3.86.18 \(\int \frac {4 x \log ^2(5)+(2 x^2+4 x^3+2 x^2 \log ^2(5)) \log (\frac {x^2}{4})+(4 \log ^2(5)+(2 x+4 x^2+2 x \log ^2(5)) \log (\frac {x^2}{4})) \log (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log (\frac {x^2}{4}))}{x \log ^2(5) \log (\frac {x^2}{4})} \, dx\)

Optimal. Leaf size=27 \[ \left (x+\log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )\right )^2 \]

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Rubi [F]  time = 1.26, 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 x \log ^2(5)+\left (2 x^2+4 x^3+2 x^2 \log ^2(5)\right ) \log \left (\frac {x^2}{4}\right )+\left (4 \log ^2(5)+\left (2 x+4 x^2+2 x \log ^2(5)\right ) \log \left (\frac {x^2}{4}\right )\right ) \log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )}{x \log ^2(5) \log \left (\frac {x^2}{4}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4*x*Log[5]^2 + (2*x^2 + 4*x^3 + 2*x^2*Log[5]^2)*Log[x^2/4] + (4*Log[5]^2 + (2*x + 4*x^2 + 2*x*Log[5]^2)*L
og[x^2/4])*Log[E^((-1 + x + x^2)/Log[5]^2)*Log[x^2/4]])/(x*Log[5]^2*Log[x^2/4]),x]

[Out]

(4*x*ExpIntegralEi[Log[x^2/4]/2])/Sqrt[x^2] + x^2*(1 + Log[5]^(-2)) - (4*x*ExpIntegralEi[Log[x^2/4]/2]*(1 + Lo
g[5]^(-2)))/Sqrt[x^2] - (2*x^3)/(3*Log[5]^4) - x^4/Log[5]^4 + (4*x^3)/(3*Log[5]^2) - (x^2*(1 + Log[5]^2))/Log[
5]^4 - (4*x^3*(1 + Log[5]^2))/(3*Log[5]^4) + 2*x*(1 + Log[5]^(-2))*Log[Log[x^2/4]/E^((1 - x - x^2)/Log[5]^2)]
+ (2*x^2*Log[Log[x^2/4]/E^((1 - x - x^2)/Log[5]^2)])/Log[5]^2 - (8*LogIntegral[x^2/4])/Log[5]^2 + 4*Defer[Int]
[Log[E^((-1 + x + x^2)/Log[5]^2)*Log[x^2/4]]/(x*Log[x^2/4]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {4 x \log ^2(5)+\left (2 x^2+4 x^3+2 x^2 \log ^2(5)\right ) \log \left (\frac {x^2}{4}\right )+\left (4 \log ^2(5)+\left (2 x+4 x^2+2 x \log ^2(5)\right ) \log \left (\frac {x^2}{4}\right )\right ) \log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )}{x \log \left (\frac {x^2}{4}\right )} \, dx}{\log ^2(5)}\\ &=\frac {\int \frac {2 \left (2 \log ^2(5)+x \left (1+2 x+\log ^2(5)\right ) \log \left (\frac {x^2}{4}\right )\right ) \left (x+\log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )\right )}{x \log \left (\frac {x^2}{4}\right )} \, dx}{\log ^2(5)}\\ &=\frac {2 \int \frac {\left (2 \log ^2(5)+x \left (1+2 x+\log ^2(5)\right ) \log \left (\frac {x^2}{4}\right )\right ) \left (x+\log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )\right )}{x \log \left (\frac {x^2}{4}\right )} \, dx}{\log ^2(5)}\\ &=\frac {2 \int \left (\frac {2 \log ^2(5)+2 x^2 \log \left (\frac {x^2}{4}\right )+x \left (1+\log ^2(5)\right ) \log \left (\frac {x^2}{4}\right )}{\log \left (\frac {x^2}{4}\right )}+\frac {\left (2 \log ^2(5)+2 x^2 \log \left (\frac {x^2}{4}\right )+x \left (1+\log ^2(5)\right ) \log \left (\frac {x^2}{4}\right )\right ) \log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )}{x \log \left (\frac {x^2}{4}\right )}\right ) \, dx}{\log ^2(5)}\\ &=\frac {2 \int \frac {2 \log ^2(5)+2 x^2 \log \left (\frac {x^2}{4}\right )+x \left (1+\log ^2(5)\right ) \log \left (\frac {x^2}{4}\right )}{\log \left (\frac {x^2}{4}\right )} \, dx}{\log ^2(5)}+\frac {2 \int \frac {\left (2 \log ^2(5)+2 x^2 \log \left (\frac {x^2}{4}\right )+x \left (1+\log ^2(5)\right ) \log \left (\frac {x^2}{4}\right )\right ) \log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )}{x \log \left (\frac {x^2}{4}\right )} \, dx}{\log ^2(5)}\\ &=\frac {2 \int \left (x \left (1+2 x+\log ^2(5)\right )+\frac {2 \log ^2(5)}{\log \left (\frac {x^2}{4}\right )}\right ) \, dx}{\log ^2(5)}+\frac {2 \int \frac {\left (2 \log ^2(5)+x \left (1+2 x+\log ^2(5)\right ) \log \left (\frac {x^2}{4}\right )\right ) \log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )}{x \log \left (\frac {x^2}{4}\right )} \, dx}{\log ^2(5)}\\ &=4 \int \frac {1}{\log \left (\frac {x^2}{4}\right )} \, dx+\frac {2 \int x \left (1+2 x+\log ^2(5)\right ) \, dx}{\log ^2(5)}+\frac {2 \int \left (2 x \log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )+\left (1+\log ^2(5)\right ) \log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )+\frac {2 \log ^2(5) \log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )}{x \log \left (\frac {x^2}{4}\right )}\right ) \, dx}{\log ^2(5)}\\ &=4 \int \frac {\log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )}{x \log \left (\frac {x^2}{4}\right )} \, dx+\frac {(4 x) \operatorname {Subst}\left (\int \frac {e^{x/2}}{x} \, dx,x,\log \left (\frac {x^2}{4}\right )\right )}{\sqrt {x^2}}+\left (2 \left (1+\frac {1}{\log ^2(5)}\right )\right ) \int \log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right ) \, dx+\frac {2 \int \left (2 x^2+x \left (1+\log ^2(5)\right )\right ) \, dx}{\log ^2(5)}+\frac {4 \int x \log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right ) \, dx}{\log ^2(5)}\\ &=\frac {4 x \text {Ei}\left (\frac {1}{2} \log \left (\frac {x^2}{4}\right )\right )}{\sqrt {x^2}}+x^2 \left (1+\frac {1}{\log ^2(5)}\right )+\frac {4 x^3}{3 \log ^2(5)}+2 x \left (1+\frac {1}{\log ^2(5)}\right ) \log \left (e^{-\frac {1-x-x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )+\frac {2 x^2 \log \left (e^{-\frac {1-x-x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )}{\log ^2(5)}+4 \int \frac {\log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )}{x \log \left (\frac {x^2}{4}\right )} \, dx-\left (2 \left (1+\frac {1}{\log ^2(5)}\right )\right ) \int \left (\frac {x (1+2 x)}{\log ^2(5)}+\frac {2}{\log \left (\frac {x^2}{4}\right )}\right ) \, dx-\frac {4 \int \left (\frac {x^2 (1+2 x)}{2 \log ^2(5)}+\frac {x}{\log \left (\frac {x^2}{4}\right )}\right ) \, dx}{\log ^2(5)}\\ &=\frac {4 x \text {Ei}\left (\frac {1}{2} \log \left (\frac {x^2}{4}\right )\right )}{\sqrt {x^2}}+x^2 \left (1+\frac {1}{\log ^2(5)}\right )+\frac {4 x^3}{3 \log ^2(5)}+2 x \left (1+\frac {1}{\log ^2(5)}\right ) \log \left (e^{-\frac {1-x-x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )+\frac {2 x^2 \log \left (e^{-\frac {1-x-x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )}{\log ^2(5)}+4 \int \frac {\log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )}{x \log \left (\frac {x^2}{4}\right )} \, dx-\left (4 \left (1+\frac {1}{\log ^2(5)}\right )\right ) \int \frac {1}{\log \left (\frac {x^2}{4}\right )} \, dx-\frac {2 \int x^2 (1+2 x) \, dx}{\log ^4(5)}-\frac {4 \int \frac {x}{\log \left (\frac {x^2}{4}\right )} \, dx}{\log ^2(5)}-\frac {\left (2 \left (1+\log ^2(5)\right )\right ) \int x (1+2 x) \, dx}{\log ^4(5)}\\ &=\frac {4 x \text {Ei}\left (\frac {1}{2} \log \left (\frac {x^2}{4}\right )\right )}{\sqrt {x^2}}+x^2 \left (1+\frac {1}{\log ^2(5)}\right )+\frac {4 x^3}{3 \log ^2(5)}+2 x \left (1+\frac {1}{\log ^2(5)}\right ) \log \left (e^{-\frac {1-x-x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )+\frac {2 x^2 \log \left (e^{-\frac {1-x-x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )}{\log ^2(5)}+4 \int \frac {\log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )}{x \log \left (\frac {x^2}{4}\right )} \, dx-\frac {\left (4 x \left (1+\frac {1}{\log ^2(5)}\right )\right ) \operatorname {Subst}\left (\int \frac {e^{x/2}}{x} \, dx,x,\log \left (\frac {x^2}{4}\right )\right )}{\sqrt {x^2}}-\frac {2 \int \left (x^2+2 x^3\right ) \, dx}{\log ^4(5)}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{\log \left (\frac {x}{4}\right )} \, dx,x,x^2\right )}{\log ^2(5)}-\frac {\left (2 \left (1+\log ^2(5)\right )\right ) \int \left (x+2 x^2\right ) \, dx}{\log ^4(5)}\\ &=\frac {4 x \text {Ei}\left (\frac {1}{2} \log \left (\frac {x^2}{4}\right )\right )}{\sqrt {x^2}}+x^2 \left (1+\frac {1}{\log ^2(5)}\right )-\frac {4 x \text {Ei}\left (\frac {1}{2} \log \left (\frac {x^2}{4}\right )\right ) \left (1+\frac {1}{\log ^2(5)}\right )}{\sqrt {x^2}}-\frac {2 x^3}{3 \log ^4(5)}-\frac {x^4}{\log ^4(5)}+\frac {4 x^3}{3 \log ^2(5)}-\frac {x^2 \left (1+\log ^2(5)\right )}{\log ^4(5)}-\frac {4 x^3 \left (1+\log ^2(5)\right )}{3 \log ^4(5)}+2 x \left (1+\frac {1}{\log ^2(5)}\right ) \log \left (e^{-\frac {1-x-x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )+\frac {2 x^2 \log \left (e^{-\frac {1-x-x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )}{\log ^2(5)}-\frac {8 \text {li}\left (\frac {x^2}{4}\right )}{\log ^2(5)}+4 \int \frac {\log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )}{x \log \left (\frac {x^2}{4}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.23, size = 85, normalized size = 3.15 \begin {gather*} -\frac {\left (x \left (1+x+\log ^2(5)\right )+\log ^2(5) \log \left (\log \left (\frac {x^2}{4}\right )\right )\right ) \left (x \left (1+x-\log ^2(5)\right )+\log ^2(5) \log \left (\log \left (\frac {x^2}{4}\right )\right )-2 \log ^2(5) \log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )\right )}{\log ^4(5)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.68, size = 49, normalized size = 1.81 \begin {gather*} x^{2} + 2 \, x \log \left (e^{\left (\frac {x^{2} + x - 1}{\log \relax (5)^{2}}\right )} \log \left (\frac {1}{4} \, x^{2}\right )\right ) + \log \left (e^{\left (\frac {x^{2} + x - 1}{\log \relax (5)^{2}}\right )} \log \left (\frac {1}{4} \, x^{2}\right )\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.72, size = 132, normalized size = 4.89 \begin {gather*} -\frac {\log \relax (5)^{2} \log \left (-2 \, \log \relax (2) + \log \left (x^{2}\right )\right )^{2} - \frac {2 \, {\left (\log \relax (5)^{2} + 1\right )} x^{3}}{\log \relax (5)^{2}} - \frac {x^{4}}{\log \relax (5)^{2}} - 2 \, {\left (\log \relax (5)^{2} \log \left (-2 \, \log \relax (2) + \log \left (x^{2}\right )\right ) + {\left (\log \relax (5)^{2} + 1\right )} x + x^{2}\right )} \log \left (\log \left (\frac {1}{4} \, x^{2}\right )\right ) - \frac {{\left (\log \relax (5)^{4} + 2 \, \log \relax (5)^{2} - 1\right )} x^{2}}{\log \relax (5)^{2}} + \frac {2 \, {\left (\log \relax (5)^{2} + 1\right )} x}{\log \relax (5)^{2}} + 2 \, \log \left (2 \, \log \relax (2) - \log \left (x^{2}\right )\right )}{\log \relax (5)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 1.45, size = 4657, normalized size = 172.48

method	result	size
risch	\(\text {Expression too large to display}\)	\(4657\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x*ln(5)^2+4*x^2+2*x)*ln(1/4*x^2)+4*ln(5)^2)*ln(exp((x^2+x-1)/ln(5)^2)*ln(1/4*x^2))+(2*x^2*ln(5)^2+4*x
^3+2*x^2)*ln(1/4*x^2)+4*x*ln(5)^2)/x/ln(5)^2/ln(1/4*x^2),x,method=_RETURNVERBOSE)

[Out]

x^2-2*x*ln(2)-I*Pi*x+2*x*ln(4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^3
-4*I*ln(2))+I*Pi*ln(ln(x)-1/4*I*(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^3-4*I*
ln(2)))*csgn(I*exp((x^2+x-1)/ln(5)^2)*(-4*I*ln(x)-Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*c
sgn(I*x^2)^3+4*I*ln(2)))^3+I*Pi*x*csgn(I*exp((x^2+x-1)/ln(5)^2)*(-4*I*ln(x)-Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*cs
gn(I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2)^3+4*I*ln(2)))^3+I*Pi*ln(ln(x)-1/4*I*(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn
(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^3-4*I*ln(2)))*csgn(exp((x^2+x-1)/ln(5)^2)*(-4*I*ln(x)-Pi*csgn(I*x)^2*csgn(I
*x^2)+2*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2)^3+4*I*ln(2)))^3+I*Pi*x*csgn(exp((x^2+x-1)/ln(5)^2)*(-4*I*ln(
x)-Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2)^3+4*I*ln(2)))^3+I*Pi*ln(ln(x)-1/4*I*
(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^3-4*I*ln(2)))*csgn(exp((x^2+x-1)/ln(5)
^2)*(-4*I*ln(x)-Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2)^3+4*I*ln(2)))^2+I*Pi*x*
csgn(exp((x^2+x-1)/ln(5)^2)*(-4*I*ln(x)-Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2)
^3+4*I*ln(2)))^2-I/ln(5)^2*Pi*x^2-I/ln(5)^2*Pi*x+ln(4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I
*x^2)^2+Pi*csgn(I*x^2)^3-4*I*ln(2))^2+I/ln(5)^2*Pi*x^2*csgn(I*exp((x^2+x-1)/ln(5)^2))*csgn(I*exp((x^2+x-1)/ln(
5)^2)*(-4*I*ln(x)-Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2)^3+4*I*ln(2)))^2+I/ln(
5)^2*Pi*x*csgn(I*exp((x^2+x-1)/ln(5)^2))*csgn(I*exp((x^2+x-1)/ln(5)^2)*(-4*I*ln(x)-Pi*csgn(I*x)^2*csgn(I*x^2)+
2*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2)^3+4*I*ln(2)))^2-I/ln(5)^2*Pi*x^2*csgn(I*(-4*I*ln(x)-Pi*csgn(I*x)^2
*csgn(I*x^2)+2*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2)^3+4*I*ln(2)))*csgn(I*exp((x^2+x-1)/ln(5)^2)*(-4*I*ln(
x)-Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2)^3+4*I*ln(2)))^2-I/ln(5)^2*Pi*x^2*csg
n(I*exp((x^2+x-1)/ln(5)^2)*(-4*I*ln(x)-Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2)^
3+4*I*ln(2)))*csgn(exp((x^2+x-1)/ln(5)^2)*(-4*I*ln(x)-Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*csgn(I*x)*csgn(I*x^2)^2-
Pi*csgn(I*x^2)^3+4*I*ln(2)))^2-I/ln(5)^2*Pi*x*csgn(I*(-4*I*ln(x)-Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*csgn(I*x)*csg
n(I*x^2)^2-Pi*csgn(I*x^2)^3+4*I*ln(2)))*csgn(I*exp((x^2+x-1)/ln(5)^2)*(-4*I*ln(x)-Pi*csgn(I*x)^2*csgn(I*x^2)+2
*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2)^3+4*I*ln(2)))^2-2/ln(5)^2*ln(2)*x^2-2/ln(5)^2*ln(2)*x-I*Pi*ln(ln(x)
-1/4*I*(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^3-4*I*ln(2)))-2*ln(2)*ln(ln(x)-
1/4*I*(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^3-4*I*ln(2)))-1/ln(5)^4*x^4-2/ln
(5)^4*x^3-1/ln(5)^4*x^2+I/ln(5)^2*Pi*x^2*csgn(I*exp((x^2+x-1)/ln(5)^2)*(-4*I*ln(x)-Pi*csgn(I*x)^2*csgn(I*x^2)+
2*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2)^3+4*I*ln(2)))^3+I/ln(5)^2*Pi*x^2*csgn(exp((x^2+x-1)/ln(5)^2)*(-4*I
*ln(x)-Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2)^3+4*I*ln(2)))^3+I/ln(5)^2*Pi*x*c
sgn(I*exp((x^2+x-1)/ln(5)^2)*(-4*I*ln(x)-Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2
)^3+4*I*ln(2)))^3+I/ln(5)^2*Pi*x*csgn(exp((x^2+x-1)/ln(5)^2)*(-4*I*ln(x)-Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*csgn(
I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2)^3+4*I*ln(2)))^3+I/ln(5)^2*Pi*x^2*csgn(exp((x^2+x-1)/ln(5)^2)*(-4*I*ln(x)-Pi*
csgn(I*x)^2*csgn(I*x^2)+2*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2)^3+4*I*ln(2)))^2+I/ln(5)^2*Pi*x*csgn(exp((x
^2+x-1)/ln(5)^2)*(-4*I*ln(x)-Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2)^3+4*I*ln(2
)))^2-I*Pi*ln(ln(x)-1/4*I*(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^3-4*I*ln(2))
)*csgn(I*(-4*I*ln(x)-Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2)^3+4*I*ln(2)))*csgn
(I*exp((x^2+x-1)/ln(5)^2)*(-4*I*ln(x)-Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2)^3
+4*I*ln(2)))^2-I*Pi*x*csgn(I*(-4*I*ln(x)-Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2
)^3+4*I*ln(2)))*csgn(I*exp((x^2+x-1)/ln(5)^2)*(-4*I*ln(x)-Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*csgn(I*x)*csgn(I*x^2
)^2-Pi*csgn(I*x^2)^3+4*I*ln(2)))^2-I*Pi*ln(ln(x)-1/4*I*(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^
2+Pi*csgn(I*x^2)^3-4*I*ln(2)))*csgn(I*exp((x^2+x-1)/ln(5)^2)*(-4*I*ln(x)-Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*csgn(
I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2)^3+4*I*ln(2)))*csgn(exp((x^2+x-1)/ln(5)^2)*(-4*I*ln(x)-Pi*csgn(I*x)^2*csgn(I*
x^2)+2*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2)^3+4*I*ln(2)))^2-I*Pi*x*csgn(I*exp((x^2+x-1)/ln(5)^2)*(-4*I*ln
(x)-Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2)^3+4*I*ln(2)))*csgn(exp((x^2+x-1)/ln
(5)^2)*(-4*I*ln(x)-Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2)^3+4*I*ln(2)))^2-I*Pi
*ln(ln(x)-1/4*I*(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^3-4*I*ln(2)))*csgn(I*e
xp((x^2+x-1)/ln(5)^2)*(-4*I*ln(x)-Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2)^3+4*I
*ln(2)))*csgn(exp((x^2+x-1)/ln(5)^2)*(-4*I*ln(x)-Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*cs
gn(I*x^2)^3+4*I*ln(2)))-I*Pi*x*csgn(I*exp((x^2+x-1)/ln(5)^2)*(-4*I*ln(x)-Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*csgn(
I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2)^3+4*I*ln(2)))*csgn(exp((x^2+x-1)/ln(5)^2)*(-4*I*ln(x)-Pi*csgn(I*x)^2*csgn(I*
x^2)+2*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2)^3+4*I*ln(2)))-I/ln(5)^2*Pi*x^2*csgn(I*(-4*I*ln(x)-Pi*csgn(I*x
)^2*csgn(I*x^2)+2*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2)^3+4*I*ln(2)))*csgn(I*exp((x^2+x-1)/ln(5)^2))*csgn(
I*exp((x^2+x-1)/ln(5)^2)*(-4*I*ln(x)-Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2)^3+
4*I*ln(2)))-I/ln(5)^2*Pi*x*csgn(I*(-4*I*ln(x)-Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*csgn(
I*x^2)^3+4*I*ln(2)))*csgn(I*exp((x^2+x-1)/ln(5)^2))*csgn(I*exp((x^2+x-1)/ln(5)^2)*(-4*I*ln(x)-Pi*csgn(I*x)^2*c
sgn(I*x^2)+2*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2)^3+4*I*ln(2)))+1/ln(5)^2*(2*x*ln(5)^2+2*ln(5)^2*ln(4*I*l
n(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^3-4*I*ln(2))+2*x^2+2*x)*ln(exp((x^
2+x-1)/ln(5)^2))-I/ln(5)^2*Pi*x*csgn(I*exp((x^2+x-1)/ln(5)^2)*(-4*I*ln(x)-Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*csgn
(I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2)^3+4*I*ln(2)))*csgn(exp((x^2+x-1)/ln(5)^2)*(-4*I*ln(x)-Pi*csgn(I*x)^2*csgn(I
*x^2)+2*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2)^3+4*I*ln(2)))^2-I/ln(5)^2*Pi*x^2*csgn(I*exp((x^2+x-1)/ln(5)^
2)*(-4*I*ln(x)-Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2)^3+4*I*ln(2)))*csgn(exp((
x^2+x-1)/ln(5)^2)*(-4*I*ln(x)-Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2)^3+4*I*ln(
2)))-I/ln(5)^2*Pi*x*csgn(I*exp((x^2+x-1)/ln(5)^2)*(-4*I*ln(x)-Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*csgn(I*x)*csgn(I
*x^2)^2-Pi*csgn(I*x^2)^3+4*I*ln(2)))*csgn(exp((x^2+x-1)/ln(5)^2)*(-4*I*ln(x)-Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*c
sgn(I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2)^3+4*I*ln(2)))-I*Pi*ln(ln(x)-1/4*I*(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(
I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^3-4*I*ln(2)))*csgn(I*(-4*I*ln(x)-Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*csgn(I*x)*c
sgn(I*x^2)^2-Pi*csgn(I*x^2)^3+4*I*ln(2)))*csgn(I*exp((x^2+x-1)/ln(5)^2))*csgn(I*exp((x^2+x-1)/ln(5)^2)*(-4*I*l
n(x)-Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2)^3+4*I*ln(2)))-I*Pi*x*csgn(I*(-4*I*
ln(x)-Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2)^3+4*I*ln(2)))*csgn(I*exp((x^2+x-1
)/ln(5)^2))*csgn(I*exp((x^2+x-1)/ln(5)^2)*(-4*I*ln(x)-Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*csgn(I*x)*csgn(I*x^2)^2-
Pi*csgn(I*x^2)^3+4*I*ln(2)))+I*Pi*ln(ln(x)-1/4*I*(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*c
sgn(I*x^2)^3-4*I*ln(2)))*csgn(I*exp((x^2+x-1)/ln(5)^2))*csgn(I*exp((x^2+x-1)/ln(5)^2)*(-4*I*ln(x)-Pi*csgn(I*x)
^2*csgn(I*x^2)+2*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2)^3+4*I*ln(2)))^2+I*Pi*x*csgn(I*exp((x^2+x-1)/ln(5)^2
))*csgn(I*exp((x^2+x-1)/ln(5)^2)*(-4*I*ln(x)-Pi*csgn(I*x)^2*csgn(I*x^2)+2*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*csgn(I
*x^2)^3+4*I*ln(2)))^2

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maxima [B]  time = 0.49, size = 148, normalized size = 5.48 \begin {gather*} \frac {3 \, x^{2} \log \relax (5)^{2} + 4 \, x^{3} + 3 \, x^{2} + \frac {3 \, \log \relax (5)^{4} \log \left (-\log \relax (2) + \log \relax (x)\right )^{2} + 2 \, {\left (\log \relax (5)^{2} + 3\right )} x^{3} + 3 \, x^{4} + 3 \, {\left (2 \, \log \relax (5)^{2} \log \relax (2) + \log \relax (5)^{2} - 1\right )} x^{2} - 6 \, {\left (\log \relax (5)^{2} - {\left (\log \relax (5)^{4} + \log \relax (5)^{2}\right )} \log \relax (2) + 1\right )} x + 6 \, {\left (\log \relax (5)^{4} \log \relax (2) + x^{2} \log \relax (5)^{2} + {\left (\log \relax (5)^{4} + \log \relax (5)^{2}\right )} x - \log \relax (5)^{2}\right )} \log \left (-\log \relax (2) + \log \relax (x)\right )}{\log \relax (5)^{2}}}{3 \, \log \relax (5)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 5.83, size = 35, normalized size = 1.30 \begin {gather*} {\left (x+\ln \left ({\mathrm {e}}^{\frac {x}{{\ln \relax (5)}^2}}\,{\mathrm {e}}^{\frac {x^2}{{\ln \relax (5)}^2}}\,{\mathrm {e}}^{-\frac {1}{{\ln \relax (5)}^2}}\,\ln \left (\frac {x^2}{4}\right )\right )\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x*log(5)^2 + log(exp((x + x^2 - 1)/log(5)^2)*log(x^2/4))*(log(x^2/4)*(2*x + 2*x*log(5)^2 + 4*x^2) + 4*l
og(5)^2) + log(x^2/4)*(2*x^2*log(5)^2 + 2*x^2 + 4*x^3))/(x*log(5)^2*log(x^2/4)),x)

[Out]

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

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sympy [B]  time = 0.67, size = 51, normalized size = 1.89 \begin {gather*} x^{2} + 2 x \log {\left (e^{\frac {x^{2} + x - 1}{\log {\relax (5 )}^{2}}} \log {\left (\frac {x^{2}}{4} \right )} \right )} + \log {\left (e^{\frac {x^{2} + x - 1}{\log {\relax (5 )}^{2}}} \log {\left (\frac {x^{2}}{4} \right )} \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x*ln(5)**2+4*x**2+2*x)*ln(1/4*x**2)+4*ln(5)**2)*ln(exp((x**2+x-1)/ln(5)**2)*ln(1/4*x**2))+(2*x*
*2*ln(5)**2+4*x**3+2*x**2)*ln(1/4*x**2)+4*x*ln(5)**2)/x/ln(5)**2/ln(1/4*x**2),x)

[Out]

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

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