3.53.85 \(\int \frac {(20 x^2-4 x^4) \log (5-x^2)+(-10 x^3+2 x^5) \log (5-x^2) \log (\log (5-x^2))+\sqrt {2-x \log (\log (5-x^2))} (2 x^3+(-20+4 x^2) \log (5-x^2)+(5 x-x^3) \log (5-x^2) \log (\log (5-x^2)))}{(20 x^2-4 x^4) \log (5-x^2)+(-10 x^3+2 x^5) \log (5-x^2) \log (\log (5-x^2))} \, dx\) [5285]

Optimal. Leaf size=25 \[ -3+x+\frac {\sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}}{x} \]

[Out]

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

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Rubi [F]
time = 7.83, 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 {\left (20 x^2-4 x^4\right ) \log \left (5-x^2\right )+\left (-10 x^3+2 x^5\right ) \log \left (5-x^2\right ) \log \left (\log \left (5-x^2\right )\right )+\sqrt {2-x \log \left (\log \left (5-x^2\right )\right )} \left (2 x^3+\left (-20+4 x^2\right ) \log \left (5-x^2\right )+\left (5 x-x^3\right ) \log \left (5-x^2\right ) \log \left (\log \left (5-x^2\right )\right )\right )}{\left (20 x^2-4 x^4\right ) \log \left (5-x^2\right )+\left (-10 x^3+2 x^5\right ) \log \left (5-x^2\right ) \log \left (\log \left (5-x^2\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((20*x^2 - 4*x^4)*Log[5 - x^2] + (-10*x^3 + 2*x^5)*Log[5 - x^2]*Log[Log[5 - x^2]] + Sqrt[2 - x*Log[Log[5 -
x^2]]]*(2*x^3 + (-20 + 4*x^2)*Log[5 - x^2] + (5*x - x^3)*Log[5 - x^2]*Log[Log[5 - x^2]]))/((20*x^2 - 4*x^4)*L
og[5 - x^2] + (-10*x^3 + 2*x^5)*Log[5 - x^2]*Log[Log[5 - x^2]]),x]

[Out]

x - Defer[Int][Log[Log[5 - x^2]]^2/Sqrt[2 - x*Log[Log[5 - x^2]]], x]/4 - Defer[Int][Sqrt[2 - x*Log[Log[5 - x^2
]]]/x^2, x] - Defer[Int][(Log[Log[5 - x^2]]*Sqrt[2 - x*Log[Log[5 - x^2]]])/x, x]/4 + 2*Defer[Int][Log[Log[5 -
x^2]]/(Log[5 - x^2]*Sqrt[2 - x*Log[Log[5 - x^2]]]*(-4 + 5*Log[Log[5 - x^2]]^2)), x] - Defer[Int][Sqrt[2 - x*Lo
g[Log[5 - x^2]]]/((Sqrt[5] - x)*Log[5 - x^2]*(-4 + 5*Log[Log[5 - x^2]]^2)), x] + Defer[Int][Sqrt[2 - x*Log[Log
[5 - x^2]]]/((Sqrt[5] + x)*Log[5 - x^2]*(-4 + 5*Log[Log[5 - x^2]]^2)), x] - (Sqrt[5]*Defer[Int][(Log[Log[5 - x
^2]]*Sqrt[2 - x*Log[Log[5 - x^2]]])/((Sqrt[5] - x)*Log[5 - x^2]*(-4 + 5*Log[Log[5 - x^2]]^2)), x])/2 - (Sqrt[5
]*Defer[Int][(Log[Log[5 - x^2]]*Sqrt[2 - x*Log[Log[5 - x^2]]])/((Sqrt[5] + x)*Log[5 - x^2]*(-4 + 5*Log[Log[5 -
x^2]]^2)), x])/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 x^2-4 \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}-\frac {2 x^3 \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}}{\left (-5+x^2\right ) \log \left (5-x^2\right )}-\log \left (\log \left (5-x^2\right )\right ) \left (2 x^3-x \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}\right )}{2 x^2 \left (2-x \log \left (\log \left (5-x^2\right )\right )\right )} \, dx\\ &=\frac {1}{2} \int \frac {4 x^2-4 \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}-\frac {2 x^3 \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}}{\left (-5+x^2\right ) \log \left (5-x^2\right )}-\log \left (\log \left (5-x^2\right )\right ) \left (2 x^3-x \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}\right )}{x^2 \left (2-x \log \left (\log \left (5-x^2\right )\right )\right )} \, dx\\ &=\frac {1}{2} \int \left (2-\frac {2 \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}}{x^2}-\frac {\log \left (\log \left (5-x^2\right )\right ) \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}}{2 x}+\frac {4 \log \left (\log \left (5-x^2\right )\right )}{\log \left (5-x^2\right ) \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )} \left (-4+5 \log ^2\left (\log \left (5-x^2\right )\right )\right )}+\frac {2 \log ^2\left (\log \left (5-x^2\right )\right )}{\sqrt {2-x \log \left (\log \left (5-x^2\right )\right )} \left (-4+5 \log ^2\left (\log \left (5-x^2\right )\right )\right )}-\frac {5 \log ^4\left (\log \left (5-x^2\right )\right )}{2 \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )} \left (-4+5 \log ^2\left (\log \left (5-x^2\right )\right )\right )}+\frac {4 x \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}}{\left (-5+x^2\right ) \log \left (5-x^2\right ) \left (-4+5 \log ^2\left (\log \left (5-x^2\right )\right )\right )}+\frac {10 \log \left (\log \left (5-x^2\right )\right ) \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}}{\left (-5+x^2\right ) \log \left (5-x^2\right ) \left (-4+5 \log ^2\left (\log \left (5-x^2\right )\right )\right )}\right ) \, dx\\ &=x-\frac {1}{4} \int \frac {\log \left (\log \left (5-x^2\right )\right ) \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}}{x} \, dx-\frac {5}{4} \int \frac {\log ^4\left (\log \left (5-x^2\right )\right )}{\sqrt {2-x \log \left (\log \left (5-x^2\right )\right )} \left (-4+5 \log ^2\left (\log \left (5-x^2\right )\right )\right )} \, dx+2 \int \frac {\log \left (\log \left (5-x^2\right )\right )}{\log \left (5-x^2\right ) \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )} \left (-4+5 \log ^2\left (\log \left (5-x^2\right )\right )\right )} \, dx+2 \int \frac {x \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}}{\left (-5+x^2\right ) \log \left (5-x^2\right ) \left (-4+5 \log ^2\left (\log \left (5-x^2\right )\right )\right )} \, dx+5 \int \frac {\log \left (\log \left (5-x^2\right )\right ) \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}}{\left (-5+x^2\right ) \log \left (5-x^2\right ) \left (-4+5 \log ^2\left (\log \left (5-x^2\right )\right )\right )} \, dx-\int \frac {\sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}}{x^2} \, dx+\int \frac {\log ^2\left (\log \left (5-x^2\right )\right )}{\sqrt {2-x \log \left (\log \left (5-x^2\right )\right )} \left (-4+5 \log ^2\left (\log \left (5-x^2\right )\right )\right )} \, dx\\ &=x-\frac {1}{4} \int \frac {\log \left (\log \left (5-x^2\right )\right ) \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}}{x} \, dx-\frac {5}{4} \int \left (\frac {4}{25 \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}}+\frac {\log ^2\left (\log \left (5-x^2\right )\right )}{5 \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}}+\frac {16}{25 \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )} \left (-4+5 \log ^2\left (\log \left (5-x^2\right )\right )\right )}\right ) \, dx+2 \int \frac {\log \left (\log \left (5-x^2\right )\right )}{\log \left (5-x^2\right ) \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )} \left (-4+5 \log ^2\left (\log \left (5-x^2\right )\right )\right )} \, dx+2 \int \left (-\frac {\sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}}{2 \left (\sqrt {5}-x\right ) \log \left (5-x^2\right ) \left (-4+5 \log ^2\left (\log \left (5-x^2\right )\right )\right )}+\frac {\sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}}{2 \left (\sqrt {5}+x\right ) \log \left (5-x^2\right ) \left (-4+5 \log ^2\left (\log \left (5-x^2\right )\right )\right )}\right ) \, dx+5 \int \left (-\frac {\log \left (\log \left (5-x^2\right )\right ) \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}}{2 \sqrt {5} \left (\sqrt {5}-x\right ) \log \left (5-x^2\right ) \left (-4+5 \log ^2\left (\log \left (5-x^2\right )\right )\right )}-\frac {\log \left (\log \left (5-x^2\right )\right ) \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}}{2 \sqrt {5} \left (\sqrt {5}+x\right ) \log \left (5-x^2\right ) \left (-4+5 \log ^2\left (\log \left (5-x^2\right )\right )\right )}\right ) \, dx-\int \frac {\sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}}{x^2} \, dx+\int \left (\frac {1}{5 \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}}+\frac {4}{5 \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )} \left (-4+5 \log ^2\left (\log \left (5-x^2\right )\right )\right )}\right ) \, dx\\ &=x-\frac {1}{4} \int \frac {\log ^2\left (\log \left (5-x^2\right )\right )}{\sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}} \, dx-\frac {1}{4} \int \frac {\log \left (\log \left (5-x^2\right )\right ) \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}}{x} \, dx+2 \int \frac {\log \left (\log \left (5-x^2\right )\right )}{\log \left (5-x^2\right ) \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )} \left (-4+5 \log ^2\left (\log \left (5-x^2\right )\right )\right )} \, dx-\frac {1}{2} \sqrt {5} \int \frac {\log \left (\log \left (5-x^2\right )\right ) \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}}{\left (\sqrt {5}-x\right ) \log \left (5-x^2\right ) \left (-4+5 \log ^2\left (\log \left (5-x^2\right )\right )\right )} \, dx-\frac {1}{2} \sqrt {5} \int \frac {\log \left (\log \left (5-x^2\right )\right ) \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}}{\left (\sqrt {5}+x\right ) \log \left (5-x^2\right ) \left (-4+5 \log ^2\left (\log \left (5-x^2\right )\right )\right )} \, dx-\int \frac {\sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}}{x^2} \, dx-\int \frac {\sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}}{\left (\sqrt {5}-x\right ) \log \left (5-x^2\right ) \left (-4+5 \log ^2\left (\log \left (5-x^2\right )\right )\right )} \, dx+\int \frac {\sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}}{\left (\sqrt {5}+x\right ) \log \left (5-x^2\right ) \left (-4+5 \log ^2\left (\log \left (5-x^2\right )\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.13, size = 31, normalized size = 1.24 \begin {gather*} \frac {1}{2} \left (2 x+\frac {2 \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((20*x^2 - 4*x^4)*Log[5 - x^2] + (-10*x^3 + 2*x^5)*Log[5 - x^2]*Log[Log[5 - x^2]] + Sqrt[2 - x*Log[L
og[5 - x^2]]]*(2*x^3 + (-20 + 4*x^2)*Log[5 - x^2] + (5*x - x^3)*Log[5 - x^2]*Log[Log[5 - x^2]]))/((20*x^2 - 4*
x^4)*Log[5 - x^2] + (-10*x^3 + 2*x^5)*Log[5 - x^2]*Log[Log[5 - x^2]]),x]

[Out]

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

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (-x^{3}+5 x \right ) \ln \left (-x^{2}+5\right ) \ln \left (\ln \left (-x^{2}+5\right )\right )+\left (4 x^{2}-20\right ) \ln \left (-x^{2}+5\right )+2 x^{3}\right ) \sqrt {-x \ln \left (\ln \left (-x^{2}+5\right )\right )+2}+\left (2 x^{5}-10 x^{3}\right ) \ln \left (-x^{2}+5\right ) \ln \left (\ln \left (-x^{2}+5\right )\right )+\left (-4 x^{4}+20 x^{2}\right ) \ln \left (-x^{2}+5\right )}{\left (2 x^{5}-10 x^{3}\right ) \ln \left (-x^{2}+5\right ) \ln \left (\ln \left (-x^{2}+5\right )\right )+\left (-4 x^{4}+20 x^{2}\right ) \ln \left (-x^{2}+5\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-x^3+5*x)*ln(-x^2+5)*ln(ln(-x^2+5))+(4*x^2-20)*ln(-x^2+5)+2*x^3)*(-x*ln(ln(-x^2+5))+2)^(1/2)+(2*x^5-10*
x^3)*ln(-x^2+5)*ln(ln(-x^2+5))+(-4*x^4+20*x^2)*ln(-x^2+5))/((2*x^5-10*x^3)*ln(-x^2+5)*ln(ln(-x^2+5))+(-4*x^4+2
0*x^2)*ln(-x^2+5)),x)

[Out]

int((((-x^3+5*x)*ln(-x^2+5)*ln(ln(-x^2+5))+(4*x^2-20)*ln(-x^2+5)+2*x^3)*(-x*ln(ln(-x^2+5))+2)^(1/2)+(2*x^5-10*
x^3)*ln(-x^2+5)*ln(ln(-x^2+5))+(-4*x^4+20*x^2)*ln(-x^2+5))/((2*x^5-10*x^3)*ln(-x^2+5)*ln(ln(-x^2+5))+(-4*x^4+2
0*x^2)*ln(-x^2+5)),x)

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Maxima [A]
time = 0.35, size = 24, normalized size = 0.96 \begin {gather*} \frac {x^{2} + \sqrt {-x \log \left (\log \left (-x^{2} + 5\right )\right ) + 2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^3+5*x)*log(-x^2+5)*log(log(-x^2+5))+(4*x^2-20)*log(-x^2+5)+2*x^3)*(-x*log(log(-x^2+5))+2)^(1/2
)+(2*x^5-10*x^3)*log(-x^2+5)*log(log(-x^2+5))+(-4*x^4+20*x^2)*log(-x^2+5))/((2*x^5-10*x^3)*log(-x^2+5)*log(log
(-x^2+5))+(-4*x^4+20*x^2)*log(-x^2+5)),x, algorithm="maxima")

[Out]

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

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^3+5*x)*log(-x^2+5)*log(log(-x^2+5))+(4*x^2-20)*log(-x^2+5)+2*x^3)*(-x*log(log(-x^2+5))+2)^(1/2
)+(2*x^5-10*x^3)*log(-x^2+5)*log(log(-x^2+5))+(-4*x^4+20*x^2)*log(-x^2+5))/((2*x^5-10*x^3)*log(-x^2+5)*log(log
(-x^2+5))+(-4*x^4+20*x^2)*log(-x^2+5)),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x**3+5*x)*ln(-x**2+5)*ln(ln(-x**2+5))+(4*x**2-20)*ln(-x**2+5)+2*x**3)*(-x*ln(ln(-x**2+5))+2)**(1
/2)+(2*x**5-10*x**3)*ln(-x**2+5)*ln(ln(-x**2+5))+(-4*x**4+20*x**2)*ln(-x**2+5))/((2*x**5-10*x**3)*ln(-x**2+5)*
ln(ln(-x**2+5))+(-4*x**4+20*x**2)*ln(-x**2+5)),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^3+5*x)*log(-x^2+5)*log(log(-x^2+5))+(4*x^2-20)*log(-x^2+5)+2*x^3)*(-x*log(log(-x^2+5))+2)^(1/2
)+(2*x^5-10*x^3)*log(-x^2+5)*log(log(-x^2+5))+(-4*x^4+20*x^2)*log(-x^2+5))/((2*x^5-10*x^3)*log(-x^2+5)*log(log
(-x^2+5))+(-4*x^4+20*x^2)*log(-x^2+5)),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 5.44, size = 22, normalized size = 0.88 \begin {gather*} x+\frac {\sqrt {2-x\,\ln \left (\ln \left (5-x^2\right )\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(5 - x^2)*(20*x^2 - 4*x^4) + (2 - x*log(log(5 - x^2)))^(1/2)*(2*x^3 + log(5 - x^2)*(4*x^2 - 20) + log(
5 - x^2)*log(log(5 - x^2))*(5*x - x^3)) - log(5 - x^2)*log(log(5 - x^2))*(10*x^3 - 2*x^5))/(log(5 - x^2)*(20*x
^2 - 4*x^4) - log(5 - x^2)*log(log(5 - x^2))*(10*x^3 - 2*x^5)),x)

[Out]

x + (2 - x*log(log(5 - x^2)))^(1/2)/x

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