\(\int x^2 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}} \, dx\) [1921]

Optimal result

Integrand size = 30, antiderivative size = 133 \[ \int x^2 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}} \, dx=\frac {x \sqrt {1+x^4} \left (28 x^2+16 x^6\right ) \sqrt {x^2+\sqrt {1+x^4}}+x \left (9+36 x^4+16 x^8\right ) \sqrt {x^2+\sqrt {1+x^4}}}{96 x^2 \sqrt {1+x^4}+48 \left (1+2 x^4\right )}-\frac {3 \arctan \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}\right )}{16 \sqrt {2}} \]

[Out]

(x*(x^4+1)^(1/2)*(16*x^6+28*x^2)*(x^2+(x^4+1)^(1/2))^(1/2)+x*(16*x^8+36*x^4+9)*(x^2+(x^4+1)^(1/2))^(1/2))/(96*
(x^4+1)^(1/2)*x^2+96*x^4+48)-3/32*arctan(2^(1/2)*x*(x^2+(x^4+1)^(1/2))^(1/2))*2^(1/2)

Rubi [F]

\[ \int x^2 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}} \, dx=\int x^2 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}} \, dx \]

[In]

Int[x^2*Sqrt[1 + x^4]*Sqrt[x^2 + Sqrt[1 + x^4]],x]

[Out]

Defer[Int][x^2*Sqrt[1 + x^4]*Sqrt[x^2 + Sqrt[1 + x^4]], x]

Rubi steps \begin{align*} \text {integral}& = \int x^2 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.90 \[ \int x^2 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}} \, dx=\frac {1}{96} \left (\frac {2 x \sqrt {x^2+\sqrt {1+x^4}} \left (9+36 x^4+16 x^8+28 x^2 \sqrt {1+x^4}+16 x^6 \sqrt {1+x^4}\right )}{1+2 x^4+2 x^2 \sqrt {1+x^4}}-9 \sqrt {2} \arctan \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}\right )\right ) \]

[In]

Integrate[x^2*Sqrt[1 + x^4]*Sqrt[x^2 + Sqrt[1 + x^4]],x]

[Out]

((2*x*Sqrt[x^2 + Sqrt[1 + x^4]]*(9 + 36*x^4 + 16*x^8 + 28*x^2*Sqrt[1 + x^4] + 16*x^6*Sqrt[1 + x^4]))/(1 + 2*x^
4 + 2*x^2*Sqrt[1 + x^4]) - 9*Sqrt[2]*ArcTan[Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]]])/96

Maple [F]

\[\int x^{2} \sqrt {x^{4}+1}\, \sqrt {x^{2}+\sqrt {x^{4}+1}}d x\]

[In]

int(x^2*(x^4+1)^(1/2)*(x^2+(x^4+1)^(1/2))^(1/2),x)

[Out]

int(x^2*(x^4+1)^(1/2)*(x^2+(x^4+1)^(1/2))^(1/2),x)

Fricas [A] (verification not implemented)

none

Time = 0.56 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.61 \[ \int x^2 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}} \, dx=-\frac {1}{48} \, {\left (2 \, x^{5} - 10 \, \sqrt {x^{4} + 1} x^{3} - 9 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + \frac {3}{32} \, \sqrt {2} \arctan \left (-\frac {{\left (\sqrt {2} x^{2} - \sqrt {2} \sqrt {x^{4} + 1}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{2 \, x}\right ) \]

[In]

integrate(x^2*(x^4+1)^(1/2)*(x^2+(x^4+1)^(1/2))^(1/2),x, algorithm="fricas")

[Out]

-1/48*(2*x^5 - 10*sqrt(x^4 + 1)*x^3 - 9*x)*sqrt(x^2 + sqrt(x^4 + 1)) + 3/32*sqrt(2)*arctan(-1/2*(sqrt(2)*x^2 -
sqrt(2)*sqrt(x^4 + 1))*sqrt(x^2 + sqrt(x^4 + 1))/x)

Sympy [F]

\[ \int x^2 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}} \, dx=\int x^{2} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {x^{4} + 1}\, dx \]

[In]

integrate(x**2*(x**4+1)**(1/2)*(x**2+(x**4+1)**(1/2))**(1/2),x)

[Out]

Integral(x**2*sqrt(x**2 + sqrt(x**4 + 1))*sqrt(x**4 + 1), x)

Maxima [F]

\[ \int x^2 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}} \, dx=\int { \sqrt {x^{4} + 1} \sqrt {x^{2} + \sqrt {x^{4} + 1}} x^{2} \,d x } \]

[In]

integrate(x^2*(x^4+1)^(1/2)*(x^2+(x^4+1)^(1/2))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(x^4 + 1)*sqrt(x^2 + sqrt(x^4 + 1))*x^2, x)

Giac [F]

\[ \int x^2 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}} \, dx=\int { \sqrt {x^{4} + 1} \sqrt {x^{2} + \sqrt {x^{4} + 1}} x^{2} \,d x } \]

[In]

integrate(x^2*(x^4+1)^(1/2)*(x^2+(x^4+1)^(1/2))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(x^4 + 1)*sqrt(x^2 + sqrt(x^4 + 1))*x^2, x)

Mupad [F(-1)]

Timed out. \[ \int x^2 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}} \, dx=\int x^2\,\sqrt {x^4+1}\,\sqrt {\sqrt {x^4+1}+x^2} \,d x \]

[In]

int(x^2*(x^4 + 1)^(1/2)*((x^4 + 1)^(1/2) + x^2)^(1/2),x)

[Out]

int(x^2*(x^4 + 1)^(1/2)*((x^4 + 1)^(1/2) + x^2)^(1/2), x)