\(\int \frac {\sqrt {-1+x^2} \sqrt {x^2+x \sqrt {-1+x^2}}}{1+x^2} \, dx\) [2045]

Optimal result

Integrand size = 36, antiderivative size = 146 \[ \int \frac {\sqrt {-1+x^2} \sqrt {x^2+x \sqrt {-1+x^2}}}{1+x^2} \, dx=\sqrt {x^2+x \sqrt {-1+x^2}}-\sqrt {2 \left (1+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {x^2+x \sqrt {-1+x^2}}}{\sqrt {1+\sqrt {2}}}\right )-\frac {\text {arctanh}\left (\sqrt {2} \sqrt {x^2+x \sqrt {-1+x^2}}\right )}{\sqrt {2}}+\sqrt {2 \left (-1+\sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {x^2+x \sqrt {-1+x^2}}}{\sqrt {-1+\sqrt {2}}}\right ) \]

[Out]

(x^2+x*(x^2-1)^(1/2))^(1/2)-(2+2*2^(1/2))^(1/2)*arctan((x^2+x*(x^2-1)^(1/2))^(1/2)/(1+2^(1/2))^(1/2))-1/2*arct
anh(2^(1/2)*(x^2+x*(x^2-1)^(1/2))^(1/2))*2^(1/2)+(-2+2*2^(1/2))^(1/2)*arctanh((x^2+x*(x^2-1)^(1/2))^(1/2)/(2^(
1/2)-1)^(1/2))

Rubi [F]

\[ \int \frac {\sqrt {-1+x^2} \sqrt {x^2+x \sqrt {-1+x^2}}}{1+x^2} \, dx=\int \frac {\sqrt {-1+x^2} \sqrt {x^2+x \sqrt {-1+x^2}}}{1+x^2} \, dx \]

[In]

Int[(Sqrt[-1 + x^2]*Sqrt[x^2 + x*Sqrt[-1 + x^2]])/(1 + x^2),x]

[Out]

(I/2)*Defer[Int][(Sqrt[-1 + x^2]*Sqrt[x^2 + x*Sqrt[-1 + x^2]])/(I - x), x] + (I/2)*Defer[Int][(Sqrt[-1 + x^2]*
Sqrt[x^2 + x*Sqrt[-1 + x^2]])/(I + x), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {i \sqrt {-1+x^2} \sqrt {x^2+x \sqrt {-1+x^2}}}{2 (i-x)}+\frac {i \sqrt {-1+x^2} \sqrt {x^2+x \sqrt {-1+x^2}}}{2 (i+x)}\right ) \, dx \\ & = \frac {1}{2} i \int \frac {\sqrt {-1+x^2} \sqrt {x^2+x \sqrt {-1+x^2}}}{i-x} \, dx+\frac {1}{2} i \int \frac {\sqrt {-1+x^2} \sqrt {x^2+x \sqrt {-1+x^2}}}{i+x} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.33 \[ \int \frac {\sqrt {-1+x^2} \sqrt {x^2+x \sqrt {-1+x^2}}}{1+x^2} \, dx=\frac {\sqrt {x} \sqrt {x+\sqrt {-1+x^2}} \left (2 \sqrt {x} \sqrt {x+\sqrt {-1+x^2}}-2 \sqrt {2 \left (1+\sqrt {2}\right )} \arctan \left (\sqrt {-1+\sqrt {2}} \sqrt {x} \sqrt {x+\sqrt {-1+x^2}}\right )-\sqrt {2} \text {arctanh}\left (\sqrt {2} \sqrt {x} \sqrt {x+\sqrt {-1+x^2}}\right )+2 \sqrt {2 \left (-1+\sqrt {2}\right )} \text {arctanh}\left (\sqrt {1+\sqrt {2}} \sqrt {x} \sqrt {x+\sqrt {-1+x^2}}\right )\right )}{2 \sqrt {x \left (x+\sqrt {-1+x^2}\right )}} \]

[In]

Integrate[(Sqrt[-1 + x^2]*Sqrt[x^2 + x*Sqrt[-1 + x^2]])/(1 + x^2),x]

[Out]

(Sqrt[x]*Sqrt[x + Sqrt[-1 + x^2]]*(2*Sqrt[x]*Sqrt[x + Sqrt[-1 + x^2]] - 2*Sqrt[2*(1 + Sqrt[2])]*ArcTan[Sqrt[-1
+ Sqrt[2]]*Sqrt[x]*Sqrt[x + Sqrt[-1 + x^2]]] - Sqrt[2]*ArcTanh[Sqrt[2]*Sqrt[x]*Sqrt[x + Sqrt[-1 + x^2]]] + 2*
Sqrt[2*(-1 + Sqrt[2])]*ArcTanh[Sqrt[1 + Sqrt[2]]*Sqrt[x]*Sqrt[x + Sqrt[-1 + x^2]]]))/(2*Sqrt[x*(x + Sqrt[-1 +
x^2])])

Maple [F]

\[\int \frac {\sqrt {x^{2}-1}\, \sqrt {x^{2}+x \sqrt {x^{2}-1}}}{x^{2}+1}d x\]

[In]

int((x^2-1)^(1/2)*(x^2+x*(x^2-1)^(1/2))^(1/2)/(x^2+1),x)

[Out]

int((x^2-1)^(1/2)*(x^2+x*(x^2-1)^(1/2))^(1/2)/(x^2+1),x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 594 vs. \(2 (110) = 220\).

Time = 17.03 (sec) , antiderivative size = 594, normalized size of antiderivative = 4.07 \[ \int \frac {\sqrt {-1+x^2} \sqrt {x^2+x \sqrt {-1+x^2}}}{1+x^2} \, dx=\frac {1}{4} \, \sqrt {2} \log \left (4 \, x^{2} + 2 \, {\left (2 \, \sqrt {2} \sqrt {x^{2} - 1} x - \sqrt {2} {\left (2 \, x^{2} - 1\right )}\right )} \sqrt {x^{2} + \sqrt {x^{2} - 1} x} - 4 \, \sqrt {x^{2} - 1} x - 1\right ) + \frac {1}{4} \, \sqrt {-2 \, \sqrt {2} - 2} \log \left (-\frac {2 \, \sqrt {x^{2} - 1} {\left (31 \, \sqrt {2} x - 218 \, x\right )} \sqrt {-2 \, \sqrt {2} - 2} + 4 \, {\left (31 \, x^{2} - \sqrt {2} {\left (109 \, x^{2} + 78\right )} + \sqrt {x^{2} - 1} {\left (109 \, \sqrt {2} x - 31 \, x\right )} - 187\right )} \sqrt {x^{2} + \sqrt {x^{2} - 1} x} + {\left (280 \, x^{2} - \sqrt {2} {\left (249 \, x^{2} - 187\right )} + 156\right )} \sqrt {-2 \, \sqrt {2} - 2}}{x^{2} + 1}\right ) - \frac {1}{4} \, \sqrt {-2 \, \sqrt {2} - 2} \log \left (\frac {2 \, \sqrt {x^{2} - 1} {\left (31 \, \sqrt {2} x - 218 \, x\right )} \sqrt {-2 \, \sqrt {2} - 2} - 4 \, {\left (31 \, x^{2} - \sqrt {2} {\left (109 \, x^{2} + 78\right )} + \sqrt {x^{2} - 1} {\left (109 \, \sqrt {2} x - 31 \, x\right )} - 187\right )} \sqrt {x^{2} + \sqrt {x^{2} - 1} x} + {\left (280 \, x^{2} - \sqrt {2} {\left (249 \, x^{2} - 187\right )} + 156\right )} \sqrt {-2 \, \sqrt {2} - 2}}{x^{2} + 1}\right ) + \frac {1}{4} \, \sqrt {2 \, \sqrt {2} - 2} \log \left (-\frac {4 \, {\left (31 \, x^{2} + \sqrt {2} {\left (109 \, x^{2} + 78\right )} - \sqrt {x^{2} - 1} {\left (109 \, \sqrt {2} x + 31 \, x\right )} - 187\right )} \sqrt {x^{2} + \sqrt {x^{2} - 1} x} + {\left (280 \, x^{2} + \sqrt {2} {\left (249 \, x^{2} - 187\right )} - 2 \, \sqrt {x^{2} - 1} {\left (31 \, \sqrt {2} x + 218 \, x\right )} + 156\right )} \sqrt {2 \, \sqrt {2} - 2}}{x^{2} + 1}\right ) - \frac {1}{4} \, \sqrt {2 \, \sqrt {2} - 2} \log \left (-\frac {4 \, {\left (31 \, x^{2} + \sqrt {2} {\left (109 \, x^{2} + 78\right )} - \sqrt {x^{2} - 1} {\left (109 \, \sqrt {2} x + 31 \, x\right )} - 187\right )} \sqrt {x^{2} + \sqrt {x^{2} - 1} x} - {\left (280 \, x^{2} + \sqrt {2} {\left (249 \, x^{2} - 187\right )} - 2 \, \sqrt {x^{2} - 1} {\left (31 \, \sqrt {2} x + 218 \, x\right )} + 156\right )} \sqrt {2 \, \sqrt {2} - 2}}{x^{2} + 1}\right ) + \sqrt {x^{2} + \sqrt {x^{2} - 1} x} \]

[In]

integrate((x^2-1)^(1/2)*(x^2+x*(x^2-1)^(1/2))^(1/2)/(x^2+1),x, algorithm="fricas")

[Out]

1/4*sqrt(2)*log(4*x^2 + 2*(2*sqrt(2)*sqrt(x^2 - 1)*x - sqrt(2)*(2*x^2 - 1))*sqrt(x^2 + sqrt(x^2 - 1)*x) - 4*sq
rt(x^2 - 1)*x - 1) + 1/4*sqrt(-2*sqrt(2) - 2)*log(-(2*sqrt(x^2 - 1)*(31*sqrt(2)*x - 218*x)*sqrt(-2*sqrt(2) - 2
) + 4*(31*x^2 - sqrt(2)*(109*x^2 + 78) + sqrt(x^2 - 1)*(109*sqrt(2)*x - 31*x) - 187)*sqrt(x^2 + sqrt(x^2 - 1)*
x) + (280*x^2 - sqrt(2)*(249*x^2 - 187) + 156)*sqrt(-2*sqrt(2) - 2))/(x^2 + 1)) - 1/4*sqrt(-2*sqrt(2) - 2)*log
((2*sqrt(x^2 - 1)*(31*sqrt(2)*x - 218*x)*sqrt(-2*sqrt(2) - 2) - 4*(31*x^2 - sqrt(2)*(109*x^2 + 78) + sqrt(x^2
- 1)*(109*sqrt(2)*x - 31*x) - 187)*sqrt(x^2 + sqrt(x^2 - 1)*x) + (280*x^2 - sqrt(2)*(249*x^2 - 187) + 156)*sqr
t(-2*sqrt(2) - 2))/(x^2 + 1)) + 1/4*sqrt(2*sqrt(2) - 2)*log(-(4*(31*x^2 + sqrt(2)*(109*x^2 + 78) - sqrt(x^2 -
1)*(109*sqrt(2)*x + 31*x) - 187)*sqrt(x^2 + sqrt(x^2 - 1)*x) + (280*x^2 + sqrt(2)*(249*x^2 - 187) - 2*sqrt(x^2
- 1)*(31*sqrt(2)*x + 218*x) + 156)*sqrt(2*sqrt(2) - 2))/(x^2 + 1)) - 1/4*sqrt(2*sqrt(2) - 2)*log(-(4*(31*x^2
+ sqrt(2)*(109*x^2 + 78) - sqrt(x^2 - 1)*(109*sqrt(2)*x + 31*x) - 187)*sqrt(x^2 + sqrt(x^2 - 1)*x) - (280*x^2
+ sqrt(2)*(249*x^2 - 187) - 2*sqrt(x^2 - 1)*(31*sqrt(2)*x + 218*x) + 156)*sqrt(2*sqrt(2) - 2))/(x^2 + 1)) + sq
rt(x^2 + sqrt(x^2 - 1)*x)

Sympy [F]

\[ \int \frac {\sqrt {-1+x^2} \sqrt {x^2+x \sqrt {-1+x^2}}}{1+x^2} \, dx=\int \frac {\sqrt {x \left (x + \sqrt {x^{2} - 1}\right )} \sqrt {\left (x - 1\right ) \left (x + 1\right )}}{x^{2} + 1}\, dx \]

[In]

integrate((x**2-1)**(1/2)*(x**2+x*(x**2-1)**(1/2))**(1/2)/(x**2+1),x)

[Out]

Integral(sqrt(x*(x + sqrt(x**2 - 1)))*sqrt((x - 1)*(x + 1))/(x**2 + 1), x)

Maxima [F]

\[ \int \frac {\sqrt {-1+x^2} \sqrt {x^2+x \sqrt {-1+x^2}}}{1+x^2} \, dx=\int { \frac {\sqrt {x^{2} + \sqrt {x^{2} - 1} x} \sqrt {x^{2} - 1}}{x^{2} + 1} \,d x } \]

[In]

integrate((x^2-1)^(1/2)*(x^2+x*(x^2-1)^(1/2))^(1/2)/(x^2+1),x, algorithm="maxima")

[Out]

integrate(sqrt(x^2 + sqrt(x^2 - 1)*x)*sqrt(x^2 - 1)/(x^2 + 1), x)

Giac [F]

\[ \int \frac {\sqrt {-1+x^2} \sqrt {x^2+x \sqrt {-1+x^2}}}{1+x^2} \, dx=\int { \frac {\sqrt {x^{2} + \sqrt {x^{2} - 1} x} \sqrt {x^{2} - 1}}{x^{2} + 1} \,d x } \]

[In]

integrate((x^2-1)^(1/2)*(x^2+x*(x^2-1)^(1/2))^(1/2)/(x^2+1),x, algorithm="giac")

[Out]

integrate(sqrt(x^2 + sqrt(x^2 - 1)*x)*sqrt(x^2 - 1)/(x^2 + 1), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {-1+x^2} \sqrt {x^2+x \sqrt {-1+x^2}}}{1+x^2} \, dx=\int \frac {\sqrt {x^2-1}\,\sqrt {x\,\sqrt {x^2-1}+x^2}}{x^2+1} \,d x \]

[In]

int(((x^2 - 1)^(1/2)*(x*(x^2 - 1)^(1/2) + x^2)^(1/2))/(x^2 + 1),x)

[Out]

int(((x^2 - 1)^(1/2)*(x*(x^2 - 1)^(1/2) + x^2)^(1/2))/(x^2 + 1), x)