Integrand size = 22, antiderivative size = 86 \[ \int \sqrt {1-x^2+x \sqrt {-1+x^2}} \, dx=\frac {3}{4} x \sqrt {1-x^2+x \sqrt {-1+x^2}}+\frac {1}{4} \sqrt {-1+x^2} \sqrt {1-x^2+x \sqrt {-1+x^2}}+\frac {3 \arcsin \left (x-\sqrt {-1+x^2}\right )}{4 \sqrt {2}} \] Output:
3/4*x*(1-x^2+x*(x^2-1)^(1/2))^(1/2)+1/4*(x^2-1)^(1/2)*(1-x^2+x*(x^2-1)^(1/ 2))^(1/2)+3/8*arcsin(x-(x^2-1)^(1/2))*2^(1/2)
Time = 0.88 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.28 \[ \int \sqrt {1-x^2+x \sqrt {-1+x^2}} \, dx=\frac {1}{8} \left (\frac {2 \left (-1+x^2\right ) \left (3 x+\sqrt {-1+x^2}\right )}{\sqrt {1-x^2+x \sqrt {-1+x^2}} \left (-1+x^2+x \sqrt {-1+x^2}\right )}-3 \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-1+x^2}}{\sqrt {1-x^2+x \sqrt {-1+x^2}}}\right )\right ) \] Input:
Integrate[Sqrt[1 - x^2 + x*Sqrt[-1 + x^2]],x]
Output:
((2*(-1 + x^2)*(3*x + Sqrt[-1 + x^2]))/(Sqrt[1 - x^2 + x*Sqrt[-1 + x^2]]*( -1 + x^2 + x*Sqrt[-1 + x^2])) - 3*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[-1 + x^2])/ Sqrt[1 - x^2 + x*Sqrt[-1 + x^2]]])/8
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.
\(\displaystyle \int \sqrt {-x^2+\sqrt {x^2-1} x+1} \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \sqrt {-x^2+\sqrt {x^2-1} x+1}dx\) |
Input:
Int[Sqrt[1 - x^2 + x*Sqrt[-1 + x^2]],x]
Output:
$Aborted
\[\int \sqrt {1-x^{2}+x \sqrt {x^{2}-1}}d x\]
Input:
int((1-x^2+x*(x^2-1)^(1/2))^(1/2),x)
Output:
int((1-x^2+x*(x^2-1)^(1/2))^(1/2),x)
Time = 0.47 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.86 \[ \int \sqrt {1-x^2+x \sqrt {-1+x^2}} \, dx=\frac {1}{4} \, \sqrt {-x^{2} + \sqrt {x^{2} - 1} x + 1} {\left (3 \, x + \sqrt {x^{2} - 1}\right )} - \frac {3}{8} \, \sqrt {2} \arctan \left (\sqrt {-x^{2} + \sqrt {x^{2} - 1} x + 1} {\left (\sqrt {2} x + \sqrt {2} \sqrt {x^{2} - 1}\right )}\right ) \] Input:
integrate((1-x^2+x*(x^2-1)^(1/2))^(1/2),x, algorithm="fricas")
Output:
1/4*sqrt(-x^2 + sqrt(x^2 - 1)*x + 1)*(3*x + sqrt(x^2 - 1)) - 3/8*sqrt(2)*a rctan(sqrt(-x^2 + sqrt(x^2 - 1)*x + 1)*(sqrt(2)*x + sqrt(2)*sqrt(x^2 - 1)) )
\[ \int \sqrt {1-x^2+x \sqrt {-1+x^2}} \, dx=\int \sqrt {- x^{2} + x \sqrt {x^{2} - 1} + 1}\, dx \] Input:
integrate((1-x**2+x*(x**2-1)**(1/2))**(1/2),x)
Output:
Integral(sqrt(-x**2 + x*sqrt(x**2 - 1) + 1), x)
\[ \int \sqrt {1-x^2+x \sqrt {-1+x^2}} \, dx=\int { \sqrt {-x^{2} + \sqrt {x^{2} - 1} x + 1} \,d x } \] Input:
integrate((1-x^2+x*(x^2-1)^(1/2))^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(-x^2 + sqrt(x^2 - 1)*x + 1), x)
\[ \int \sqrt {1-x^2+x \sqrt {-1+x^2}} \, dx=\int { \sqrt {-x^{2} + \sqrt {x^{2} - 1} x + 1} \,d x } \] Input:
integrate((1-x^2+x*(x^2-1)^(1/2))^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(-x^2 + sqrt(x^2 - 1)*x + 1), x)
Timed out. \[ \int \sqrt {1-x^2+x \sqrt {-1+x^2}} \, dx=\int \sqrt {x\,\sqrt {x^2-1}-x^2+1} \,d x \] Input:
int((x*(x^2 - 1)^(1/2) - x^2 + 1)^(1/2),x)
Output:
int((x*(x^2 - 1)^(1/2) - x^2 + 1)^(1/2), x)
\[ \int \sqrt {1-x^2+x \sqrt {-1+x^2}} \, dx=\frac {\sqrt {2}\, \left (-2 \left (x +1\right )^{\frac {1}{4}} \left (x -1\right )^{\frac {1}{4}} x^{2}+2 \left (x +1\right )^{\frac {1}{4}} \left (x -1\right )^{\frac {1}{4}}+5 \sqrt {x -1}\, \left (\int \frac {\sqrt {x +1}}{\left (x^{2}-1\right )^{\frac {1}{4}}}d x \right )+5 \sqrt {x -1}\, \left (\int \frac {\sqrt {x +1}\, \sqrt {x^{2}-1}}{\left (x^{2}-1\right )^{\frac {1}{4}}}d x \right )\right )}{10 \sqrt {x -1}} \] Input:
int((1-x^2+x*(x^2-1)^(1/2))^(1/2),x)
Output:
(sqrt(2)*( - 2*(x + 1)**(1/4)*(x - 1)**(1/4)*x**2 + 2*(x + 1)**(1/4)*(x - 1)**(1/4) + 5*sqrt(x - 1)*int(sqrt(x + 1)/(x**2 - 1)**(1/4),x) + 5*sqrt(x - 1)*int((sqrt(x + 1)*sqrt(x**2 - 1))/(x**2 - 1)**(1/4),x)))/(10*sqrt(x - 1))