3.26.36 \(\int \frac {-2 x \log (x)+(3 x+x \log ^2(x)) \log (3+\log ^2(x))+(6+2 \log ^2(x)) \log ^2(3+\log ^2(x))+((-3 x-x \log ^2(x)) \log (3+\log ^2(x))+(-15+3 \log (5)+3 \log (\frac {3}{4 x^2})+(-5+\log (5)+\log (\frac {3}{4 x^2})) \log ^2(x)) \log ^2(3+\log ^2(x))) \log (\frac {-x+(-5+\log (5)+\log (\frac {3}{4 x^2})) \log (3+\log ^2(x))}{\log (3+\log ^2(x))})}{((-3 x-x \log ^2(x)) \log (3+\log ^2(x))+(-15+3 \log (5)+3 \log (\frac {3}{4 x^2})+(-5+\log (5)+\log (\frac {3}{4 x^2})) \log ^2(x)) \log ^2(3+\log ^2(x))) \log ^2(\frac {-x+(-5+\log (5)+\log (\frac {3}{4 x^2})) \log (3+\log ^2(x))}{\log (3+\log ^2(x))})} \, dx\)

Optimal. Leaf size=29 \[ \frac {x}{\log \left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )} \]

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Rubi [F]  time = 6.79, 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 {-2 x \log (x)+\left (3 x+x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (6+2 \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )+\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log \left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )}{\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log ^2\left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-2*x*Log[x] + (3*x + x*Log[x]^2)*Log[3 + Log[x]^2] + (6 + 2*Log[x]^2)*Log[3 + Log[x]^2]^2 + ((-3*x - x*Lo
g[x]^2)*Log[3 + Log[x]^2] + (-15 + 3*Log[5] + 3*Log[3/(4*x^2)] + (-5 + Log[5] + Log[3/(4*x^2)])*Log[x]^2)*Log[
3 + Log[x]^2]^2)*Log[(-x + (-5 + Log[5] + Log[3/(4*x^2)])*Log[3 + Log[x]^2])/Log[3 + Log[x]^2]])/(((-3*x - x*L
og[x]^2)*Log[3 + Log[x]^2] + (-15 + 3*Log[5] + 3*Log[3/(4*x^2)] + (-5 + Log[5] + Log[3/(4*x^2)])*Log[x]^2)*Log
[3 + Log[x]^2]^2)*Log[(-x + (-5 + Log[5] + Log[3/(4*x^2)])*Log[3 + Log[x]^2])/Log[3 + Log[x]^2]]^2),x]

[Out]

-3*Defer[Int][x/((3 + Log[x]^2)*(x + 5*Log[3 + Log[x]^2] - Log[15/(4*x^2)]*Log[3 + Log[x]^2])*Log[-5 + Log[15/
(4*x^2)] - x/Log[3 + Log[x]^2]]^2), x] - Defer[Int][(x*Log[x]^2)/((3 + Log[x]^2)*(x + 5*Log[3 + Log[x]^2] - Lo
g[15/(4*x^2)]*Log[3 + Log[x]^2])*Log[-5 + Log[15/(4*x^2)] - x/Log[3 + Log[x]^2]]^2), x] - 2*Defer[Int][(x*Log[
x])/((3 + Log[x]^2)*Log[3 + Log[x]^2]*(-x - 5*Log[3 + Log[x]^2] + Log[15/(4*x^2)]*Log[3 + Log[x]^2])*Log[-5 +
Log[15/(4*x^2)] - x/Log[3 + Log[x]^2]]^2), x] + 6*Defer[Int][Log[3 + Log[x]^2]/((3 + Log[x]^2)*(-x - 5*Log[3 +
Log[x]^2] + Log[15/(4*x^2)]*Log[3 + Log[x]^2])*Log[-5 + Log[15/(4*x^2)] - x/Log[3 + Log[x]^2]]^2), x] + 2*Def
er[Int][(Log[x]^2*Log[3 + Log[x]^2])/((3 + Log[x]^2)*(-x - 5*Log[3 + Log[x]^2] + Log[15/(4*x^2)]*Log[3 + Log[x
]^2])*Log[-5 + Log[15/(4*x^2)] - x/Log[3 + Log[x]^2]]^2), x] + Defer[Int][Log[-5 + Log[15/(4*x^2)] - x/Log[3 +
Log[x]^2]]^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 x \log (x)-x \left (3+\log ^2(x)\right ) \log \left (3+\log ^2(x)\right )-2 \left (3+\log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )+\left (3+\log ^2(x)\right ) \log \left (3+\log ^2(x)\right ) \left (x-\left (-5+\log \left (\frac {15}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )\right ) \log \left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )}{\left (3+\log ^2(x)\right ) \log \left (3+\log ^2(x)\right ) \left (x-\left (-5+\log \left (\frac {15}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )\right ) \log ^2\left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )} \, dx\\ &=\int \left (\frac {-2 x \log (x)+3 x \log \left (3+\log ^2(x)\right )+x \log ^2(x) \log \left (3+\log ^2(x)\right )+6 \log ^2\left (3+\log ^2(x)\right )+2 \log ^2(x) \log ^2\left (3+\log ^2(x)\right )}{\left (3+\log ^2(x)\right ) \log \left (3+\log ^2(x)\right ) \left (-x-5 \log \left (3+\log ^2(x)\right )+\log \left (\frac {15}{4 x^2}\right ) \log \left (3+\log ^2(x)\right )\right ) \log ^2\left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )}+\frac {1}{\log \left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )}\right ) \, dx\\ &=\int \frac {-2 x \log (x)+3 x \log \left (3+\log ^2(x)\right )+x \log ^2(x) \log \left (3+\log ^2(x)\right )+6 \log ^2\left (3+\log ^2(x)\right )+2 \log ^2(x) \log ^2\left (3+\log ^2(x)\right )}{\left (3+\log ^2(x)\right ) \log \left (3+\log ^2(x)\right ) \left (-x-5 \log \left (3+\log ^2(x)\right )+\log \left (\frac {15}{4 x^2}\right ) \log \left (3+\log ^2(x)\right )\right ) \log ^2\left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )} \, dx+\int \frac {1}{\log \left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )} \, dx\\ &=\int \left (-\frac {3 x}{\left (3+\log ^2(x)\right ) \left (x+5 \log \left (3+\log ^2(x)\right )-\log \left (\frac {15}{4 x^2}\right ) \log \left (3+\log ^2(x)\right )\right ) \log ^2\left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )}-\frac {x \log ^2(x)}{\left (3+\log ^2(x)\right ) \left (x+5 \log \left (3+\log ^2(x)\right )-\log \left (\frac {15}{4 x^2}\right ) \log \left (3+\log ^2(x)\right )\right ) \log ^2\left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )}-\frac {2 x \log (x)}{\left (3+\log ^2(x)\right ) \log \left (3+\log ^2(x)\right ) \left (-x-5 \log \left (3+\log ^2(x)\right )+\log \left (\frac {15}{4 x^2}\right ) \log \left (3+\log ^2(x)\right )\right ) \log ^2\left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )}+\frac {6 \log \left (3+\log ^2(x)\right )}{\left (3+\log ^2(x)\right ) \left (-x-5 \log \left (3+\log ^2(x)\right )+\log \left (\frac {15}{4 x^2}\right ) \log \left (3+\log ^2(x)\right )\right ) \log ^2\left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )}+\frac {2 \log ^2(x) \log \left (3+\log ^2(x)\right )}{\left (3+\log ^2(x)\right ) \left (-x-5 \log \left (3+\log ^2(x)\right )+\log \left (\frac {15}{4 x^2}\right ) \log \left (3+\log ^2(x)\right )\right ) \log ^2\left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )}\right ) \, dx+\int \frac {1}{\log \left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )} \, dx\\ &=-\left (2 \int \frac {x \log (x)}{\left (3+\log ^2(x)\right ) \log \left (3+\log ^2(x)\right ) \left (-x-5 \log \left (3+\log ^2(x)\right )+\log \left (\frac {15}{4 x^2}\right ) \log \left (3+\log ^2(x)\right )\right ) \log ^2\left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )} \, dx\right )+2 \int \frac {\log ^2(x) \log \left (3+\log ^2(x)\right )}{\left (3+\log ^2(x)\right ) \left (-x-5 \log \left (3+\log ^2(x)\right )+\log \left (\frac {15}{4 x^2}\right ) \log \left (3+\log ^2(x)\right )\right ) \log ^2\left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )} \, dx-3 \int \frac {x}{\left (3+\log ^2(x)\right ) \left (x+5 \log \left (3+\log ^2(x)\right )-\log \left (\frac {15}{4 x^2}\right ) \log \left (3+\log ^2(x)\right )\right ) \log ^2\left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )} \, dx+6 \int \frac {\log \left (3+\log ^2(x)\right )}{\left (3+\log ^2(x)\right ) \left (-x-5 \log \left (3+\log ^2(x)\right )+\log \left (\frac {15}{4 x^2}\right ) \log \left (3+\log ^2(x)\right )\right ) \log ^2\left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )} \, dx-\int \frac {x \log ^2(x)}{\left (3+\log ^2(x)\right ) \left (x+5 \log \left (3+\log ^2(x)\right )-\log \left (\frac {15}{4 x^2}\right ) \log \left (3+\log ^2(x)\right )\right ) \log ^2\left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )} \, dx+\int \frac {1}{\log \left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.26, size = 27, normalized size = 0.93 \begin {gather*} \frac {x}{\log \left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*x*Log[x] + (3*x + x*Log[x]^2)*Log[3 + Log[x]^2] + (6 + 2*Log[x]^2)*Log[3 + Log[x]^2]^2 + ((-3*x
- x*Log[x]^2)*Log[3 + Log[x]^2] + (-15 + 3*Log[5] + 3*Log[3/(4*x^2)] + (-5 + Log[5] + Log[3/(4*x^2)])*Log[x]^2
)*Log[3 + Log[x]^2]^2)*Log[(-x + (-5 + Log[5] + Log[3/(4*x^2)])*Log[3 + Log[x]^2])/Log[3 + Log[x]^2]])/(((-3*x
- x*Log[x]^2)*Log[3 + Log[x]^2] + (-15 + 3*Log[5] + 3*Log[3/(4*x^2)] + (-5 + Log[5] + Log[3/(4*x^2)])*Log[x]^
2)*Log[3 + Log[x]^2]^2)*Log[(-x + (-5 + Log[5] + Log[3/(4*x^2)])*Log[3 + Log[x]^2])/Log[3 + Log[x]^2]]^2),x]

[Out]

x/Log[-5 + Log[15/(4*x^2)] - x/Log[3 + Log[x]^2]]

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fricas [B]  time = 0.65, size = 81, normalized size = 2.79 \begin {gather*} \frac {x}{\log \left (\frac {{\left (\log \relax (5) + \log \left (\frac {3}{4 \, x^{2}}\right ) - 5\right )} \log \left (\frac {1}{4} \, \log \left (\frac {3}{4}\right )^{2} - \frac {1}{2} \, \log \left (\frac {3}{4}\right ) \log \left (\frac {3}{4 \, x^{2}}\right ) + \frac {1}{4} \, \log \left (\frac {3}{4 \, x^{2}}\right )^{2} + 3\right ) - x}{\log \left (\frac {1}{4} \, \log \left (\frac {3}{4}\right )^{2} - \frac {1}{2} \, \log \left (\frac {3}{4}\right ) \log \left (\frac {3}{4 \, x^{2}}\right ) + \frac {1}{4} \, \log \left (\frac {3}{4 \, x^{2}}\right )^{2} + 3\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((log(3/4/x^2)+log(5)-5)*log(x)^2+3*log(3/4/x^2)+3*log(5)-15)*log(log(x)^2+3)^2+(-x*log(x)^2-3*x)*
log(log(x)^2+3))*log(((log(3/4/x^2)+log(5)-5)*log(log(x)^2+3)-x)/log(log(x)^2+3))+(2*log(x)^2+6)*log(log(x)^2+
3)^2+(x*log(x)^2+3*x)*log(log(x)^2+3)-2*x*log(x))/(((log(3/4/x^2)+log(5)-5)*log(x)^2+3*log(3/4/x^2)+3*log(5)-1
5)*log(log(x)^2+3)^2+(-x*log(x)^2-3*x)*log(log(x)^2+3))/log(((log(3/4/x^2)+log(5)-5)*log(log(x)^2+3)-x)/log(lo
g(x)^2+3))^2,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 9.00, size = 71, normalized size = 2.45 \begin {gather*} \frac {x}{\log \left (\log \relax (5) \log \left (\log \relax (x)^{2} + 3\right ) + \log \relax (3) \log \left (\log \relax (x)^{2} + 3\right ) - 2 \, \log \relax (2) \log \left (\log \relax (x)^{2} + 3\right ) - 2 \, \log \left (\log \relax (x)^{2} + 3\right ) \log \relax (x) - x - 5 \, \log \left (\log \relax (x)^{2} + 3\right )\right ) - \log \left (\log \left (\log \relax (x)^{2} + 3\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((log(3/4/x^2)+log(5)-5)*log(x)^2+3*log(3/4/x^2)+3*log(5)-15)*log(log(x)^2+3)^2+(-x*log(x)^2-3*x)*
log(log(x)^2+3))*log(((log(3/4/x^2)+log(5)-5)*log(log(x)^2+3)-x)/log(log(x)^2+3))+(2*log(x)^2+6)*log(log(x)^2+
3)^2+(x*log(x)^2+3*x)*log(log(x)^2+3)-2*x*log(x))/(((log(3/4/x^2)+log(5)-5)*log(x)^2+3*log(3/4/x^2)+3*log(5)-1
5)*log(log(x)^2+3)^2+(-x*log(x)^2-3*x)*log(log(x)^2+3))/log(((log(3/4/x^2)+log(5)-5)*log(log(x)^2+3)-x)/log(lo
g(x)^2+3))^2,x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.87, size = 1756, normalized size = 60.55

method	result	size
risch	\(\text {Expression too large to display}\)	\(1756\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((((ln(3/4/x^2)+ln(5)-5)*ln(x)^2+3*ln(3/4/x^2)+3*ln(5)-15)*ln(ln(x)^2+3)^2+(-x*ln(x)^2-3*x)*ln(ln(x)^2+3))
*ln(((ln(3/4/x^2)+ln(5)-5)*ln(ln(x)^2+3)-x)/ln(ln(x)^2+3))+(2*ln(x)^2+6)*ln(ln(x)^2+3)^2+(x*ln(x)^2+3*x)*ln(ln
(x)^2+3)-2*x*ln(x))/(((ln(3/4/x^2)+ln(5)-5)*ln(x)^2+3*ln(3/4/x^2)+3*ln(5)-15)*ln(ln(x)^2+3)^2+(-x*ln(x)^2-3*x)
*ln(ln(x)^2+3))/ln(((ln(3/4/x^2)+ln(5)-5)*ln(ln(x)^2+3)-x)/ln(ln(x)^2+3))^2,x,method=_RETURNVERBOSE)

[Out]

2*I*x/(Pi*csgn(I/ln(ln(x)^2+3)*(-2*I*x-ln(ln(x)^2+3)*Pi*csgn(I*x^2)^3+2*I*ln(ln(x)^2+3)*ln(3/4)-4*I*ln(ln(x)^2
+3)*ln(x)+2*I*ln(ln(x)^2+3)*ln(5)+2*ln(ln(x)^2+3)*Pi*csgn(I*x)*csgn(I*x^2)^2-ln(ln(x)^2+3)*Pi*csgn(I*x)^2*csgn
(I*x^2)-10*I*ln(ln(x)^2+3)))*csgn((-2*I*x-ln(ln(x)^2+3)*Pi*csgn(I*x^2)^3+2*I*ln(ln(x)^2+3)*ln(3/4)-4*I*ln(ln(x
)^2+3)*ln(x)+2*I*ln(ln(x)^2+3)*ln(5)+2*ln(ln(x)^2+3)*Pi*csgn(I*x)*csgn(I*x^2)^2-ln(ln(x)^2+3)*Pi*csgn(I*x)^2*c
sgn(I*x^2)-10*I*ln(ln(x)^2+3))/ln(ln(x)^2+3))-Pi*csgn((-2*I*x-ln(ln(x)^2+3)*Pi*csgn(I*x^2)^3+2*I*ln(ln(x)^2+3)
*ln(3/4)-4*I*ln(ln(x)^2+3)*ln(x)+2*I*ln(ln(x)^2+3)*ln(5)+2*ln(ln(x)^2+3)*Pi*csgn(I*x)*csgn(I*x^2)^2-ln(ln(x)^2
+3)*Pi*csgn(I*x)^2*csgn(I*x^2)-10*I*ln(ln(x)^2+3))/ln(ln(x)^2+3))^2+Pi*csgn(I*(-2*I*x-ln(ln(x)^2+3)*Pi*csgn(I*
x^2)^3+2*I*ln(ln(x)^2+3)*ln(3/4)-4*I*ln(ln(x)^2+3)*ln(x)+2*I*ln(ln(x)^2+3)*ln(5)+2*ln(ln(x)^2+3)*Pi*csgn(I*x)*
csgn(I*x^2)^2-ln(ln(x)^2+3)*Pi*csgn(I*x)^2*csgn(I*x^2)-10*I*ln(ln(x)^2+3)))*csgn(I/ln(ln(x)^2+3))*csgn(I/ln(ln
(x)^2+3)*(-2*I*x-ln(ln(x)^2+3)*Pi*csgn(I*x^2)^3+2*I*ln(ln(x)^2+3)*ln(3/4)-4*I*ln(ln(x)^2+3)*ln(x)+2*I*ln(ln(x)
^2+3)*ln(5)+2*ln(ln(x)^2+3)*Pi*csgn(I*x)*csgn(I*x^2)^2-ln(ln(x)^2+3)*Pi*csgn(I*x)^2*csgn(I*x^2)-10*I*ln(ln(x)^
2+3)))-Pi*csgn(I*(-2*I*x-ln(ln(x)^2+3)*Pi*csgn(I*x^2)^3+2*I*ln(ln(x)^2+3)*ln(3/4)-4*I*ln(ln(x)^2+3)*ln(x)+2*I*
ln(ln(x)^2+3)*ln(5)+2*ln(ln(x)^2+3)*Pi*csgn(I*x)*csgn(I*x^2)^2-ln(ln(x)^2+3)*Pi*csgn(I*x)^2*csgn(I*x^2)-10*I*l
n(ln(x)^2+3)))*csgn(I/ln(ln(x)^2+3)*(-2*I*x-ln(ln(x)^2+3)*Pi*csgn(I*x^2)^3+2*I*ln(ln(x)^2+3)*ln(3/4)-4*I*ln(ln
(x)^2+3)*ln(x)+2*I*ln(ln(x)^2+3)*ln(5)+2*ln(ln(x)^2+3)*Pi*csgn(I*x)*csgn(I*x^2)^2-ln(ln(x)^2+3)*Pi*csgn(I*x)^2
*csgn(I*x^2)-10*I*ln(ln(x)^2+3)))^2-Pi*csgn(I/ln(ln(x)^2+3))*csgn(I/ln(ln(x)^2+3)*(-2*I*x-ln(ln(x)^2+3)*Pi*csg
n(I*x^2)^3+2*I*ln(ln(x)^2+3)*ln(3/4)-4*I*ln(ln(x)^2+3)*ln(x)+2*I*ln(ln(x)^2+3)*ln(5)+2*ln(ln(x)^2+3)*Pi*csgn(I
*x)*csgn(I*x^2)^2-ln(ln(x)^2+3)*Pi*csgn(I*x)^2*csgn(I*x^2)-10*I*ln(ln(x)^2+3)))^2+Pi*csgn(I/ln(ln(x)^2+3)*(-2*
I*x-ln(ln(x)^2+3)*Pi*csgn(I*x^2)^3+2*I*ln(ln(x)^2+3)*ln(3/4)-4*I*ln(ln(x)^2+3)*ln(x)+2*I*ln(ln(x)^2+3)*ln(5)+2
*ln(ln(x)^2+3)*Pi*csgn(I*x)*csgn(I*x^2)^2-ln(ln(x)^2+3)*Pi*csgn(I*x)^2*csgn(I*x^2)-10*I*ln(ln(x)^2+3)))^3-Pi*c
sgn(I/ln(ln(x)^2+3)*(-2*I*x-ln(ln(x)^2+3)*Pi*csgn(I*x^2)^3+2*I*ln(ln(x)^2+3)*ln(3/4)-4*I*ln(ln(x)^2+3)*ln(x)+2
*I*ln(ln(x)^2+3)*ln(5)+2*ln(ln(x)^2+3)*Pi*csgn(I*x)*csgn(I*x^2)^2-ln(ln(x)^2+3)*Pi*csgn(I*x)^2*csgn(I*x^2)-10*
I*ln(ln(x)^2+3)))*csgn((-2*I*x-ln(ln(x)^2+3)*Pi*csgn(I*x^2)^3+2*I*ln(ln(x)^2+3)*ln(3/4)-4*I*ln(ln(x)^2+3)*ln(x
)+2*I*ln(ln(x)^2+3)*ln(5)+2*ln(ln(x)^2+3)*Pi*csgn(I*x)*csgn(I*x^2)^2-ln(ln(x)^2+3)*Pi*csgn(I*x)^2*csgn(I*x^2)-
10*I*ln(ln(x)^2+3))/ln(ln(x)^2+3))^2+Pi*csgn((-2*I*x-ln(ln(x)^2+3)*Pi*csgn(I*x^2)^3+2*I*ln(ln(x)^2+3)*ln(3/4)-
4*I*ln(ln(x)^2+3)*ln(x)+2*I*ln(ln(x)^2+3)*ln(5)+2*ln(ln(x)^2+3)*Pi*csgn(I*x)*csgn(I*x^2)^2-ln(ln(x)^2+3)*Pi*cs
gn(I*x)^2*csgn(I*x^2)-10*I*ln(ln(x)^2+3))/ln(ln(x)^2+3))^3+Pi-2*I*ln(2)-2*I*ln(ln(ln(x)^2+3))+2*I*ln(-2*I*x-ln
(ln(x)^2+3)*Pi*csgn(I*x^2)^3+2*I*ln(ln(x)^2+3)*(ln(3)-2*ln(2))-4*I*ln(ln(x)^2+3)*ln(x)+2*I*ln(ln(x)^2+3)*ln(5)
+2*ln(ln(x)^2+3)*Pi*csgn(I*x)*csgn(I*x^2)^2-ln(ln(x)^2+3)*Pi*csgn(I*x)^2*csgn(I*x^2)-10*I*ln(ln(x)^2+3)))

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maxima [A]  time = 0.75, size = 49, normalized size = 1.69 \begin {gather*} \frac {x}{\log \left ({\left (\log \relax (5) + \log \relax (3) - 2 \, \log \relax (2) - 5\right )} \log \left (\log \relax (x)^{2} + 3\right ) - 2 \, \log \left (\log \relax (x)^{2} + 3\right ) \log \relax (x) - x\right ) - \log \left (\log \left (\log \relax (x)^{2} + 3\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((log(3/4/x^2)+log(5)-5)*log(x)^2+3*log(3/4/x^2)+3*log(5)-15)*log(log(x)^2+3)^2+(-x*log(x)^2-3*x)*
log(log(x)^2+3))*log(((log(3/4/x^2)+log(5)-5)*log(log(x)^2+3)-x)/log(log(x)^2+3))+(2*log(x)^2+6)*log(log(x)^2+
3)^2+(x*log(x)^2+3*x)*log(log(x)^2+3)-2*x*log(x))/(((log(3/4/x^2)+log(5)-5)*log(x)^2+3*log(3/4/x^2)+3*log(5)-1
5)*log(log(x)^2+3)^2+(-x*log(x)^2-3*x)*log(log(x)^2+3))/log(((log(3/4/x^2)+log(5)-5)*log(log(x)^2+3)-x)/log(lo
g(x)^2+3))^2,x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\ln \left (-\frac {x-\ln \left ({\ln \relax (x)}^2+3\right )\,\left (\ln \relax (5)+\ln \left (\frac {3}{4\,x^2}\right )-5\right )}{\ln \left ({\ln \relax (x)}^2+3\right )}\right )\,\left ({\ln \left ({\ln \relax (x)}^2+3\right )}^2\,\left (\left (\ln \relax (5)+\ln \left (\frac {3}{4\,x^2}\right )-5\right )\,{\ln \relax (x)}^2+3\,\ln \relax (5)+3\,\ln \left (\frac {3}{4\,x^2}\right )-15\right )-\ln \left ({\ln \relax (x)}^2+3\right )\,\left (x\,{\ln \relax (x)}^2+3\,x\right )\right )-2\,x\,\ln \relax (x)+\ln \left ({\ln \relax (x)}^2+3\right )\,\left (x\,{\ln \relax (x)}^2+3\,x\right )+{\ln \left ({\ln \relax (x)}^2+3\right )}^2\,\left (2\,{\ln \relax (x)}^2+6\right )}{{\ln \left (-\frac {x-\ln \left ({\ln \relax (x)}^2+3\right )\,\left (\ln \relax (5)+\ln \left (\frac {3}{4\,x^2}\right )-5\right )}{\ln \left ({\ln \relax (x)}^2+3\right )}\right )}^2\,\left ({\ln \left ({\ln \relax (x)}^2+3\right )}^2\,\left (\left (\ln \relax (5)+\ln \left (\frac {3}{4\,x^2}\right )-5\right )\,{\ln \relax (x)}^2+3\,\ln \relax (5)+3\,\ln \left (\frac {3}{4\,x^2}\right )-15\right )-\ln \left ({\ln \relax (x)}^2+3\right )\,\left (x\,{\ln \relax (x)}^2+3\,x\right )\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(-(x - log(log(x)^2 + 3)*(log(5) + log(3/(4*x^2)) - 5))/log(log(x)^2 + 3))*(log(log(x)^2 + 3)^2*(3*log
(5) + 3*log(3/(4*x^2)) + log(x)^2*(log(5) + log(3/(4*x^2)) - 5) - 15) - log(log(x)^2 + 3)*(3*x + x*log(x)^2))
- 2*x*log(x) + log(log(x)^2 + 3)*(3*x + x*log(x)^2) + log(log(x)^2 + 3)^2*(2*log(x)^2 + 6))/(log(-(x - log(log
(x)^2 + 3)*(log(5) + log(3/(4*x^2)) - 5))/log(log(x)^2 + 3))^2*(log(log(x)^2 + 3)^2*(3*log(5) + 3*log(3/(4*x^2
)) + log(x)^2*(log(5) + log(3/(4*x^2)) - 5) - 15) - log(log(x)^2 + 3)*(3*x + x*log(x)^2))),x)

[Out]

int((log(-(x - log(log(x)^2 + 3)*(log(5) + log(3/(4*x^2)) - 5))/log(log(x)^2 + 3))*(log(log(x)^2 + 3)^2*(3*log
(5) + 3*log(3/(4*x^2)) + log(x)^2*(log(5) + log(3/(4*x^2)) - 5) - 15) - log(log(x)^2 + 3)*(3*x + x*log(x)^2))
- 2*x*log(x) + log(log(x)^2 + 3)*(3*x + x*log(x)^2) + log(log(x)^2 + 3)^2*(2*log(x)^2 + 6))/(log(-(x - log(log
(x)^2 + 3)*(log(5) + log(3/(4*x^2)) - 5))/log(log(x)^2 + 3))^2*(log(log(x)^2 + 3)^2*(3*log(5) + 3*log(3/(4*x^2
)) + log(x)^2*(log(5) + log(3/(4*x^2)) - 5) - 15) - log(log(x)^2 + 3)*(3*x + x*log(x)^2))), x)

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sympy [A]  time = 152.44, size = 36, normalized size = 1.24 \begin {gather*} \frac {x}{\log {\left (\frac {- x + \left (- 2 \log {\relax (x )} - 5 + \log {\left (\frac {3}{4} \right )} + \log {\relax (5 )}\right ) \log {\left (\log {\relax (x )}^{2} + 3 \right )}}{\log {\left (\log {\relax (x )}^{2} + 3 \right )}} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((ln(3/4/x**2)+ln(5)-5)*ln(x)**2+3*ln(3/4/x**2)+3*ln(5)-15)*ln(ln(x)**2+3)**2+(-x*ln(x)**2-3*x)*ln
(ln(x)**2+3))*ln(((ln(3/4/x**2)+ln(5)-5)*ln(ln(x)**2+3)-x)/ln(ln(x)**2+3))+(2*ln(x)**2+6)*ln(ln(x)**2+3)**2+(x
*ln(x)**2+3*x)*ln(ln(x)**2+3)-2*x*ln(x))/(((ln(3/4/x**2)+ln(5)-5)*ln(x)**2+3*ln(3/4/x**2)+3*ln(5)-15)*ln(ln(x)
**2+3)**2+(-x*ln(x)**2-3*x)*ln(ln(x)**2+3))/ln(((ln(3/4/x**2)+ln(5)-5)*ln(ln(x)**2+3)-x)/ln(ln(x)**2+3))**2,x)

[Out]

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

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