fricas")

[Out]

(x^3 + 25*x*log(3)^2 + x*log(5/2/(x*log(x)^2))^2 - 2*(x^2 + 5*x*log(3))*log(5/2/(x*log(x)^2)) + 40*log(3))/((x
^2 - 4)*log(3)^2)

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giac [B]  time = 0.52, size = 150, normalized size = 4.29 \begin {gather*} \frac {2 \, {\left (\frac {x \log \relax (x)}{x^{2} - 4} - \frac {x \log \relax (5) - 5 \, x \log \relax (3) - 4}{x^{2} - 4}\right )} \log \left (2 \, \log \relax (x)^{2}\right ) + \frac {x \log \left (2 \, \log \relax (x)^{2}\right )^{2}}{x^{2} - 4} + \frac {x \log \relax (x)^{2}}{x^{2} - 4} + x - \frac {2 \, {\left (x \log \relax (5) - 5 \, x \log \relax (3) - 4\right )} \log \relax (x)}{x^{2} - 4} + \frac {x \log \relax (5)^{2} - 10 \, x \log \relax (5) \log \relax (3) + 25 \, x \log \relax (3)^{2} + 4 \, x - 8 \, \log \relax (5) + 40 \, \log \relax (3)}{x^{2} - 4} + 2 \, \log \relax (x) + 4 \, \log \left (\log \relax (x)\right )}{\log \relax (3)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2-4)*log(x)*log(5/2/x/log(x)^2)^2+(((10*x^2+40)*log(3)-2*x^2+16*x+8)*log(x)-4*x^2+16)*log(5/2/x
/log(x)^2)+((-25*x^2-100)*log(3)^2+(10*x^2-80*x-40)*log(3)+x^4+2*x^3-12*x^2-8*x)*log(x)+(20*x^2-80)*log(3)+4*x
^3-16*x)/(x^4-8*x^2+16)/log(3)^2/log(x),x, algorithm="giac")

[Out]

(2*(x*log(x)/(x^2 - 4) - (x*log(5) - 5*x*log(3) - 4)/(x^2 - 4))*log(2*log(x)^2) + x*log(2*log(x)^2)^2/(x^2 - 4
) + x*log(x)^2/(x^2 - 4) + x - 2*(x*log(5) - 5*x*log(3) - 4)*log(x)/(x^2 - 4) + (x*log(5)^2 - 10*x*log(5)*log(
3) + 25*x*log(3)^2 + 4*x - 8*log(5) + 40*log(3))/(x^2 - 4) + 2*log(x) + 4*log(log(x)))/log(3)^2

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maple [C]  time = 0.84, size = 2034, normalized size = 58.11

method	result	size
risch	\(\text {Expression too large to display}\)	\(2034\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^2-4)*ln(x)*ln(5/2/x/ln(x)^2)^2+(((10*x^2+40)*ln(3)-2*x^2+16*x+8)*ln(x)-4*x^2+16)*ln(5/2/x/ln(x)^2)+((
-25*x^2-100)*ln(3)^2+(10*x^2-80*x-40)*ln(3)+x^4+2*x^3-12*x^2-8*x)*ln(x)+(20*x^2-80)*ln(3)+4*x^3-16*x)/(x^4-8*x
^2+16)/ln(3)^2/ln(x),x,method=_RETURNVERBOSE)

[Out]

4/ln(3)^2*x/(x^2-4)*ln(ln(x))^2+2/ln(3)^2*(I*Pi*x*csgn(I/x)*csgn(I/ln(x)^2)*csgn(I/x/ln(x)^2)-I*Pi*x*csgn(I*ln
(x)^2)^3+I*Pi*x*csgn(I/x/ln(x)^2)^3-I*Pi*x*csgn(I/ln(x)^2)*csgn(I/x/ln(x)^2)^2-I*Pi*x*csgn(I/x)*csgn(I/x/ln(x)
^2)^2+2*I*Pi*x*csgn(I*ln(x))*csgn(I*ln(x)^2)^2-I*Pi*x*csgn(I*ln(x))^2*csgn(I*ln(x)^2)+8+2*x*ln(2)+10*x*ln(3)-2
*x*ln(5)+2*x*ln(x))/(x^2-4)*ln(ln(x))+1/4/ln(3)^2*(40*x*ln(3)*ln(x)+8*x^2*ln(x)+8*x*ln(2)*ln(x)-8*x*ln(2)*ln(5
)+16*x^2*ln(ln(x))-32*ln(5)+32*ln(2)-64*ln(ln(x))+160*ln(3)+4*x^3-2*Pi^2*x*csgn(I/x)*csgn(I*ln(x))^2*csgn(I*ln
(x)^2)*csgn(I/x/ln(x)^2)^2+4*Pi^2*x*csgn(I/x)*csgn(I*ln(x))*csgn(I*ln(x)^2)^2*csgn(I/x/ln(x)^2)^2+2*Pi^2*x*csg
n(I/x)*csgn(I/ln(x)^2)*csgn(I*ln(x)^2)^3*csgn(I/x/ln(x)^2)-2*Pi^2*x*csgn(I*ln(x))^2*csgn(I/ln(x)^2)*csgn(I*ln(
x)^2)*csgn(I/x/ln(x)^2)^2+4*Pi^2*x*csgn(I*ln(x))*csgn(I/ln(x)^2)*csgn(I*ln(x)^2)^2*csgn(I/x/ln(x)^2)^2-4*I*Pi*
ln(2)*x*csgn(I/x)*csgn(I/x/ln(x)^2)^2-16*I*Pi*csgn(I*ln(x)^2)^3+16*I*Pi*csgn(I/x/ln(x)^2)^3-Pi^2*x*csgn(I*ln(x
)^2)^6+4*x*ln(2)^2+100*x*ln(3)^2+4*x*ln(x)^2-16*I*Pi*csgn(I/ln(x)^2)*csgn(I/x/ln(x)^2)^2-16*I*Pi*csgn(I*ln(x))
^2*csgn(I*ln(x)^2)-Pi^2*x*csgn(I/ln(x)^2)^2*csgn(I/x/ln(x)^2)^4+2*Pi^2*x*csgn(I/ln(x)^2)*csgn(I/x/ln(x)^2)^5+2
*Pi^2*x*csgn(I*ln(x)^2)^3*csgn(I/x/ln(x)^2)^3-Pi^2*x*csgn(I/x/ln(x)^2)^6-Pi^2*x*csgn(I/x)^2*csgn(I/x/ln(x)^2)^
4+2*Pi^2*x*csgn(I/x)*csgn(I/x/ln(x)^2)^5-Pi^2*x*csgn(I*ln(x))^4*csgn(I*ln(x)^2)^2+4*Pi^2*x*csgn(I*ln(x))^3*csg
n(I*ln(x)^2)^3-6*Pi^2*x*csgn(I*ln(x))^2*csgn(I*ln(x)^2)^4+4*Pi^2*x*csgn(I*ln(x))*csgn(I*ln(x)^2)^5+32*I*Pi*csg
n(I*ln(x))*csgn(I*ln(x)^2)^2-16*I*Pi*csgn(I/x)*csgn(I/x/ln(x)^2)^2-40*x*ln(3)*ln(5)+4*x*ln(5)^2-8*x*ln(5)*ln(x
)+40*x*ln(2)*ln(3)-4*I*Pi*x*csgn(I*ln(x)^2)^3*ln(x)+4*I*Pi*x*csgn(I/x/ln(x)^2)^3*ln(x)-4*I*Pi*ln(2)*x*csgn(I*l
n(x))^2*csgn(I*ln(x)^2)+8*I*Pi*ln(2)*x*csgn(I*ln(x))*csgn(I*ln(x)^2)^2-4*I*Pi*ln(2)*x*csgn(I/ln(x)^2)*csgn(I/x
/ln(x)^2)^2-20*I*Pi*ln(3)*x*csgn(I/x)*csgn(I/x/ln(x)^2)^2-20*I*Pi*ln(3)*x*csgn(I*ln(x))^2*csgn(I*ln(x)^2)+40*I
*Pi*ln(3)*x*csgn(I*ln(x))*csgn(I*ln(x)^2)^2-20*I*Pi*ln(3)*x*csgn(I/ln(x)^2)*csgn(I/x/ln(x)^2)^2+4*I*Pi*ln(5)*x
*csgn(I/x)*csgn(I/x/ln(x)^2)^2+4*I*Pi*ln(5)*x*csgn(I*ln(x))^2*csgn(I*ln(x)^2)-8*I*Pi*ln(5)*x*csgn(I*ln(x))*csg
n(I*ln(x)^2)^2+4*I*Pi*ln(5)*x*csgn(I/ln(x)^2)*csgn(I/x/ln(x)^2)^2-4*I*Pi*x*csgn(I/x)*csgn(I/x/ln(x)^2)^2*ln(x)
-4*I*Pi*x*csgn(I*ln(x))^2*csgn(I*ln(x)^2)*ln(x)+8*I*Pi*x*csgn(I*ln(x))*csgn(I*ln(x)^2)^2*ln(x)-4*I*Pi*x*csgn(I
/ln(x)^2)*csgn(I/x/ln(x)^2)^2*ln(x)+2*Pi^2*x*csgn(I/x)*csgn(I*ln(x))^2*csgn(I/ln(x)^2)*csgn(I*ln(x)^2)*csgn(I/
x/ln(x)^2)-4*Pi^2*x*csgn(I/x)*csgn(I*ln(x))*csgn(I/ln(x)^2)*csgn(I*ln(x)^2)^2*csgn(I/x/ln(x)^2)+4*I*Pi*ln(2)*x
*csgn(I/x)*csgn(I/ln(x)^2)*csgn(I/x/ln(x)^2)+20*I*Pi*ln(3)*x*csgn(I/x)*csgn(I/ln(x)^2)*csgn(I/x/ln(x)^2)-4*I*P
i*ln(5)*x*csgn(I/x)*csgn(I/ln(x)^2)*csgn(I/x/ln(x)^2)+4*I*Pi*x*csgn(I/x)*csgn(I/ln(x)^2)*csgn(I/x/ln(x)^2)*ln(
x)-Pi^2*x*csgn(I/x)^2*csgn(I/ln(x)^2)^2*csgn(I/x/ln(x)^2)^2+2*Pi^2*x*csgn(I/x)^2*csgn(I/ln(x)^2)*csgn(I/x/ln(x
)^2)^3+2*Pi^2*x*csgn(I/x)*csgn(I/ln(x)^2)^2*csgn(I/x/ln(x)^2)^3-4*Pi^2*x*csgn(I/x)*csgn(I/ln(x)^2)*csgn(I/x/ln
(x)^2)^4-2*Pi^2*x*csgn(I/x)*csgn(I*ln(x)^2)^3*csgn(I/x/ln(x)^2)^2+2*Pi^2*x*csgn(I*ln(x))^2*csgn(I*ln(x)^2)*csg
n(I/x/ln(x)^2)^3-4*Pi^2*x*csgn(I*ln(x))*csgn(I*ln(x)^2)^2*csgn(I/x/ln(x)^2)^3-2*Pi^2*x*csgn(I/ln(x)^2)*csgn(I*
ln(x)^2)^3*csgn(I/x/ln(x)^2)^2-4*I*Pi*ln(2)*x*csgn(I*ln(x)^2)^3+4*I*Pi*ln(2)*x*csgn(I/x/ln(x)^2)^3-20*I*Pi*ln(
3)*x*csgn(I*ln(x)^2)^3+20*I*Pi*ln(3)*x*csgn(I/x/ln(x)^2)^3+4*I*Pi*ln(5)*x*csgn(I*ln(x)^2)^3-4*I*Pi*ln(5)*x*csg
n(I/x/ln(x)^2)^3+16*I*Pi*csgn(I/x)*csgn(I/ln(x)^2)*csgn(I/x/ln(x)^2))/(x^2-4)

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maxima [B]  time = 0.74, size = 125, normalized size = 3.57 \begin {gather*} \frac {x^{3} + x \log \relax (x)^{2} + 4 \, x \log \left (\log \relax (x)\right )^{2} + {\left (\log \relax (5)^{2} - 10 \, \log \relax (5) \log \relax (3) + 25 \, \log \relax (3)^{2} - 2 \, {\left (\log \relax (5) - 5 \, \log \relax (3)\right )} \log \relax (2) + \log \relax (2)^{2}\right )} x + 2 \, {\left (x^{2} - x {\left (\log \relax (5) - 5 \, \log \relax (3) - \log \relax (2)\right )}\right )} \log \relax (x) + 4 \, {\left (x^{2} - x {\left (\log \relax (5) - 5 \, \log \relax (3) - \log \relax (2)\right )} + x \log \relax (x)\right )} \log \left (\log \relax (x)\right ) - 8 \, \log \relax (5) + 40 \, \log \relax (3) + 8 \, \log \relax (2)}{{\left (x^{2} - 4\right )} \log \relax (3)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2-4)*log(x)*log(5/2/x/log(x)^2)^2+(((10*x^2+40)*log(3)-2*x^2+16*x+8)*log(x)-4*x^2+16)*log(5/2/x
/log(x)^2)+((-25*x^2-100)*log(3)^2+(10*x^2-80*x-40)*log(3)+x^4+2*x^3-12*x^2-8*x)*log(x)+(20*x^2-80)*log(3)+4*x
^3-16*x)/(x^4-8*x^2+16)/log(3)^2/log(x),x, algorithm="maxima")

[Out]

(x^3 + x*log(x)^2 + 4*x*log(log(x))^2 + (log(5)^2 - 10*log(5)*log(3) + 25*log(3)^2 - 2*(log(5) - 5*log(3))*log
(2) + log(2)^2)*x + 2*(x^2 - x*(log(5) - 5*log(3) - log(2)))*log(x) + 4*(x^2 - x*(log(5) - 5*log(3) - log(2))
+ x*log(x))*log(log(x)) - 8*log(5) + 40*log(3) + 8*log(2))/((x^2 - 4)*log(3)^2)

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mupad [B]  time = 3.88, size = 124, normalized size = 3.54 \begin {gather*} \frac {2\,\ln \relax (x)}{{\ln \relax (3)}^2}+\frac {4\,\ln \left (\ln \relax (x)\right )}{{\ln \relax (3)}^2}+\frac {x}{{\ln \relax (3)}^2}+\frac {40\,\ln \relax (3)+x\,\left (25\,{\ln \relax (3)}^2+4\right )}{x^2\,{\ln \relax (3)}^2-4\,{\ln \relax (3)}^2}+\frac {x\,{\ln \left (\frac {5}{2\,x\,{\ln \relax (x)}^2}\right )}^2}{x^2\,{\ln \relax (3)}^2-4\,{\ln \relax (3)}^2}-\frac {\ln \left (\frac {5}{2\,x\,{\ln \relax (x)}^2}\right )\,\left (10\,x\,\ln \relax (3)+8\right )}{x^2\,{\ln \relax (3)}^2-4\,{\ln \relax (3)}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(16*x + log(x)*(8*x + log(3)*(80*x - 10*x^2 + 40) + log(3)^2*(25*x^2 + 100) + 12*x^2 - 2*x^3 - x^4) - log
(3)*(20*x^2 - 80) - 4*x^3 - log(5/(2*x*log(x)^2))*(log(x)*(16*x + log(3)*(10*x^2 + 40) - 2*x^2 + 8) - 4*x^2 +
16) + log(x)*log(5/(2*x*log(x)^2))^2*(x^2 + 4))/(log(3)^2*log(x)*(x^4 - 8*x^2 + 16)),x)

[Out]

(2*log(x))/log(3)^2 + (4*log(log(x)))/log(3)^2 + x/log(3)^2 + (40*log(3) + x*(25*log(3)^2 + 4))/(x^2*log(3)^2
- 4*log(3)^2) + (x*log(5/(2*x*log(x)^2))^2)/(x^2*log(3)^2 - 4*log(3)^2) - (log(5/(2*x*log(x)^2))*(10*x*log(3)
+ 8))/(x^2*log(3)^2 - 4*log(3)^2)

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sympy [B]  time = 1.75, size = 126, normalized size = 3.60 \begin {gather*} \frac {x}{\log {\relax (3 )}^{2}} + \frac {x \log {\left (\frac {5}{2 x \log {\relax (x )}^{2}} \right )}^{2}}{x^{2} \log {\relax (3 )}^{2} - 4 \log {\relax (3 )}^{2}} + \frac {x \left (4 + 25 \log {\relax (3 )}^{2}\right ) + 40 \log {\relax (3 )}}{x^{2} \log {\relax (3 )}^{2} - 4 \log {\relax (3 )}^{2}} + \frac {\left (- 10 x \log {\relax (3 )} - 8\right ) \log {\left (\frac {5}{2 x \log {\relax (x )}^{2}} \right )}}{x^{2} \log {\relax (3 )}^{2} - 4 \log {\relax (3 )}^{2}} + \frac {2 \log {\relax (x )}}{\log {\relax (3 )}^{2}} + \frac {4 \log {\left (\log {\relax (x )} \right )}}{\log {\relax (3 )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**2-4)*ln(x)*ln(5/2/x/ln(x)**2)**2+(((10*x**2+40)*ln(3)-2*x**2+16*x+8)*ln(x)-4*x**2+16)*ln(5/2/x
/ln(x)**2)+((-25*x**2-100)*ln(3)**2+(10*x**2-80*x-40)*ln(3)+x**4+2*x**3-12*x**2-8*x)*ln(x)+(20*x**2-80)*ln(3)+
4*x**3-16*x)/(x**4-8*x**2+16)/ln(3)**2/ln(x),x)

[Out]

x/log(3)**2 + x*log(5/(2*x*log(x)**2))**2/(x**2*log(3)**2 - 4*log(3)**2) + (x*(4 + 25*log(3)**2) + 40*log(3))/
(x**2*log(3)**2 - 4*log(3)**2) + (-10*x*log(3) - 8)*log(5/(2*x*log(x)**2))/(x**2*log(3)**2 - 4*log(3)**2) + 2*
log(x)/log(3)**2 + 4*log(log(x))/log(3)**2

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