\(\int \frac {(-1+x^4)^2 \sqrt {x^2+\sqrt {1+x^4}}}{(1+x^4)^2} \, dx\) [2255]

Optimal result

Integrand size = 32, antiderivative size = 170 \[ \int \frac {\left (-1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^4\right )^2} \, dx=\frac {x \sqrt {1+x^4} \left (7 x^2+2 x^6\right ) \sqrt {x^2+\sqrt {1+x^4}}+x \left (4+8 x^4+2 x^8\right ) \sqrt {x^2+\sqrt {1+x^4}}}{2 \sqrt {1+x^4} \left (2 x^2+2 x^6\right )+2 \left (1+3 x^4+2 x^8\right )}-\frac {3}{2} \arctan \left (x \sqrt {x^2+\sqrt {1+x^4}}\right )+\frac {\arctan \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}\right )}{2 \sqrt {2}} \]

[Out]

(x*(x^4+1)^(1/2)*(2*x^6+7*x^2)*(x^2+(x^4+1)^(1/2))^(1/2)+x*(2*x^8+8*x^4+4)*(x^2+(x^4+1)^(1/2))^(1/2))/(2*(x^4+
1)^(1/2)*(2*x^6+2*x^2)+4*x^8+6*x^4+2)-3/2*arctan(x*(x^2+(x^4+1)^(1/2))^(1/2))+1/4*arctan(2^(1/2)*x*(x^2+(x^4+1
)^(1/2))^(1/2))*2^(1/2)

Rubi [F]

\[ \int \frac {\left (-1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^4\right )^2} \, dx=\int \frac {\left (-1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^4\right )^2} \, dx \]

[In]

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

[Out]

Defer[Int][Sqrt[x^2 + Sqrt[1 + x^4]], x] + (I/4)*Defer[Int][Sqrt[x^2 + Sqrt[1 + x^4]]/((-1)^(1/4) - x)^2, x] -
((-1)^(1/4)*Defer[Int][Sqrt[x^2 + Sqrt[1 + x^4]]/((-1)^(1/4) - x), x])/4 - (I/4)*Defer[Int][Sqrt[x^2 + Sqrt[1
+ x^4]]/(-(-1)^(3/4) - x)^2, x] + ((-1)^(3/4)*Defer[Int][Sqrt[x^2 + Sqrt[1 + x^4]]/(-(-1)^(3/4) - x), x])/4 +
(I/4)*Defer[Int][Sqrt[x^2 + Sqrt[1 + x^4]]/((-1)^(1/4) + x)^2, x] - ((-1)^(1/4)*Defer[Int][Sqrt[x^2 + Sqrt[1
+ x^4]]/((-1)^(1/4) + x), x])/4 - (I/4)*Defer[Int][Sqrt[x^2 + Sqrt[1 + x^4]]/(-(-1)^(3/4) + x)^2, x] + ((-1)^(
3/4)*Defer[Int][Sqrt[x^2 + Sqrt[1 + x^4]]/(-(-1)^(3/4) + x), x])/4

Rubi steps \begin{align*} \text {integral}& = \int \left (\sqrt {x^2+\sqrt {1+x^4}}+\frac {4 \sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^4\right )^2}-\frac {4 \sqrt {x^2+\sqrt {1+x^4}}}{1+x^4}\right ) \, dx \\ & = 4 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^4\right )^2} \, dx-4 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+x^4} \, dx+\int \sqrt {x^2+\sqrt {1+x^4}} \, dx \\ & = -\left (4 \int \left (\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (i-x^2\right )}+\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (i+x^2\right )}\right ) \, dx\right )+4 \int \left (-\frac {\sqrt {x^2+\sqrt {1+x^4}}}{4 \left (i-x^2\right )^2}-\frac {\sqrt {x^2+\sqrt {1+x^4}}}{4 \left (i+x^2\right )^2}-\frac {\sqrt {x^2+\sqrt {1+x^4}}}{2 \left (-1-x^4\right )}\right ) \, dx+\int \sqrt {x^2+\sqrt {1+x^4}} \, dx \\ & = -\left (2 i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{i-x^2} \, dx\right )-2 i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{i+x^2} \, dx-2 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{-1-x^4} \, dx+\int \sqrt {x^2+\sqrt {1+x^4}} \, dx-\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (i-x^2\right )^2} \, dx-\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (i+x^2\right )^2} \, dx \\ & = -\left (2 i \int \left (-\frac {(-1)^{3/4} \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (\sqrt [4]{-1}-x\right )}-\frac {(-1)^{3/4} \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (\sqrt [4]{-1}+x\right )}\right ) \, dx\right )-2 i \int \left (-\frac {\sqrt [4]{-1} \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (-(-1)^{3/4}-x\right )}-\frac {\sqrt [4]{-1} \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (-(-1)^{3/4}+x\right )}\right ) \, dx-2 \int \left (-\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (i-x^2\right )}-\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (i+x^2\right )}\right ) \, dx+\int \sqrt {x^2+\sqrt {1+x^4}} \, dx-\int \left (\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{4 \left (-(-1)^{3/4}-x\right )^2}+\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{4 \left (-(-1)^{3/4}+x\right )^2}+\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (-i-x^2\right )}\right ) \, dx-\int \left (-\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{4 \left (\sqrt [4]{-1}-x\right )^2}-\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{4 \left (\sqrt [4]{-1}+x\right )^2}-\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (i-x^2\right )}\right ) \, dx \\ & = \frac {1}{4} i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (\sqrt [4]{-1}-x\right )^2} \, dx-\frac {1}{4} i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-(-1)^{3/4}-x\right )^2} \, dx+\frac {1}{4} i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (\sqrt [4]{-1}+x\right )^2} \, dx-\frac {1}{4} i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-(-1)^{3/4}+x\right )^2} \, dx-\frac {1}{2} i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{-i-x^2} \, dx+\frac {1}{2} i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{i-x^2} \, dx+i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{i-x^2} \, dx+i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{i+x^2} \, dx-\sqrt [4]{-1} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt [4]{-1}-x} \, dx-\sqrt [4]{-1} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt [4]{-1}+x} \, dx+(-1)^{3/4} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{-(-1)^{3/4}-x} \, dx+(-1)^{3/4} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{-(-1)^{3/4}+x} \, dx+\int \sqrt {x^2+\sqrt {1+x^4}} \, dx \\ & = \frac {1}{4} i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (\sqrt [4]{-1}-x\right )^2} \, dx-\frac {1}{4} i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-(-1)^{3/4}-x\right )^2} \, dx+\frac {1}{4} i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (\sqrt [4]{-1}+x\right )^2} \, dx-\frac {1}{4} i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-(-1)^{3/4}+x\right )^2} \, dx+\frac {1}{2} i \int \left (-\frac {(-1)^{3/4} \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (\sqrt [4]{-1}-x\right )}-\frac {(-1)^{3/4} \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (\sqrt [4]{-1}+x\right )}\right ) \, dx-\frac {1}{2} i \int \left (\frac {\sqrt [4]{-1} \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (-(-1)^{3/4}-x\right )}+\frac {\sqrt [4]{-1} \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (-(-1)^{3/4}+x\right )}\right ) \, dx+i \int \left (-\frac {(-1)^{3/4} \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (\sqrt [4]{-1}-x\right )}-\frac {(-1)^{3/4} \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (\sqrt [4]{-1}+x\right )}\right ) \, dx+i \int \left (-\frac {\sqrt [4]{-1} \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (-(-1)^{3/4}-x\right )}-\frac {\sqrt [4]{-1} \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (-(-1)^{3/4}+x\right )}\right ) \, dx-\sqrt [4]{-1} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt [4]{-1}-x} \, dx-\sqrt [4]{-1} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt [4]{-1}+x} \, dx+(-1)^{3/4} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{-(-1)^{3/4}-x} \, dx+(-1)^{3/4} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{-(-1)^{3/4}+x} \, dx+\int \sqrt {x^2+\sqrt {1+x^4}} \, dx \\ & = \frac {1}{4} i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (\sqrt [4]{-1}-x\right )^2} \, dx-\frac {1}{4} i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-(-1)^{3/4}-x\right )^2} \, dx+\frac {1}{4} i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (\sqrt [4]{-1}+x\right )^2} \, dx-\frac {1}{4} i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-(-1)^{3/4}+x\right )^2} \, dx+\frac {1}{4} \sqrt [4]{-1} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt [4]{-1}-x} \, dx+\frac {1}{4} \sqrt [4]{-1} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt [4]{-1}+x} \, dx+\frac {1}{2} \sqrt [4]{-1} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt [4]{-1}-x} \, dx+\frac {1}{2} \sqrt [4]{-1} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt [4]{-1}+x} \, dx-\sqrt [4]{-1} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt [4]{-1}-x} \, dx-\sqrt [4]{-1} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt [4]{-1}+x} \, dx-\frac {1}{4} (-1)^{3/4} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{-(-1)^{3/4}-x} \, dx-\frac {1}{4} (-1)^{3/4} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{-(-1)^{3/4}+x} \, dx-\frac {1}{2} (-1)^{3/4} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{-(-1)^{3/4}-x} \, dx-\frac {1}{2} (-1)^{3/4} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{-(-1)^{3/4}+x} \, dx+(-1)^{3/4} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{-(-1)^{3/4}-x} \, dx+(-1)^{3/4} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{-(-1)^{3/4}+x} \, dx+\int \sqrt {x^2+\sqrt {1+x^4}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.87 \[ \int \frac {\left (-1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^4\right )^2} \, dx=\frac {1}{4} \left (\frac {2 x \sqrt {x^2+\sqrt {1+x^4}} \left (4+8 x^4+2 x^8+7 x^2 \sqrt {1+x^4}+2 x^6 \sqrt {1+x^4}\right )}{\left (1+x^4\right ) \left (1+2 x^4+2 x^2 \sqrt {1+x^4}\right )}-6 \arctan \left (x \sqrt {x^2+\sqrt {1+x^4}}\right )+\sqrt {2} \arctan \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}\right )\right ) \]

[In]

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

[Out]

((2*x*Sqrt[x^2 + Sqrt[1 + x^4]]*(4 + 8*x^4 + 2*x^8 + 7*x^2*Sqrt[1 + x^4] + 2*x^6*Sqrt[1 + x^4]))/((1 + x^4)*(1
+ 2*x^4 + 2*x^2*Sqrt[1 + x^4])) - 6*ArcTan[x*Sqrt[x^2 + Sqrt[1 + x^4]]] + Sqrt[2]*ArcTan[Sqrt[2]*x*Sqrt[x^2 +
Sqrt[1 + x^4]]])/4

Maple [F]

\[\int \frac {\left (x^{4}-1\right )^{2} \sqrt {x^{2}+\sqrt {x^{4}+1}}}{\left (x^{4}+1\right )^{2}}d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.98 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.96 \[ \int \frac {\left (-1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^4\right )^2} \, dx=-\frac {2 \, \sqrt {2} {\left (x^{4} + 1\right )} \arctan \left (-\frac {{\left (\sqrt {2} x^{2} - \sqrt {2} \sqrt {x^{4} + 1}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{2 \, x}\right ) - 3 \, {\left (x^{4} + 1\right )} \arctan \left (-\frac {4 \, {\left (3 \, x^{9} - 12 \, x^{5} - {\left (3 \, x^{7} - 5 \, x^{3}\right )} \sqrt {x^{4} + 1} + x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{17 \, x^{8} - 46 \, x^{4} + 1}\right ) - 4 \, {\left (2 \, x^{5} - \sqrt {x^{4} + 1} x^{3} + 4 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{8 \, {\left (x^{4} + 1\right )}} \]

[In]

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

[Out]

-1/8*(2*sqrt(2)*(x^4 + 1)*arctan(-1/2*(sqrt(2)*x^2 - sqrt(2)*sqrt(x^4 + 1))*sqrt(x^2 + sqrt(x^4 + 1))/x) - 3*(
x^4 + 1)*arctan(-4*(3*x^9 - 12*x^5 - (3*x^7 - 5*x^3)*sqrt(x^4 + 1) + x)*sqrt(x^2 + sqrt(x^4 + 1))/(17*x^8 - 46
*x^4 + 1)) - 4*(2*x^5 - sqrt(x^4 + 1)*x^3 + 4*x)*sqrt(x^2 + sqrt(x^4 + 1)))/(x^4 + 1)

Sympy [F]

\[ \int \frac {\left (-1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^4\right )^2} \, dx=\int \frac {\left (x - 1\right )^{2} \left (x + 1\right )^{2} \left (x^{2} + 1\right )^{2} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{\left (x^{4} + 1\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\left (-1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^4\right )^2} \, dx=\int { \frac {{\left (x^{4} - 1\right )}^{2} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{{\left (x^{4} + 1\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\left (-1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^4\right )^2} \, dx=\int { \frac {{\left (x^{4} - 1\right )}^{2} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{{\left (x^{4} + 1\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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