\(\int x \sqrt {-1+x^2} \sqrt {x^2+x \sqrt {-1+x^2}} \, dx\) [1774]

Optimal result

Integrand size = 30, antiderivative size = 119 \[ \int x \sqrt {-1+x^2} \sqrt {x^2+x \sqrt {-1+x^2}} \, dx=\frac {1}{192} \sqrt {-1+x^2} \left (-39+56 x^2\right ) \sqrt {x^2+x \sqrt {-1+x^2}}+\frac {1}{192} \left (13 x-8 x^3\right ) \sqrt {x^2+x \sqrt {-1+x^2}}+\frac {13 \log \left (x+\sqrt {-1+x^2}-\sqrt {2} \sqrt {x^2+x \sqrt {-1+x^2}}\right )}{64 \sqrt {2}} \]

[Out]

1/192*(x^2-1)^(1/2)*(56*x^2-39)*(x^2+x*(x^2-1)^(1/2))^(1/2)+1/192*(-8*x^3+13*x)*(x^2+x*(x^2-1)^(1/2))^(1/2)+13
/128*2^(1/2)*ln(x+(x^2-1)^(1/2)-2^(1/2)*(x^2+x*(x^2-1)^(1/2))^(1/2))

Rubi [F]

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

[In]

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

[Out]

Defer[Int][x*Sqrt[-1 + x^2]*Sqrt[x^2 + x*Sqrt[-1 + x^2]], x]

Rubi steps \begin{align*} \text {integral}& = \int x \sqrt {-1+x^2} \sqrt {x^2+x \sqrt {-1+x^2}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.50 \[ \int x \sqrt {-1+x^2} \sqrt {x^2+x \sqrt {-1+x^2}} \, dx=\frac {1}{384} \sqrt {x \left (x+\sqrt {-1+x^2}\right )} \left (\frac {78-424 x^2+720 x^4-384 x^6-208 x \sqrt {-1+x^2}+528 x^3 \sqrt {-1+x^2}-384 x^5 \sqrt {-1+x^2}}{3 x-4 x^3+\sqrt {-1+x^2}-4 x^2 \sqrt {-1+x^2}}-\frac {39 \sqrt {2} \log \left (x+\sqrt {-1+x^2}+\sqrt {2} \sqrt {x} \sqrt {x+\sqrt {-1+x^2}}\right )}{\sqrt {x} \sqrt {x+\sqrt {-1+x^2}}}\right ) \]

[In]

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

[Out]

(Sqrt[x*(x + Sqrt[-1 + x^2])]*((78 - 424*x^2 + 720*x^4 - 384*x^6 - 208*x*Sqrt[-1 + x^2] + 528*x^3*Sqrt[-1 + x^
2] - 384*x^5*Sqrt[-1 + x^2])/(3*x - 4*x^3 + Sqrt[-1 + x^2] - 4*x^2*Sqrt[-1 + x^2]) - (39*Sqrt[2]*Log[x + Sqrt[
-1 + x^2] + Sqrt[2]*Sqrt[x]*Sqrt[x + Sqrt[-1 + x^2]]])/(Sqrt[x]*Sqrt[x + Sqrt[-1 + x^2]])))/384

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.63 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.84 \[ \int x \sqrt {-1+x^2} \sqrt {x^2+x \sqrt {-1+x^2}} \, dx=-\frac {1}{192} \, {\left (8 \, x^{3} - {\left (56 \, x^{2} - 39\right )} \sqrt {x^{2} - 1} - 13 \, x\right )} \sqrt {x^{2} + \sqrt {x^{2} - 1} x} + \frac {13}{256} \, \sqrt {2} \log \left (-4 \, x^{2} + 2 \, \sqrt {x^{2} + \sqrt {x^{2} - 1} x} {\left (\sqrt {2} x + \sqrt {2} \sqrt {x^{2} - 1}\right )} - 4 \, \sqrt {x^{2} - 1} x + 1\right ) \]

[In]

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

[Out]

-1/192*(8*x^3 - (56*x^2 - 39)*sqrt(x^2 - 1) - 13*x)*sqrt(x^2 + sqrt(x^2 - 1)*x) + 13/256*sqrt(2)*log(-4*x^2 +
2*sqrt(x^2 + sqrt(x^2 - 1)*x)*(sqrt(2)*x + sqrt(2)*sqrt(x^2 - 1)) - 4*sqrt(x^2 - 1)*x + 1)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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