Integrand size = 171, antiderivative size = 25 \[ \int \frac {\left (20 x^2-4 x^4\right ) \log \left (5-x^2\right )+\left (-10 x^3+2 x^5\right ) \log \left (5-x^2\right ) \log \left (\log \left (5-x^2\right )\right )+\sqrt {2-x \log \left (\log \left (5-x^2\right )\right )} \left (2 x^3+\left (-20+4 x^2\right ) \log \left (5-x^2\right )+\left (5 x-x^3\right ) \log \left (5-x^2\right ) \log \left (\log \left (5-x^2\right )\right )\right )}{\left (20 x^2-4 x^4\right ) \log \left (5-x^2\right )+\left (-10 x^3+2 x^5\right ) \log \left (5-x^2\right ) \log \left (\log \left (5-x^2\right )\right )} \, dx=-3+x+\frac {\sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}}{x} \] Output:
x-3+(-x*ln(ln(-x^2+5))+2)^(1/2)/x
Time = 0.13 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {\left (20 x^2-4 x^4\right ) \log \left (5-x^2\right )+\left (-10 x^3+2 x^5\right ) \log \left (5-x^2\right ) \log \left (\log \left (5-x^2\right )\right )+\sqrt {2-x \log \left (\log \left (5-x^2\right )\right )} \left (2 x^3+\left (-20+4 x^2\right ) \log \left (5-x^2\right )+\left (5 x-x^3\right ) \log \left (5-x^2\right ) \log \left (\log \left (5-x^2\right )\right )\right )}{\left (20 x^2-4 x^4\right ) \log \left (5-x^2\right )+\left (-10 x^3+2 x^5\right ) \log \left (5-x^2\right ) \log \left (\log \left (5-x^2\right )\right )} \, dx=\frac {1}{2} \left (2 x+\frac {2 \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}}{x}\right ) \] Input:
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[Log[5 - x^2]]]*(2*x^3 + (-20 + 4*x^2)*L og[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]
Output:
(2*x + (2*Sqrt[2 - x*Log[Log[5 - x^2]]])/x)/2
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (20 x^2-4 x^4\right ) \log \left (5-x^2\right )+\sqrt {2-x \log \left (\log \left (5-x^2\right )\right )} \left (2 x^3+\left (4 x^2-20\right ) \log \left (5-x^2\right )+\left (5 x-x^3\right ) \log \left (5-x^2\right ) \log \left (\log \left (5-x^2\right )\right )\right )+\left (2 x^5-10 x^3\right ) \log \left (\log \left (5-x^2\right )\right ) \log \left (5-x^2\right )}{\left (20 x^2-4 x^4\right ) \log \left (5-x^2\right )+\left (2 x^5-10 x^3\right ) \log \left (\log \left (5-x^2\right )\right ) \log \left (5-x^2\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (20 x^2-4 x^4\right ) \log \left (5-x^2\right )+\sqrt {2-x \log \left (\log \left (5-x^2\right )\right )} \left (2 x^3+\left (4 x^2-20\right ) \log \left (5-x^2\right )+\left (5 x-x^3\right ) \log \left (5-x^2\right ) \log \left (\log \left (5-x^2\right )\right )\right )+\left (2 x^5-10 x^3\right ) \log \left (\log \left (5-x^2\right )\right ) \log \left (5-x^2\right )}{2 x^2 \left (5-x^2\right ) \log \left (5-x^2\right ) \left (2-x \log \left (\log \left (5-x^2\right )\right )\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int \frac {4 \left (5 x^2-x^4\right ) \log \left (5-x^2\right )-2 \left (5 x^3-x^5\right ) \log \left (\log \left (5-x^2\right )\right ) \log \left (5-x^2\right )+\sqrt {2-x \log \left (\log \left (5-x^2\right )\right )} \left (2 x^3-4 \left (5-x^2\right ) \log \left (5-x^2\right )+\left (5 x-x^3\right ) \log \left (5-x^2\right ) \log \left (\log \left (5-x^2\right )\right )\right )}{x^2 \left (5-x^2\right ) \log \left (5-x^2\right ) \left (2-x \log \left (\log \left (5-x^2\right )\right )\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {1}{2} \int \left (-\frac {5 \log ^4\left (\log \left (5-x^2\right )\right )}{2 \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )} \left (5 \log ^2\left (\log \left (5-x^2\right )\right )-4\right )}+\frac {2 \log ^2\left (\log \left (5-x^2\right )\right )}{\sqrt {2-x \log \left (\log \left (5-x^2\right )\right )} \left (5 \log ^2\left (\log \left (5-x^2\right )\right )-4\right )}-\frac {\sqrt {2-x \log \left (\log \left (5-x^2\right )\right )} \log \left (\log \left (5-x^2\right )\right )}{2 x}+\frac {10 \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )} \log \left (\log \left (5-x^2\right )\right )}{\left (x^2-5\right ) \log \left (5-x^2\right ) \left (5 \log ^2\left (\log \left (5-x^2\right )\right )-4\right )}+\frac {4 \log \left (\log \left (5-x^2\right )\right )}{\log \left (5-x^2\right ) \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )} \left (5 \log ^2\left (\log \left (5-x^2\right )\right )-4\right )}-\frac {2 \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}}{x^2}+\frac {4 x \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}}{\left (x^2-5\right ) \log \left (5-x^2\right ) \left (5 \log ^2\left (\log \left (5-x^2\right )\right )-4\right )}+2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (-\frac {1}{2} \int \frac {\log ^2\left (\log \left (5-x^2\right )\right )}{\sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}}dx+4 \int \frac {\log \left (\log \left (5-x^2\right )\right )}{\log \left (5-x^2\right ) \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )} \left (5 \log ^2\left (\log \left (5-x^2\right )\right )-4\right )}dx-2 \int \frac {\sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}}{\left (\sqrt {5}-x\right ) \log \left (5-x^2\right ) \left (5 \log ^2\left (\log \left (5-x^2\right )\right )-4\right )}dx+2 \int \frac {\sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}}{\left (x+\sqrt {5}\right ) \log \left (5-x^2\right ) \left (5 \log ^2\left (\log \left (5-x^2\right )\right )-4\right )}dx-\sqrt {5} \int \frac {\log \left (\log \left (5-x^2\right )\right ) \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}}{\left (\sqrt {5}-x\right ) \log \left (5-x^2\right ) \left (5 \log ^2\left (\log \left (5-x^2\right )\right )-4\right )}dx-\sqrt {5} \int \frac {\log \left (\log \left (5-x^2\right )\right ) \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}}{\left (x+\sqrt {5}\right ) \log \left (5-x^2\right ) \left (5 \log ^2\left (\log \left (5-x^2\right )\right )-4\right )}dx-2 \int \frac {\sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}}{x^2}dx-\frac {1}{2} \int \frac {\log \left (\log \left (5-x^2\right )\right ) \sqrt {2-x \log \left (\log \left (5-x^2\right )\right )}}{x}dx+2 x\right )\) |
Input:
Int[((20*x^2 - 4*x^4)*Log[5 - x^2] + (-10*x^3 + 2*x^5)*Log[5 - x^2]*Log[Lo g[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]]),x]
Output:
$Aborted
\[\int \frac {\left (\left (-x^{3}+5 x \right ) \ln \left (-x^{2}+5\right ) \ln \left (\ln \left (-x^{2}+5\right )\right )+\left (4 x^{2}-20\right ) \ln \left (-x^{2}+5\right )+2 x^{3}\right ) \sqrt {-x \ln \left (\ln \left (-x^{2}+5\right )\right )+2}+\left (2 x^{5}-10 x^{3}\right ) \ln \left (-x^{2}+5\right ) \ln \left (\ln \left (-x^{2}+5\right )\right )+\left (-4 x^{4}+20 x^{2}\right ) \ln \left (-x^{2}+5\right )}{\left (2 x^{5}-10 x^{3}\right ) \ln \left (-x^{2}+5\right ) \ln \left (\ln \left (-x^{2}+5\right )\right )+\left (-4 x^{4}+20 x^{2}\right ) \ln \left (-x^{2}+5\right )}d x\]
Input:
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+20* x^2)*ln(-x^2+5)),x)
Output:
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+20* x^2)*ln(-x^2+5)),x)
Exception generated. \[ \int \frac {\left (20 x^2-4 x^4\right ) \log \left (5-x^2\right )+\left (-10 x^3+2 x^5\right ) \log \left (5-x^2\right ) \log \left (\log \left (5-x^2\right )\right )+\sqrt {2-x \log \left (\log \left (5-x^2\right )\right )} \left (2 x^3+\left (-20+4 x^2\right ) \log \left (5-x^2\right )+\left (5 x-x^3\right ) \log \left (5-x^2\right ) \log \left (\log \left (5-x^2\right )\right )\right )}{\left (20 x^2-4 x^4\right ) \log \left (5-x^2\right )+\left (-10 x^3+2 x^5\right ) \log \left (5-x^2\right ) \log \left (\log \left (5-x^2\right )\right )} \, dx=\text {Exception raised: TypeError} \] Input:
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")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
Timed out. \[ \int \frac {\left (20 x^2-4 x^4\right ) \log \left (5-x^2\right )+\left (-10 x^3+2 x^5\right ) \log \left (5-x^2\right ) \log \left (\log \left (5-x^2\right )\right )+\sqrt {2-x \log \left (\log \left (5-x^2\right )\right )} \left (2 x^3+\left (-20+4 x^2\right ) \log \left (5-x^2\right )+\left (5 x-x^3\right ) \log \left (5-x^2\right ) \log \left (\log \left (5-x^2\right )\right )\right )}{\left (20 x^2-4 x^4\right ) \log \left (5-x^2\right )+\left (-10 x^3+2 x^5\right ) \log \left (5-x^2\right ) \log \left (\log \left (5-x^2\right )\right )} \, dx=\text {Timed out} \] Input:
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)
Output:
Timed out
Time = 0.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {\left (20 x^2-4 x^4\right ) \log \left (5-x^2\right )+\left (-10 x^3+2 x^5\right ) \log \left (5-x^2\right ) \log \left (\log \left (5-x^2\right )\right )+\sqrt {2-x \log \left (\log \left (5-x^2\right )\right )} \left (2 x^3+\left (-20+4 x^2\right ) \log \left (5-x^2\right )+\left (5 x-x^3\right ) \log \left (5-x^2\right ) \log \left (\log \left (5-x^2\right )\right )\right )}{\left (20 x^2-4 x^4\right ) \log \left (5-x^2\right )+\left (-10 x^3+2 x^5\right ) \log \left (5-x^2\right ) \log \left (\log \left (5-x^2\right )\right )} \, dx=\frac {x^{2} + \sqrt {-x \log \left (\log \left (-x^{2} + 5\right )\right ) + 2}}{x} \] Input:
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")
Output:
(x^2 + sqrt(-x*log(log(-x^2 + 5)) + 2))/x
\[ \int \frac {\left (20 x^2-4 x^4\right ) \log \left (5-x^2\right )+\left (-10 x^3+2 x^5\right ) \log \left (5-x^2\right ) \log \left (\log \left (5-x^2\right )\right )+\sqrt {2-x \log \left (\log \left (5-x^2\right )\right )} \left (2 x^3+\left (-20+4 x^2\right ) \log \left (5-x^2\right )+\left (5 x-x^3\right ) \log \left (5-x^2\right ) \log \left (\log \left (5-x^2\right )\right )\right )}{\left (20 x^2-4 x^4\right ) \log \left (5-x^2\right )+\left (-10 x^3+2 x^5\right ) \log \left (5-x^2\right ) \log \left (\log \left (5-x^2\right )\right )} \, dx=\int { \frac {2 \, {\left (x^{5} - 5 \, x^{3}\right )} \log \left (-x^{2} + 5\right ) \log \left (\log \left (-x^{2} + 5\right )\right ) - 4 \, {\left (x^{4} - 5 \, x^{2}\right )} \log \left (-x^{2} + 5\right ) + {\left (2 \, x^{3} - {\left (x^{3} - 5 \, x\right )} \log \left (-x^{2} + 5\right ) \log \left (\log \left (-x^{2} + 5\right )\right ) + 4 \, {\left (x^{2} - 5\right )} \log \left (-x^{2} + 5\right )\right )} \sqrt {-x \log \left (\log \left (-x^{2} + 5\right )\right ) + 2}}{2 \, {\left ({\left (x^{5} - 5 \, x^{3}\right )} \log \left (-x^{2} + 5\right ) \log \left (\log \left (-x^{2} + 5\right )\right ) - 2 \, {\left (x^{4} - 5 \, x^{2}\right )} \log \left (-x^{2} + 5\right )\right )}} \,d x } \] Input:
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")
Output:
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)
Time = 3.68 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {\left (20 x^2-4 x^4\right ) \log \left (5-x^2\right )+\left (-10 x^3+2 x^5\right ) \log \left (5-x^2\right ) \log \left (\log \left (5-x^2\right )\right )+\sqrt {2-x \log \left (\log \left (5-x^2\right )\right )} \left (2 x^3+\left (-20+4 x^2\right ) \log \left (5-x^2\right )+\left (5 x-x^3\right ) \log \left (5-x^2\right ) \log \left (\log \left (5-x^2\right )\right )\right )}{\left (20 x^2-4 x^4\right ) \log \left (5-x^2\right )+\left (-10 x^3+2 x^5\right ) \log \left (5-x^2\right ) \log \left (\log \left (5-x^2\right )\right )} \, dx=x+\frac {\sqrt {2-x\,\ln \left (\ln \left (5-x^2\right )\right )}}{x} \] Input:
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)
Output:
x + (2 - x*log(log(5 - x^2)))^(1/2)/x
Time = 0.21 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {\left (20 x^2-4 x^4\right ) \log \left (5-x^2\right )+\left (-10 x^3+2 x^5\right ) \log \left (5-x^2\right ) \log \left (\log \left (5-x^2\right )\right )+\sqrt {2-x \log \left (\log \left (5-x^2\right )\right )} \left (2 x^3+\left (-20+4 x^2\right ) \log \left (5-x^2\right )+\left (5 x-x^3\right ) \log \left (5-x^2\right ) \log \left (\log \left (5-x^2\right )\right )\right )}{\left (20 x^2-4 x^4\right ) \log \left (5-x^2\right )+\left (-10 x^3+2 x^5\right ) \log \left (5-x^2\right ) \log \left (\log \left (5-x^2\right )\right )} \, dx=\frac {\sqrt {-\mathrm {log}\left (\mathrm {log}\left (-x^{2}+5\right )\right ) x +2}+x^{2}}{x} \] Input:
int((((-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)
Output:
(sqrt( - log(log( - x**2 + 5))*x + 2) + x**2)/x