Integrand size = 16, antiderivative size = 185 \[ \int \log \left (x^2+\sqrt {1-x^2}\right ) \, dx=-2 x-\arcsin (x)+\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \arctan \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )+\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \arctan \left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} x}{\sqrt {1-x^2}}\right )+\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \text {arctanh}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )-\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} x}{\sqrt {1-x^2}}\right )+x \log \left (x^2+\sqrt {1-x^2}\right ) \] Output:
-2*x-arcsin(x)+x*ln(x^2+(-x^2+1)^(1/2))+1/2*arctanh(x*2^(1/2)/(5^(1/2)-1)^ (1/2))*(-2+2*5^(1/2))^(1/2)-1/2*arctanh(1/2*x*(-2+2*5^(1/2))^(1/2)/(-x^2+1 )^(1/2))*(-2+2*5^(1/2))^(1/2)+1/2*arctan(x*2^(1/2)/(5^(1/2)+1)^(1/2))*(2+2 *5^(1/2))^(1/2)+1/2*arctan(1/2*x*(2+2*5^(1/2))^(1/2)/(-x^2+1)^(1/2))*(2+2* 5^(1/2))^(1/2)
Result contains complex when optimal does not.
Time = 0.36 (sec) , antiderivative size = 920, normalized size of antiderivative = 4.97 \[ \int \log \left (x^2+\sqrt {1-x^2}\right ) \, dx =\text {Too large to display} \] Input:
Integrate[Log[x^2 + Sqrt[1 - x^2]],x]
Output:
(-8*Sqrt[5]*x - 4*Sqrt[5]*ArcSin[x] + 5*Sqrt[2*(-1 + Sqrt[5])]*ArcTan[Sqrt [2/(1 + Sqrt[5])]*x] + Sqrt[10*(-1 + Sqrt[5])]*ArcTan[Sqrt[2/(1 + Sqrt[5]) ]*x] - (-5 + Sqrt[5])*Sqrt[2*(1 + Sqrt[5])]*ArcTanh[Sqrt[2/(-1 + Sqrt[5])] *x] - 5*Sqrt[2 + Sqrt[5]]*Log[-Sqrt[2*(-1 + Sqrt[5])] + 2*x] + 3*Sqrt[5*(2 + Sqrt[5])]*Log[-Sqrt[2*(-1 + Sqrt[5])] + 2*x] + 5*Sqrt[2 + Sqrt[5]]*Log[ Sqrt[2*(-1 + Sqrt[5])] + 2*x] - 3*Sqrt[5*(2 + Sqrt[5])]*Log[Sqrt[2*(-1 + S qrt[5])] + 2*x] - (5*I)*Sqrt[-2 + Sqrt[5]]*Log[(-I)*Sqrt[2*(1 + Sqrt[5])] + 2*x] - (3*I)*Sqrt[5*(-2 + Sqrt[5])]*Log[(-I)*Sqrt[2*(1 + Sqrt[5])] + 2*x ] + (5*I)*Sqrt[-2 + Sqrt[5]]*Log[I*Sqrt[2*(1 + Sqrt[5])] + 2*x] + (3*I)*Sq rt[5*(-2 + Sqrt[5])]*Log[I*Sqrt[2*(1 + Sqrt[5])] + 2*x] + 4*Sqrt[5]*x*Log[ x^2 + Sqrt[1 - x^2]] + (5*I)*Sqrt[-2 + Sqrt[5]]*Log[4 - (2*I)*Sqrt[2*(1 + Sqrt[5])]*x + 2*Sqrt[2*(3 + Sqrt[5])]*Sqrt[1 - x^2]] + (3*I)*Sqrt[5*(-2 + Sqrt[5])]*Log[4 - (2*I)*Sqrt[2*(1 + Sqrt[5])]*x + 2*Sqrt[2*(3 + Sqrt[5])]* Sqrt[1 - x^2]] - (5*I)*Sqrt[-2 + Sqrt[5]]*Log[4 + (2*I)*Sqrt[2*(1 + Sqrt[5 ])]*x + 2*Sqrt[2*(3 + Sqrt[5])]*Sqrt[1 - x^2]] - (3*I)*Sqrt[5*(-2 + Sqrt[5 ])]*Log[4 + (2*I)*Sqrt[2*(1 + Sqrt[5])]*x + 2*Sqrt[2*(3 + Sqrt[5])]*Sqrt[1 - x^2]] - 5*Sqrt[2 + Sqrt[5]]*Log[2*(2 + Sqrt[2*(-1 + Sqrt[5])]*x + Sqrt[ 2]*Sqrt[(-3 + Sqrt[5])*(-1 + x^2)])] + 3*Sqrt[5*(2 + Sqrt[5])]*Log[2*(2 + Sqrt[2*(-1 + Sqrt[5])]*x + Sqrt[2]*Sqrt[(-3 + Sqrt[5])*(-1 + x^2)])] + 5*S qrt[2 + Sqrt[5]]*Log[4 - 2*Sqrt[2*(-1 + Sqrt[5])]*x + 2*Sqrt[2]*Sqrt[(-...
Time = 1.11 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.89, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3028, 7293, 2009}
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.
| \(\displaystyle \int \log \left (x^2+\sqrt {1-x^2}\right ) \, dx\) |
| \(\Big \downarrow \) 3028 |
| \(\displaystyle x \log \left (x^2+\sqrt {1-x^2}\right )-\int \frac {x^2 \left (2-\frac {1}{\sqrt {1-x^2}}\right )}{x^2+\sqrt {1-x^2}}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle x \log \left (x^2+\sqrt {1-x^2}\right )-\int \left (\frac {2 x^2}{x^2+\sqrt {1-x^2}}-\frac {x^2}{\sqrt {1-x^2} x^2-x^2+1}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\arcsin (x)+2 \sqrt {\frac {1}{5} \left (2+\sqrt {5}\right )} \arctan \left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} x}{\sqrt {1-x^2}}\right )-\sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \arctan \left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} x}{\sqrt {1-x^2}}\right )+2 \sqrt {\frac {1}{5} \left (2+\sqrt {5}\right )} \arctan \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )-\sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \arctan \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )-\sqrt {\frac {1}{10} \left (\sqrt {5}-1\right )} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2} \left (\sqrt {5}-1\right )} x}{\sqrt {1-x^2}}\right )-2 \sqrt {\frac {1}{5} \left (\sqrt {5}-2\right )} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2} \left (\sqrt {5}-1\right )} x}{\sqrt {1-x^2}}\right )+\sqrt {\frac {1}{10} \left (\sqrt {5}-1\right )} \text {arctanh}\left (\sqrt {\frac {2}{\sqrt {5}-1}} x\right )+2 \sqrt {\frac {1}{5} \left (\sqrt {5}-2\right )} \text {arctanh}\left (\sqrt {\frac {2}{\sqrt {5}-1}} x\right )+x \log \left (x^2+\sqrt {1-x^2}\right )-2 x\) |
Input:
Int[Log[x^2 + Sqrt[1 - x^2]],x]
Output:
-2*x - ArcSin[x] - Sqrt[(1 + Sqrt[5])/10]*ArcTan[Sqrt[2/(1 + Sqrt[5])]*x] + 2*Sqrt[(2 + Sqrt[5])/5]*ArcTan[Sqrt[2/(1 + Sqrt[5])]*x] - Sqrt[(1 + Sqrt [5])/10]*ArcTan[(Sqrt[(1 + Sqrt[5])/2]*x)/Sqrt[1 - x^2]] + 2*Sqrt[(2 + Sqr t[5])/5]*ArcTan[(Sqrt[(1 + Sqrt[5])/2]*x)/Sqrt[1 - x^2]] + 2*Sqrt[(-2 + Sq rt[5])/5]*ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x] + Sqrt[(-1 + Sqrt[5])/10]*ArcT anh[Sqrt[2/(-1 + Sqrt[5])]*x] - 2*Sqrt[(-2 + Sqrt[5])/5]*ArcTanh[(Sqrt[(-1 + Sqrt[5])/2]*x)/Sqrt[1 - x^2]] - Sqrt[(-1 + Sqrt[5])/10]*ArcTanh[(Sqrt[( -1 + Sqrt[5])/2]*x)/Sqrt[1 - x^2]] + x*Log[x^2 + Sqrt[1 - x^2]]
Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[x*(D[u, x]/u), x], x] /; InverseFunctionFreeQ[u, x]
Leaf count of result is larger than twice the leaf count of optimal. \(468\) vs. \(2(138)=276\).
Time = 0.18 (sec) , antiderivative size = 469, normalized size of antiderivative = 2.54
| method | result | size |
| parts | \(x \ln \left (x^{2}+\sqrt {-x^{2}+1}\right )-\frac {\sqrt {5}\, \left (\sqrt {5}+1\right ) \arctan \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{5 \sqrt {2+2 \sqrt {5}}}+\frac {\left (\sqrt {5}-1\right ) \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{5 \sqrt {-2+2 \sqrt {5}}}+\arcsin \left (x \right )-\frac {\left (\sqrt {5}-3\right ) \sqrt {5}\, \operatorname {arctanh}\left (\frac {\sqrt {-x^{2}+1}-1}{x \sqrt {-2+\sqrt {5}}}\right )}{10 \sqrt {-2+\sqrt {5}}}+\frac {\left (3+\sqrt {5}\right ) \sqrt {5}\, \arctan \left (\frac {\sqrt {-x^{2}+1}-1}{x \sqrt {2+\sqrt {5}}}\right )}{10 \sqrt {2+\sqrt {5}}}+\frac {\left (\sqrt {5}-1\right ) \sqrt {5}\, \arctan \left (\frac {\sqrt {-x^{2}+1}-1}{x \sqrt {-2+\sqrt {5}}}\right )}{10 \sqrt {-2+\sqrt {5}}}-\frac {\sqrt {5}\, \left (\sqrt {5}+1\right ) \operatorname {arctanh}\left (\frac {\sqrt {-x^{2}+1}-1}{x \sqrt {2+\sqrt {5}}}\right )}{10 \sqrt {2+\sqrt {5}}}+\frac {2 \sqrt {-2+\sqrt {5}}\, \sqrt {5}\, \operatorname {arctanh}\left (\frac {\sqrt {-x^{2}+1}-1}{x \sqrt {-2+\sqrt {5}}}\right )}{5}-\frac {2 \sqrt {2+\sqrt {5}}\, \sqrt {5}\, \arctan \left (\frac {\sqrt {-x^{2}+1}-1}{x \sqrt {2+\sqrt {5}}}\right )}{5}+4 \arctan \left (\frac {\sqrt {-x^{2}+1}-1}{x}\right )-\frac {2 \sqrt {5}\, \arctan \left (\frac {\sqrt {-x^{2}+1}-1}{x \sqrt {-2+\sqrt {5}}}\right )}{5 \sqrt {-2+\sqrt {5}}}-\frac {2 \sqrt {5}\, \operatorname {arctanh}\left (\frac {\sqrt {-x^{2}+1}-1}{x \sqrt {2+\sqrt {5}}}\right )}{5 \sqrt {2+\sqrt {5}}}-2 x +\frac {2 \left (3+\sqrt {5}\right ) \sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{5 \sqrt {2+2 \sqrt {5}}}-\frac {2 \left (\sqrt {5}-3\right ) \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{5 \sqrt {-2+2 \sqrt {5}}}\) | \(469\) |
| default | \(x \ln \left (x^{2}+\sqrt {-x^{2}+1}\right )-\frac {\sqrt {5}\, \left (\sqrt {5}+1\right ) \arctan \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{5 \sqrt {2+2 \sqrt {5}}}+\frac {\left (\sqrt {5}-1\right ) \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{5 \sqrt {-2+2 \sqrt {5}}}-2 x +\frac {2 \left (3+\sqrt {5}\right ) \sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{5 \sqrt {2+2 \sqrt {5}}}-\frac {2 \left (\sqrt {5}-3\right ) \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{5 \sqrt {-2+2 \sqrt {5}}}+\frac {\left (\sqrt {5}-1\right ) \sqrt {5}\, \operatorname {arctanh}\left (\frac {\sqrt {-x^{2}+1}-1}{x \sqrt {-2+\sqrt {5}}}\right )}{5 \sqrt {-2+\sqrt {5}}}-\frac {\sqrt {5}\, \left (\sqrt {5}+1\right ) \arctan \left (\frac {\sqrt {-x^{2}+1}-1}{x \sqrt {2+\sqrt {5}}}\right )}{5 \sqrt {2+\sqrt {5}}}+\frac {\left (\sqrt {5}-3\right ) \sqrt {5}\, \arctan \left (\frac {\sqrt {-x^{2}+1}-1}{x \sqrt {-2+\sqrt {5}}}\right )}{5 \sqrt {-2+\sqrt {5}}}-\frac {\left (3+\sqrt {5}\right ) \sqrt {5}\, \operatorname {arctanh}\left (\frac {\sqrt {-x^{2}+1}-1}{x \sqrt {2+\sqrt {5}}}\right )}{5 \sqrt {2+\sqrt {5}}}-\arcsin \left (x \right )+\frac {\left (\sqrt {5}-3\right ) \sqrt {5}\, \operatorname {arctanh}\left (\frac {\sqrt {-x^{2}+1}-1}{x \sqrt {-2+\sqrt {5}}}\right )}{10 \sqrt {-2+\sqrt {5}}}-\frac {\left (3+\sqrt {5}\right ) \sqrt {5}\, \arctan \left (\frac {\sqrt {-x^{2}+1}-1}{x \sqrt {2+\sqrt {5}}}\right )}{10 \sqrt {2+\sqrt {5}}}-\frac {\left (\sqrt {5}-1\right ) \sqrt {5}\, \arctan \left (\frac {\sqrt {-x^{2}+1}-1}{x \sqrt {-2+\sqrt {5}}}\right )}{10 \sqrt {-2+\sqrt {5}}}+\frac {\sqrt {5}\, \left (\sqrt {5}+1\right ) \operatorname {arctanh}\left (\frac {\sqrt {-x^{2}+1}-1}{x \sqrt {2+\sqrt {5}}}\right )}{10 \sqrt {2+\sqrt {5}}}\) | \(473\) |
Input:
int(ln(x^2+(-x^2+1)^(1/2)),x,method=_RETURNVERBOSE)
Output:
x*ln(x^2+(-x^2+1)^(1/2))-1/5*5^(1/2)*(5^(1/2)+1)/(2+2*5^(1/2))^(1/2)*arcta n(2*x/(2+2*5^(1/2))^(1/2))+1/5*(5^(1/2)-1)*5^(1/2)/(-2+2*5^(1/2))^(1/2)*ar ctanh(2*x/(-2+2*5^(1/2))^(1/2))+arcsin(x)-1/10*(5^(1/2)-3)*5^(1/2)/(-2+5^( 1/2))^(1/2)*arctanh(((-x^2+1)^(1/2)-1)/x/(-2+5^(1/2))^(1/2))+1/10*(3+5^(1/ 2))*5^(1/2)/(2+5^(1/2))^(1/2)*arctan(((-x^2+1)^(1/2)-1)/x/(2+5^(1/2))^(1/2 ))+1/10*(5^(1/2)-1)*5^(1/2)/(-2+5^(1/2))^(1/2)*arctan(((-x^2+1)^(1/2)-1)/x /(-2+5^(1/2))^(1/2))-1/10*5^(1/2)*(5^(1/2)+1)/(2+5^(1/2))^(1/2)*arctanh((( -x^2+1)^(1/2)-1)/x/(2+5^(1/2))^(1/2))+2/5*(-2+5^(1/2))^(1/2)*5^(1/2)*arcta nh(((-x^2+1)^(1/2)-1)/x/(-2+5^(1/2))^(1/2))-2/5*(2+5^(1/2))^(1/2)*5^(1/2)* arctan(((-x^2+1)^(1/2)-1)/x/(2+5^(1/2))^(1/2))+4*arctan(((-x^2+1)^(1/2)-1) /x)-2/5*5^(1/2)/(-2+5^(1/2))^(1/2)*arctan(((-x^2+1)^(1/2)-1)/x/(-2+5^(1/2) )^(1/2))-2/5*5^(1/2)/(2+5^(1/2))^(1/2)*arctanh(((-x^2+1)^(1/2)-1)/x/(2+5^( 1/2))^(1/2))-2*x+2/5*(3+5^(1/2))*5^(1/2)/(2+2*5^(1/2))^(1/2)*arctan(2*x/(2 +2*5^(1/2))^(1/2))-2/5*(5^(1/2)-3)*5^(1/2)/(-2+2*5^(1/2))^(1/2)*arctanh(2* x/(-2+2*5^(1/2))^(1/2))
Time = 0.11 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.46 \[ \int \log \left (x^2+\sqrt {1-x^2}\right ) \, dx=x \log \left (x^{2} + \sqrt {-x^{2} + 1}\right ) + \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \arctan \left (\frac {1}{2} \, {\left (\sqrt {5} x - x\right )} \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}\right ) - \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \arctan \left (\frac {\sqrt {-x^{2} + 1} {\left (\sqrt {5} - 1\right )} \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}}{2 \, x}\right ) + \frac {1}{2} \, \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \log \left (x + \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}\right ) - \frac {1}{2} \, \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \log \left (x - \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}\right ) + \frac {1}{2} \, \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \log \left (-\frac {x^{2} + {\left (\sqrt {-x^{2} + 1} x - x\right )} \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} + \sqrt {-x^{2} + 1} - 1}{x^{2}}\right ) - \frac {1}{2} \, \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \log \left (-\frac {x^{2} - {\left (\sqrt {-x^{2} + 1} x - x\right )} \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} + \sqrt {-x^{2} + 1} - 1}{x^{2}}\right ) - 2 \, x + 2 \, \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \] Input:
integrate(log(x^2+(-x^2+1)^(1/2)),x, algorithm="fricas")
Output:
x*log(x^2 + sqrt(-x^2 + 1)) + sqrt(1/2*sqrt(5) + 1/2)*arctan(1/2*(sqrt(5)* x - x)*sqrt(1/2*sqrt(5) + 1/2)) - sqrt(1/2*sqrt(5) + 1/2)*arctan(1/2*sqrt( -x^2 + 1)*(sqrt(5) - 1)*sqrt(1/2*sqrt(5) + 1/2)/x) + 1/2*sqrt(1/2*sqrt(5) - 1/2)*log(x + sqrt(1/2*sqrt(5) - 1/2)) - 1/2*sqrt(1/2*sqrt(5) - 1/2)*log( x - sqrt(1/2*sqrt(5) - 1/2)) + 1/2*sqrt(1/2*sqrt(5) - 1/2)*log(-(x^2 + (sq rt(-x^2 + 1)*x - x)*sqrt(1/2*sqrt(5) - 1/2) + sqrt(-x^2 + 1) - 1)/x^2) - 1 /2*sqrt(1/2*sqrt(5) - 1/2)*log(-(x^2 - (sqrt(-x^2 + 1)*x - x)*sqrt(1/2*sqr t(5) - 1/2) + sqrt(-x^2 + 1) - 1)/x^2) - 2*x + 2*arctan((sqrt(-x^2 + 1) - 1)/x)
\[ \int \log \left (x^2+\sqrt {1-x^2}\right ) \, dx=\int \log {\left (x^{2} + \sqrt {1 - x^{2}} \right )}\, dx \] Input:
integrate(ln(x**2+(-x**2+1)**(1/2)),x)
Output:
Integral(log(x**2 + sqrt(1 - x**2)), x)
\[ \int \log \left (x^2+\sqrt {1-x^2}\right ) \, dx=\int { \log \left (x^{2} + \sqrt {-x^{2} + 1}\right ) \,d x } \] Input:
integrate(log(x^2+(-x^2+1)^(1/2)),x, algorithm="maxima")
Output:
x*log(x^2 + sqrt(x + 1)*sqrt(-x + 1)) - x - integrate((x^4 - 2*x^2)/(x^4 - x^2 + (x^2 - 1)*e^(1/2*log(x + 1) + 1/2*log(-x + 1))), x) + 1/2*log(x + 1 ) - 1/2*log(-x + 1)
Leaf count of result is larger than twice the leaf count of optimal. 301 vs. \(2 (138) = 276\).
Time = 0.19 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.63 \[ \int \log \left (x^2+\sqrt {1-x^2}\right ) \, dx=x \log \left (x^{2} + \sqrt {-x^{2} + 1}\right ) - \frac {1}{2} \, \pi \mathrm {sgn}\left (x\right ) + \frac {1}{2} \, \sqrt {2 \, \sqrt {5} + 2} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}}\right ) - \frac {1}{2} \, \sqrt {2 \, \sqrt {5} + 2} \arctan \left (-\frac {\frac {x}{\sqrt {-x^{2} + 1} - 1} - \frac {\sqrt {-x^{2} + 1} - 1}{x}}{\sqrt {2 \, \sqrt {5} + 2}}\right ) + \frac {1}{4} \, \sqrt {2 \, \sqrt {5} - 2} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) - \frac {1}{4} \, \sqrt {2 \, \sqrt {5} - 2} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) - \frac {1}{4} \, \sqrt {2 \, \sqrt {5} - 2} \log \left ({\left | \sqrt {2 \, \sqrt {5} - 2} - \frac {x}{\sqrt {-x^{2} + 1} - 1} + \frac {\sqrt {-x^{2} + 1} - 1}{x} \right |}\right ) + \frac {1}{4} \, \sqrt {2 \, \sqrt {5} - 2} \log \left ({\left | -\sqrt {2 \, \sqrt {5} - 2} - \frac {x}{\sqrt {-x^{2} + 1} - 1} + \frac {\sqrt {-x^{2} + 1} - 1}{x} \right |}\right ) - 2 \, x - \arctan \left (-\frac {x {\left (\frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{2 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}\right ) \] Input:
integrate(log(x^2+(-x^2+1)^(1/2)),x, algorithm="giac")
Output:
x*log(x^2 + sqrt(-x^2 + 1)) - 1/2*pi*sgn(x) + 1/2*sqrt(2*sqrt(5) + 2)*arct an(x/sqrt(1/2*sqrt(5) + 1/2)) - 1/2*sqrt(2*sqrt(5) + 2)*arctan(-(x/(sqrt(- x^2 + 1) - 1) - (sqrt(-x^2 + 1) - 1)/x)/sqrt(2*sqrt(5) + 2)) + 1/4*sqrt(2* sqrt(5) - 2)*log(abs(x + sqrt(1/2*sqrt(5) - 1/2))) - 1/4*sqrt(2*sqrt(5) - 2)*log(abs(x - sqrt(1/2*sqrt(5) - 1/2))) - 1/4*sqrt(2*sqrt(5) - 2)*log(abs (sqrt(2*sqrt(5) - 2) - x/(sqrt(-x^2 + 1) - 1) + (sqrt(-x^2 + 1) - 1)/x)) + 1/4*sqrt(2*sqrt(5) - 2)*log(abs(-sqrt(2*sqrt(5) - 2) - x/(sqrt(-x^2 + 1) - 1) + (sqrt(-x^2 + 1) - 1)/x)) - 2*x - arctan(-1/2*x*((sqrt(-x^2 + 1) - 1 )^2/x^2 - 1)/(sqrt(-x^2 + 1) - 1))
Time = 1.19 (sec) , antiderivative size = 608, normalized size of antiderivative = 3.29 \[ \int \log \left (x^2+\sqrt {1-x^2}\right ) \, dx =\text {Too large to display} \] Input:
int(log(x^2 + (1 - x^2)^(1/2)),x)
Output:
x*log(x^2 + (1 - x^2)^(1/2)) - asin(x) - 2*x + (log(x - (2^(1/2)*(5^(1/2) - 1)^(1/2))/2)*(5^(1/2)/2 - 5/2))/(2*(5^(1/2)/2 - 1/2)^(1/2) + 4*(5^(1/2)/ 2 - 1/2)^(3/2)) - (log(x + (2^(1/2)*(5^(1/2) - 1)^(1/2))/2)*(5^(1/2)/2 - 5 /2))/(2*(5^(1/2)/2 - 1/2)^(1/2) + 4*(5^(1/2)/2 - 1/2)^(3/2)) - (log(x - (2 ^(1/2)*(- 5^(1/2) - 1)^(1/2))/2)*(5^(1/2)/2 + 5/2))/(2*(- 5^(1/2)/2 - 1/2) ^(1/2) + 4*(- 5^(1/2)/2 - 1/2)^(3/2)) + (log(x + (2^(1/2)*(- 5^(1/2) - 1)^ (1/2))/2)*(5^(1/2)/2 + 5/2))/(2*(- 5^(1/2)/2 - 1/2)^(1/2) + 4*(- 5^(1/2)/2 - 1/2)^(3/2)) + (log((((x*(5^(1/2)/2 - 1/2)^(1/2) + 1)*1i)/(3/2 - 5^(1/2) /2)^(1/2) + (1 - x^2)^(1/2)*1i)/(x + (5^(1/2)/2 - 1/2)^(1/2)))*((3*5^(1/2) )/2 - 5/2))/((2*(5^(1/2)/2 - 1/2)^(1/2) + 4*(5^(1/2)/2 - 1/2)^(3/2))*(3/2 - 5^(1/2)/2)^(1/2)) - (log((((x*(- 5^(1/2)/2 - 1/2)^(1/2) + 1)*1i)/(5^(1/2 )/2 + 3/2)^(1/2) + (1 - x^2)^(1/2)*1i)/(x + (- 5^(1/2)/2 - 1/2)^(1/2)))*(( 3*5^(1/2))/2 + 5/2))/((2*(- 5^(1/2)/2 - 1/2)^(1/2) + 4*(- 5^(1/2)/2 - 1/2) ^(3/2))*(5^(1/2)/2 + 3/2)^(1/2)) - (log((((x*(5^(1/2)/2 - 1/2)^(1/2) - 1)* 1i)/(3/2 - 5^(1/2)/2)^(1/2) - (1 - x^2)^(1/2)*1i)/(x - (5^(1/2)/2 - 1/2)^( 1/2)))*((3*5^(1/2))/2 - 5/2))/((2*(5^(1/2)/2 - 1/2)^(1/2) + 4*(5^(1/2)/2 - 1/2)^(3/2))*(3/2 - 5^(1/2)/2)^(1/2)) + (log((((x*(- 5^(1/2)/2 - 1/2)^(1/2 ) - 1)*1i)/(5^(1/2)/2 + 3/2)^(1/2) - (1 - x^2)^(1/2)*1i)/(x - (- 5^(1/2)/2 - 1/2)^(1/2)))*((3*5^(1/2))/2 + 5/2))/((2*(- 5^(1/2)/2 - 1/2)^(1/2) + 4*( - 5^(1/2)/2 - 1/2)^(3/2))*(5^(1/2)/2 + 3/2)^(1/2))
Time = 0.18 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.97 \[ \int \log \left (x^2+\sqrt {1-x^2}\right ) \, dx=-\mathit {asin} \left (x \right )+\sqrt {\sqrt {5}-2}\, \sqrt {5}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )}{\sqrt {\sqrt {5}-2}}\right )+3 \sqrt {\sqrt {5}-2}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )}{\sqrt {\sqrt {5}-2}}\right )+\frac {\sqrt {\sqrt {5}+2}\, \sqrt {5}\, \mathrm {log}\left (-\sqrt {\sqrt {5}+2}+\tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )\right )}{2}-\frac {\sqrt {\sqrt {5}+2}\, \sqrt {5}\, \mathrm {log}\left (\sqrt {\sqrt {5}+2}+\tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )\right )}{2}-\frac {3 \sqrt {\sqrt {5}+2}\, \mathrm {log}\left (-\sqrt {\sqrt {5}+2}+\tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )\right )}{2}+\frac {3 \sqrt {\sqrt {5}+2}\, \mathrm {log}\left (\sqrt {\sqrt {5}+2}+\tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )\right )}{2}+\mathrm {log}\left (\frac {-\tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )^{4}+4 \tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )^{2}+1}{\tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )^{4}+2 \tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )^{2}+1}\right ) x -2 x \] Input:
int(log(x^2+(-x^2+1)^(1/2)),x)
Output:
( - 2*asin(x) + 2*sqrt(sqrt(5) - 2)*sqrt(5)*atan(tan(asin(x)/2)/sqrt(sqrt( 5) - 2)) + 6*sqrt(sqrt(5) - 2)*atan(tan(asin(x)/2)/sqrt(sqrt(5) - 2)) + sq rt(sqrt(5) + 2)*sqrt(5)*log( - sqrt(sqrt(5) + 2) + tan(asin(x)/2)) - sqrt( sqrt(5) + 2)*sqrt(5)*log(sqrt(sqrt(5) + 2) + tan(asin(x)/2)) - 3*sqrt(sqrt (5) + 2)*log( - sqrt(sqrt(5) + 2) + tan(asin(x)/2)) + 3*sqrt(sqrt(5) + 2)* log(sqrt(sqrt(5) + 2) + tan(asin(x)/2)) + 2*log(( - tan(asin(x)/2)**4 + 4* tan(asin(x)/2)**2 + 1)/(tan(asin(x)/2)**4 + 2*tan(asin(x)/2)**2 + 1))*x - 4*x)/2