\(\int \frac {(-1+x^4) \sqrt {1+\sqrt {1+x^2}}}{1+x^4} \, dx\) [1678]

Optimal result

Integrand size = 28, antiderivative size = 112 \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x^2}}}{1+x^4} \, dx=\frac {4 x}{3 \sqrt {1+\sqrt {1+x^2}}}+\frac {2 x \sqrt {1+x^2}}{3 \sqrt {1+\sqrt {1+x^2}}}-\frac {1}{2} \text {RootSum}\left [1+4 \text {$\#$1}^4+4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\frac {x}{\sqrt {1+\sqrt {1+x^2}}}-\text {$\#$1}\right )}{2 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ] \]

[Out]

Unintegrable

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.84 \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x^2}}}{1+x^4} \, dx=\frac {2 x \left (2+\sqrt {1+x^2}\right )}{3 \sqrt {1+\sqrt {1+x^2}}}-\frac {1}{2} \text {RootSum}\left [1+4 \text {$\#$1}^4+4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\frac {x}{\sqrt {1+\sqrt {1+x^2}}}-\text {$\#$1}\right )}{2 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ] \]

[In]

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

[Out]

(2*x*(2 + Sqrt[1 + x^2]))/(3*Sqrt[1 + Sqrt[1 + x^2]]) - RootSum[1 + 4*#1^4 + 4*#1^6 + #1^8 & , Log[x/Sqrt[1 +
Sqrt[1 + x^2]] - #1]/(2*#1^3 + #1^5) & ]/2

Maple [N/A] (verified)

Not integrable

Time = 0.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.21

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

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 9.66 (sec) , antiderivative size = 6296, normalized size of antiderivative = 56.21 \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x^2}}}{1+x^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [N/A]

Not integrable

Time = 10.64 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.28 \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x^2}}}{1+x^4} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \sqrt {\sqrt {x^{2} + 1} + 1}}{x^{4} + 1}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.23 \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x^2}}}{1+x^4} \, dx=\int { \frac {{\left (x^{4} - 1\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}{x^{4} + 1} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.23 \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x^2}}}{1+x^4} \, dx=\int { \frac {{\left (x^{4} - 1\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}{x^{4} + 1} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.23 \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x^2}}}{1+x^4} \, dx=\int \frac {\left (x^4-1\right )\,\sqrt {\sqrt {x^2+1}+1}}{x^4+1} \,d x \]

[In]

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

[Out]

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