Integrand size = 26, antiderivative size = 129 \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x}}}{1+x^4} \, dx=\frac {4}{15} \sqrt {1+x} \sqrt {1+\sqrt {1+x}}+\frac {4}{15} (1+3 x) \sqrt {1+\sqrt {1+x}}-\frac {1}{2} \text {RootSum}\left [1+16 \text {$\#$1}^8-32 \text {$\#$1}^{10}+24 \text {$\#$1}^{12}-8 \text {$\#$1}^{14}+\text {$\#$1}^{16}\&,\frac {\log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right )}{-8 \text {$\#$1}^5+12 \text {$\#$1}^7-6 \text {$\#$1}^9+\text {$\#$1}^{11}}\&\right ] \] Output:
Unintegrable
Time = 0.00 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.88 \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x}}}{1+x^4} \, dx=\frac {4}{15} \sqrt {1+\sqrt {1+x}} \left (-2+\sqrt {1+x}+3 (1+x)\right )-\frac {1}{2} \text {RootSum}\left [1+16 \text {$\#$1}^8-32 \text {$\#$1}^{10}+24 \text {$\#$1}^{12}-8 \text {$\#$1}^{14}+\text {$\#$1}^{16}\&,\frac {\log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right )}{-8 \text {$\#$1}^5+12 \text {$\#$1}^7-6 \text {$\#$1}^9+\text {$\#$1}^{11}}\&\right ] \] Input:
Integrate[((-1 + x^4)*Sqrt[1 + Sqrt[1 + x]])/(1 + x^4),x]
Output:
(4*Sqrt[1 + Sqrt[1 + x]]*(-2 + Sqrt[1 + x] + 3*(1 + x)))/15 - RootSum[1 + 16*#1^8 - 32*#1^10 + 24*#1^12 - 8*#1^14 + #1^16 & , Log[Sqrt[1 + Sqrt[1 + x]] - #1]/(-8*#1^5 + 12*#1^7 - 6*#1^9 + #1^11) & ]/2
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (x^4-1\right ) \sqrt {\sqrt {x+1}+1}}{x^4+1} \, dx\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle 2 \int -\frac {\sqrt {x+1} \left (1-x^4\right ) \sqrt {\sqrt {x+1}+1}}{x^4+1}d\sqrt {x+1}\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle 2 \int -\frac {(x+1)^{3/2} \sqrt {\sqrt {x+1}+1} \left (-(x+1)^3+4 (x+1)^2-6 (x+1)+4\right )}{x^4+1}d\sqrt {x+1}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \int \frac {(x+1)^{3/2} \sqrt {\sqrt {x+1}+1} \left (-(x+1)^3+4 (x+1)^2-6 (x+1)+4\right )}{x^4+1}d\sqrt {x+1}\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle -4 \int \frac {x^3 (x+1) \left (-x^6+4 x^4-6 x^2+4\right )}{(1-x)^4 (x+1)^4+1}d\sqrt {\sqrt {x+1}+1}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 4 \int -\frac {x^3 (x+1) \left (-x^6+4 x^4-6 x^2+4\right )}{(1-x)^4 (x+1)^4+1}d\sqrt {\sqrt {x+1}+1}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left ((x+1)^2-\frac {2 x (x+1)}{(1-x)^4 (x+1)^4+1}-x-1\right )d\sqrt {\sqrt {x+1}+1}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -4 \left (-2 \int \frac {x+1}{(x+1)^8-8 (x+1)^7+24 (x+1)^6-32 (x+1)^5+16 (x+1)^4+1}d\sqrt {\sqrt {x+1}+1}+2 \int \frac {(x+1)^2}{(x+1)^8-8 (x+1)^7+24 (x+1)^6-32 (x+1)^5+16 (x+1)^4+1}d\sqrt {\sqrt {x+1}+1}-\frac {1}{5} (x+1)^{5/2}+\frac {1}{3} (x+1)^{3/2}\right )\) |
Input:
Int[((-1 + x^4)*Sqrt[1 + Sqrt[1 + x]])/(1 + x^4),x]
Output:
$Aborted
Time = 0.09 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.81
| method | result | size |
| derivativedivides | \(\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {5}{2}}}{5}-\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {3}{2}}}{3}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{16}-8 \textit {\_Z}^{14}+24 \textit {\_Z}^{12}-32 \textit {\_Z}^{10}+16 \textit {\_Z}^{8}+1\right )}{\sum }\frac {\left (\textit {\_R}^{4}-\textit {\_R}^{2}\right ) \ln \left (\sqrt {1+\sqrt {1+x}}-\textit {\_R} \right )}{\textit {\_R}^{15}-7 \textit {\_R}^{13}+18 \textit {\_R}^{11}-20 \textit {\_R}^{9}+8 \textit {\_R}^{7}}\right )}{2}\) | \(105\) |
| default | \(\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {5}{2}}}{5}-\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {3}{2}}}{3}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{16}-8 \textit {\_Z}^{14}+24 \textit {\_Z}^{12}-32 \textit {\_Z}^{10}+16 \textit {\_Z}^{8}+1\right )}{\sum }\frac {\left (\textit {\_R}^{4}-\textit {\_R}^{2}\right ) \ln \left (\sqrt {1+\sqrt {1+x}}-\textit {\_R} \right )}{\textit {\_R}^{15}-7 \textit {\_R}^{13}+18 \textit {\_R}^{11}-20 \textit {\_R}^{9}+8 \textit {\_R}^{7}}\right )}{2}\) | \(105\) |
Input:
int((x^4-1)*(1+(1+x)^(1/2))^(1/2)/(x^4+1),x,method=_RETURNVERBOSE)
Output:
4/5*(1+(1+x)^(1/2))^(5/2)-4/3*(1+(1+x)^(1/2))^(3/2)-1/2*sum((_R^4-_R^2)/(_ R^15-7*_R^13+18*_R^11-20*_R^9+8*_R^7)*ln((1+(1+x)^(1/2))^(1/2)-_R),_R=Root Of(_Z^16-8*_Z^14+24*_Z^12-32*_Z^10+16*_Z^8+1))
Timed out. \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x}}}{1+x^4} \, dx=\text {Timed out} \] Input:
integrate((x^4-1)*(1+(1+x)^(1/2))^(1/2)/(x^4+1),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x}}}{1+x^4} \, dx=\text {Timed out} \] Input:
integrate((x**4-1)*(1+(1+x)**(1/2))**(1/2)/(x**4+1),x)
Output:
Timed out
Not integrable
Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.19 \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x}}}{1+x^4} \, dx=\int { \frac {{\left (x^{4} - 1\right )} \sqrt {\sqrt {x + 1} + 1}}{x^{4} + 1} \,d x } \] Input:
integrate((x^4-1)*(1+(1+x)^(1/2))^(1/2)/(x^4+1),x, algorithm="maxima")
Output:
integrate((x^4 - 1)*sqrt(sqrt(x + 1) + 1)/(x^4 + 1), x)
Exception generated. \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x}}}{1+x^4} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((x^4-1)*(1+(1+x)^(1/2))^(1/2)/(x^4+1),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Invalid _EXT in replace_ext Error: Bad Argument ValueDone
Not integrable
Time = 0.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.19 \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x}}}{1+x^4} \, dx=\int \frac {\left (x^4-1\right )\,\sqrt {\sqrt {x+1}+1}}{x^4+1} \,d x \] Input:
int(((x^4 - 1)*((x + 1)^(1/2) + 1)^(1/2))/(x^4 + 1),x)
Output:
int(((x^4 - 1)*((x + 1)^(1/2) + 1)^(1/2))/(x^4 + 1), x)
Not integrable
Time = 0.34 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.31 \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x}}}{1+x^4} \, dx=-\left (\int \frac {\sqrt {\sqrt {x +1}+1}}{x^{4}+1}d x \right )+\int \frac {\sqrt {\sqrt {x +1}+1}\, x^{4}}{x^{4}+1}d x \] Input:
int((x^4-1)*(1+(1+x)^(1/2))^(1/2)/(x^4+1),x)
Output:
- int(sqrt(sqrt(x + 1) + 1)/(x**4 + 1),x) + int((sqrt(sqrt(x + 1) + 1)*x* *4)/(x**4 + 1),x)