\(\int x^4 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}} \, dx\) [2247]

Optimal result

Integrand size = 30, antiderivative size = 167 \[ \int x^4 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}} \, dx=\frac {x \sqrt {1+x^4} \left (104 x^2+264 x^6+192 x^{10}\right ) \sqrt {x^2+\sqrt {1+x^4}}+x \left (39+212 x^4+360 x^8+192 x^{12}\right ) \sqrt {x^2+\sqrt {1+x^4}}}{384 \sqrt {1+x^4} \left (1+4 x^4\right )+384 \left (3 x^2+4 x^6\right )}-\frac {13 \text {arctanh}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )}{64 \sqrt {2}} \]

[Out]

(x*(x^4+1)^(1/2)*(192*x^10+264*x^6+104*x^2)*(x^2+(x^4+1)^(1/2))^(1/2)+x*(192*x^12+360*x^8+212*x^4+39)*(x^2+(x^
4+1)^(1/2))^(1/2))/(384*(x^4+1)^(1/2)*(4*x^4+1)+1536*x^6+1152*x^2)-13/128*arctanh(2^(1/2)*x*(x^2+(x^4+1)^(1/2)
)^(1/2)/(1+x^2+(x^4+1)^(1/2)))*2^(1/2)

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00 \[ \int x^4 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}} \, dx=\frac {x \sqrt {1+x^4} \left (104 x^2+264 x^6+192 x^{10}\right ) \sqrt {x^2+\sqrt {1+x^4}}+x \left (39+212 x^4+360 x^8+192 x^{12}\right ) \sqrt {x^2+\sqrt {1+x^4}}}{384 \sqrt {1+x^4} \left (1+4 x^4\right )+384 \left (3 x^2+4 x^6\right )}-\frac {13 \text {arctanh}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )}{64 \sqrt {2}} \]

[In]

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

[Out]

(x*Sqrt[1 + x^4]*(104*x^2 + 264*x^6 + 192*x^10)*Sqrt[x^2 + Sqrt[1 + x^4]] + x*(39 + 212*x^4 + 360*x^8 + 192*x^
12)*Sqrt[x^2 + Sqrt[1 + x^4]])/(384*Sqrt[1 + x^4]*(1 + 4*x^4) + 384*(3*x^2 + 4*x^6)) - (13*ArcTanh[(Sqrt[2]*x*
Sqrt[x^2 + Sqrt[1 + x^4]])/(1 + x^2 + Sqrt[1 + x^4])])/(64*Sqrt[2])

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.48 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.63 \[ \int x^4 \sqrt {1+x^4} \sqrt {x^2+\sqrt {1+x^4}} \, dx=-\frac {1}{384} \, {\left (8 \, x^{7} + 13 \, x^{3} - {\left (56 \, x^{5} + 39 \, x\right )} \sqrt {x^{4} + 1}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + \frac {13}{512} \, \sqrt {2} \log \left (4 \, x^{4} + 4 \, \sqrt {x^{4} + 1} x^{2} - 2 \, {\left (\sqrt {2} x^{3} + \sqrt {2} \sqrt {x^{4} + 1} x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + 1\right ) \]

[In]

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

[Out]

-1/384*(8*x^7 + 13*x^3 - (56*x^5 + 39*x)*sqrt(x^4 + 1))*sqrt(x^2 + sqrt(x^4 + 1)) + 13/512*sqrt(2)*log(4*x^4 +
4*sqrt(x^4 + 1)*x^2 - 2*(sqrt(2)*x^3 + sqrt(2)*sqrt(x^4 + 1)*x)*sqrt(x^2 + sqrt(x^4 + 1)) + 1)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

integrate(x^4*(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^4, x)

Giac [F]

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

[In]

integrate(x^4*(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^4, x)

Mupad [F(-1)]

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

[In]

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

[Out]

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