\(\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+x} \, dx\) [2806]

Optimal result

Integrand size = 23, antiderivative size = 274 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+x} \, dx=\sqrt {x^2+\sqrt {1+x^4}}-\sqrt {-1+\sqrt {2}} \arctan \left (\frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {-1+\sqrt {2}}}\right )+\sqrt {-1+\sqrt {2}} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-\sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+\sqrt {2}}}\right )-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )+\sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right ) \]

[Out]

(x^2+(x^4+1)^(1/2))^(1/2)-(2^(1/2)-1)^(1/2)*arctan((x^2+(x^4+1)^(1/2))^(1/2)/(2^(1/2)-1)^(1/2))+(2^(1/2)-1)^(1
/2)*arctan((-2+2*2^(1/2))^(1/2)*x*(x^2+(x^4+1)^(1/2))^(1/2)/(1+x^2+(x^4+1)^(1/2)))-(1+2^(1/2))^(1/2)*arctanh((
x^2+(x^4+1)^(1/2))^(1/2)/(1+2^(1/2))^(1/2))-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)+(1+2^(1/2))^(1/2)*arctanh((2+2*2^(1/2))^(1/2)*x*(x^2+(x^4+1)^(1/2))^(1/2)/(1+x^2+(x^4+1)^(1/2)))

Rubi [F]

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

[In]

Int[Sqrt[x^2 + Sqrt[1 + x^4]]/(1 + x),x]

[Out]

Defer[Int][Sqrt[x^2 + Sqrt[1 + x^4]]/(1 + x), x]

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

Mathematica [A] (verified)

Time = 1.22 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+x} \, dx=\sqrt {x^2+\sqrt {1+x^4}}-\sqrt {-1+\sqrt {2}} \arctan \left (\sqrt {1+\sqrt {2}} \sqrt {x^2+\sqrt {1+x^4}}\right )+\sqrt {-1+\sqrt {2}} \arctan \left (\frac {\sqrt {-2+2 \sqrt {2}} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )-\sqrt {1+\sqrt {2}} \text {arctanh}\left (\sqrt {-1+\sqrt {2}} \sqrt {x^2+\sqrt {1+x^4}}\right )-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )+\sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2+2 \sqrt {2}} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right ) \]

[In]

Integrate[Sqrt[x^2 + Sqrt[1 + x^4]]/(1 + x),x]

[Out]

Sqrt[x^2 + Sqrt[1 + x^4]] - Sqrt[-1 + Sqrt[2]]*ArcTan[Sqrt[1 + Sqrt[2]]*Sqrt[x^2 + Sqrt[1 + x^4]]] + Sqrt[-1 +
Sqrt[2]]*ArcTan[(Sqrt[-2 + 2*Sqrt[2]]*x*Sqrt[x^2 + Sqrt[1 + x^4]])/(1 + x^2 + Sqrt[1 + x^4])] - Sqrt[1 + Sqrt
[2]]*ArcTanh[Sqrt[-1 + Sqrt[2]]*Sqrt[x^2 + Sqrt[1 + x^4]]] - Sqrt[2]*ArcTanh[(Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^
4]])/(1 + x^2 + Sqrt[1 + x^4])] + Sqrt[1 + Sqrt[2]]*ArcTanh[(Sqrt[2 + 2*Sqrt[2]]*x*Sqrt[x^2 + Sqrt[1 + x^4]])/
(1 + x^2 + Sqrt[1 + x^4])]

Maple [F]

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 582 vs. \(2 (208) = 416\).

Time = 1.18 (sec) , antiderivative size = 582, normalized size of antiderivative = 2.12 \[ \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+x} \, dx=\frac {1}{4} \, \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 ) - \frac {1}{8} \, \sqrt {-4 \, \sqrt {2} + 4} \log \left (-\frac {2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )} \sqrt {-4 \, \sqrt {2} + 4} + 2 \, {\left (2 \, x^{3} + \sqrt {2} {\left (x^{3} - x^{2} - x - 1\right )} - \sqrt {x^{4} + 1} {\left (\sqrt {2} {\left (x - 1\right )} + 2 \, x\right )} - 2\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} - {\left (2 \, x^{2} + \sqrt {2} {\left (x^{2} + 1\right )} + 2\right )} \sqrt {-4 \, \sqrt {2} + 4}}{x^{2} + 2 \, x + 1}\right ) + \frac {1}{8} \, \sqrt {-4 \, \sqrt {2} + 4} \log \left (\frac {2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )} \sqrt {-4 \, \sqrt {2} + 4} - 2 \, {\left (2 \, x^{3} + \sqrt {2} {\left (x^{3} - x^{2} - x - 1\right )} - \sqrt {x^{4} + 1} {\left (\sqrt {2} {\left (x - 1\right )} + 2 \, x\right )} - 2\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} - {\left (2 \, x^{2} + \sqrt {2} {\left (x^{2} + 1\right )} + 2\right )} \sqrt {-4 \, \sqrt {2} + 4}}{x^{2} + 2 \, x + 1}\right ) + \frac {1}{4} \, \sqrt {\sqrt {2} + 1} \log \left (-\frac {2 \, {\left ({\left (2 \, x^{3} - \sqrt {2} {\left (x^{3} - x^{2} - x - 1\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} {\left (x - 1\right )} - 2 \, x\right )} - 2\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + {\left (2 \, x^{2} - \sqrt {2} {\left (x^{2} + 1\right )} + 2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} - 1\right )} + 2\right )} \sqrt {\sqrt {2} + 1}\right )}}{x^{2} + 2 \, x + 1}\right ) - \frac {1}{4} \, \sqrt {\sqrt {2} + 1} \log \left (-\frac {2 \, {\left ({\left (2 \, x^{3} - \sqrt {2} {\left (x^{3} - x^{2} - x - 1\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} {\left (x - 1\right )} - 2 \, x\right )} - 2\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} - {\left (2 \, x^{2} - \sqrt {2} {\left (x^{2} + 1\right )} + 2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} - 1\right )} + 2\right )} \sqrt {\sqrt {2} + 1}\right )}}{x^{2} + 2 \, x + 1}\right ) + \sqrt {x^{2} + \sqrt {x^{4} + 1}} \]

[In]

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

[Out]

1/4*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) - 1/8*sqrt(-4*sqrt(2) + 4)*log(-(2*sqrt(x^4 + 1)*(sqrt(2) + 1)*sqrt(-4*sqrt(2) + 4) + 2*(2*x^3 + sqrt
(2)*(x^3 - x^2 - x - 1) - sqrt(x^4 + 1)*(sqrt(2)*(x - 1) + 2*x) - 2)*sqrt(x^2 + sqrt(x^4 + 1)) - (2*x^2 + sqrt
(2)*(x^2 + 1) + 2)*sqrt(-4*sqrt(2) + 4))/(x^2 + 2*x + 1)) + 1/8*sqrt(-4*sqrt(2) + 4)*log((2*sqrt(x^4 + 1)*(sqr
t(2) + 1)*sqrt(-4*sqrt(2) + 4) - 2*(2*x^3 + sqrt(2)*(x^3 - x^2 - x - 1) - sqrt(x^4 + 1)*(sqrt(2)*(x - 1) + 2*x
) - 2)*sqrt(x^2 + sqrt(x^4 + 1)) - (2*x^2 + sqrt(2)*(x^2 + 1) + 2)*sqrt(-4*sqrt(2) + 4))/(x^2 + 2*x + 1)) + 1/
4*sqrt(sqrt(2) + 1)*log(-2*((2*x^3 - sqrt(2)*(x^3 - x^2 - x - 1) + sqrt(x^4 + 1)*(sqrt(2)*(x - 1) - 2*x) - 2)*
sqrt(x^2 + sqrt(x^4 + 1)) + (2*x^2 - sqrt(2)*(x^2 + 1) + 2*sqrt(x^4 + 1)*(sqrt(2) - 1) + 2)*sqrt(sqrt(2) + 1))
/(x^2 + 2*x + 1)) - 1/4*sqrt(sqrt(2) + 1)*log(-2*((2*x^3 - sqrt(2)*(x^3 - x^2 - x - 1) + sqrt(x^4 + 1)*(sqrt(2
)*(x - 1) - 2*x) - 2)*sqrt(x^2 + sqrt(x^4 + 1)) - (2*x^2 - sqrt(2)*(x^2 + 1) + 2*sqrt(x^4 + 1)*(sqrt(2) - 1) +
2)*sqrt(sqrt(2) + 1))/(x^2 + 2*x + 1)) + sqrt(x^2 + sqrt(x^4 + 1))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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