Integrand size = 33, antiderivative size = 161 \[ \int \frac {\sqrt {1+x} \left (-1+x^4\right ) \sqrt {1+\sqrt {1+x}}}{1+x^4} \, dx=\frac {4}{105} (11+3 x) \sqrt {1+\sqrt {1+x}}+\frac {4}{105} \sqrt {1+x} (11+15 x) \sqrt {1+\sqrt {1+x}}-\frac {1}{2} \text {RootSum}\left [1+16 \text {$\#$1}^8-32 \text {$\#$1}^{10}+24 \text {$\#$1}^{12}-8 \text {$\#$1}^{14}+\text {$\#$1}^{16}\&,\frac {-\log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right )+\log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-8 \text {$\#$1}^5+12 \text {$\#$1}^7-6 \text {$\#$1}^9+\text {$\#$1}^{11}}\&\right ] \] Output:
Unintegrable
Time = 0.17 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {1+x} \left (-1+x^4\right ) \sqrt {1+\sqrt {1+x}}}{1+x^4} \, dx=\frac {4}{105} \sqrt {1+\sqrt {1+x}} \left (8-4 \sqrt {1+x}+3 (1+x)+15 (1+x)^{3/2}\right )-\frac {1}{2} \text {RootSum}\left [1+16 \text {$\#$1}^8-32 \text {$\#$1}^{10}+24 \text {$\#$1}^{12}-8 \text {$\#$1}^{14}+\text {$\#$1}^{16}\&,\frac {-\log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right )+\log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-8 \text {$\#$1}^5+12 \text {$\#$1}^7-6 \text {$\#$1}^9+\text {$\#$1}^{11}}\&\right ] \] Input:
Integrate[(Sqrt[1 + x]*(-1 + x^4)*Sqrt[1 + Sqrt[1 + x]])/(1 + x^4),x]
Output:
(4*Sqrt[1 + Sqrt[1 + x]]*(8 - 4*Sqrt[1 + x] + 3*(1 + x) + 15*(1 + x)^(3/2) ))/105 - RootSum[1 + 16*#1^8 - 32*#1^10 + 24*#1^12 - 8*#1^14 + #1^16 & , ( -Log[Sqrt[1 + Sqrt[1 + x]] - #1] + Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^2)/( -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.
| \(\displaystyle \int \frac {\sqrt {x+1} \left (x^4-1\right ) \sqrt {\sqrt {x+1}+1}}{x^4+1} \, dx\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle 2 \int -\frac {(x+1) \left (1-x^4\right ) \sqrt {\sqrt {x+1}+1}}{x^4+1}d\sqrt {x+1}\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle 2 \int -\frac {(x+1)^2 \sqrt {\sqrt {x+1}+1} \left (-(x+1)^3+4 (x+1)^2-6 (x+1)+4\right )}{x^4+1}d\sqrt {x+1}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \int \frac {(x+1)^2 \sqrt {\sqrt {x+1}+1} \left (-(x+1)^3+4 (x+1)^2-6 (x+1)+4\right )}{x^4+1}d\sqrt {x+1}\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle -4 \int \frac {x^4 (x+1) \left (-x^6+4 x^4-6 x^2+4\right )}{(1-x)^4 (x+1)^4+1}d\sqrt {\sqrt {x+1}+1}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -4 \int \left (-(x+1)^3+2 (x+1)^2+\frac {2 \left ((x+1)^2-2 (x+1)+1\right ) (x+1)}{(1-x)^4 (x+1)^4+1}-x-1\right )d\sqrt {\sqrt {x+1}+1}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -4 \left (2 \int \frac {x+1}{(x+1)^8-8 (x+1)^7+24 (x+1)^6-32 (x+1)^5+16 (x+1)^4+1}d\sqrt {\sqrt {x+1}+1}-4 \int \frac {(x+1)^2}{(x+1)^8-8 (x+1)^7+24 (x+1)^6-32 (x+1)^5+16 (x+1)^4+1}d\sqrt {\sqrt {x+1}+1}+2 \int \frac {(x+1)^3}{(x+1)^8-8 (x+1)^7+24 (x+1)^6-32 (x+1)^5+16 (x+1)^4+1}d\sqrt {\sqrt {x+1}+1}-\frac {1}{7} (x+1)^{7/2}+\frac {2}{5} (x+1)^{5/2}-\frac {1}{3} (x+1)^{3/2}\right )\) |
Input:
Int[(Sqrt[1 + x]*(-1 + x^4)*Sqrt[1 + Sqrt[1 + x]])/(1 + x^4),x]
Output:
$Aborted
Time = 0.13 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.74
| method | result | size |
| derivativedivides | \(\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {7}{2}}}{7}-\frac {8 \left (1+\sqrt {1+x}\right )^{\frac {5}{2}}}{5}+\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {3}{2}}}{3}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{16}-8 \textit {\_Z}^{14}+24 \textit {\_Z}^{12}-32 \textit {\_Z}^{10}+16 \textit {\_Z}^{8}+1\right )}{\sum }\frac {\left (\textit {\_R}^{6}-2 \textit {\_R}^{4}+\textit {\_R}^{2}\right ) \ln \left (\sqrt {1+\sqrt {1+x}}-\textit {\_R} \right )}{\textit {\_R}^{15}-7 \textit {\_R}^{13}+18 \textit {\_R}^{11}-20 \textit {\_R}^{9}+8 \textit {\_R}^{7}}\right )}{2}\) | \(119\) |
| default | \(\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {7}{2}}}{7}-\frac {8 \left (1+\sqrt {1+x}\right )^{\frac {5}{2}}}{5}+\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {3}{2}}}{3}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{16}-8 \textit {\_Z}^{14}+24 \textit {\_Z}^{12}-32 \textit {\_Z}^{10}+16 \textit {\_Z}^{8}+1\right )}{\sum }\frac {\left (\textit {\_R}^{6}-2 \textit {\_R}^{4}+\textit {\_R}^{2}\right ) \ln \left (\sqrt {1+\sqrt {1+x}}-\textit {\_R} \right )}{\textit {\_R}^{15}-7 \textit {\_R}^{13}+18 \textit {\_R}^{11}-20 \textit {\_R}^{9}+8 \textit {\_R}^{7}}\right )}{2}\) | \(119\) |
Input:
int((1+x)^(1/2)*(x^4-1)*(1+(1+x)^(1/2))^(1/2)/(x^4+1),x,method=_RETURNVERB OSE)
Output:
4/7*(1+(1+x)^(1/2))^(7/2)-8/5*(1+(1+x)^(1/2))^(5/2)+4/3*(1+(1+x)^(1/2))^(3 /2)-1/2*sum((_R^6-2*_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=RootOf(_Z^16-8*_Z^14+24*_Z^12-32*_Z^10+16*_Z^ 8+1))
Timed out. \[ \int \frac {\sqrt {1+x} \left (-1+x^4\right ) \sqrt {1+\sqrt {1+x}}}{1+x^4} \, dx=\text {Timed out} \] Input:
integrate((1+x)^(1/2)*(x^4-1)*(1+(1+x)^(1/2))^(1/2)/(x^4+1),x, algorithm=" fricas")
Output:
Timed out
Timed out. \[ \int \frac {\sqrt {1+x} \left (-1+x^4\right ) \sqrt {1+\sqrt {1+x}}}{1+x^4} \, dx=\text {Timed out} \] Input:
integrate((1+x)**(1/2)*(x**4-1)*(1+(1+x)**(1/2))**(1/2)/(x**4+1),x)
Output:
Timed out
Not integrable
Time = 0.18 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.18 \[ \int \frac {\sqrt {1+x} \left (-1+x^4\right ) \sqrt {1+\sqrt {1+x}}}{1+x^4} \, dx=\int { \frac {{\left (x^{4} - 1\right )} \sqrt {x + 1} \sqrt {\sqrt {x + 1} + 1}}{x^{4} + 1} \,d x } \] Input:
integrate((1+x)^(1/2)*(x^4-1)*(1+(1+x)^(1/2))^(1/2)/(x^4+1),x, algorithm=" maxima")
Output:
integrate((x^4 - 1)*sqrt(x + 1)*sqrt(sqrt(x + 1) + 1)/(x^4 + 1), x)
Not integrable
Time = 0.63 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.18 \[ \int \frac {\sqrt {1+x} \left (-1+x^4\right ) \sqrt {1+\sqrt {1+x}}}{1+x^4} \, dx=\int { \frac {{\left (x^{4} - 1\right )} \sqrt {x + 1} \sqrt {\sqrt {x + 1} + 1}}{x^{4} + 1} \,d x } \] Input:
integrate((1+x)^(1/2)*(x^4-1)*(1+(1+x)^(1/2))^(1/2)/(x^4+1),x, algorithm=" giac")
Output:
integrate((x^4 - 1)*sqrt(x + 1)*sqrt(sqrt(x + 1) + 1)/(x^4 + 1), x)
Not integrable
Time = 8.43 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.18 \[ \int \frac {\sqrt {1+x} \left (-1+x^4\right ) \sqrt {1+\sqrt {1+x}}}{1+x^4} \, dx=\int \frac {\left (x^4-1\right )\,\sqrt {\sqrt {x+1}+1}\,\sqrt {x+1}}{x^4+1} \,d x \] Input:
int(((x^4 - 1)*((x + 1)^(1/2) + 1)^(1/2)*(x + 1)^(1/2))/(x^4 + 1),x)
Output:
int(((x^4 - 1)*((x + 1)^(1/2) + 1)^(1/2)*(x + 1)^(1/2))/(x^4 + 1), x)
Not integrable
Time = 2.23 (sec) , antiderivative size = 368, normalized size of antiderivative = 2.29 \[ \int \frac {\sqrt {1+x} \left (-1+x^4\right ) \sqrt {1+\sqrt {1+x}}}{1+x^4} \, dx=\frac {4 \sqrt {x +1}\, \sqrt {\sqrt {x +1}+1}\, x}{7}+\frac {8 \sqrt {x +1}\, \sqrt {\sqrt {x +1}+1}}{21}+\frac {\left (\int \frac {\sqrt {\sqrt {x +1}+1}}{x^{6}+x^{5}+x^{2}+x}d x \right )}{7}+\frac {5 \left (\int \frac {\sqrt {\sqrt {x +1}+1}}{x^{5}+x^{4}+x +1}d x \right )}{21}+\frac {\left (\int \frac {\sqrt {\sqrt {x +1}+1}\, x^{5}}{x^{5}+x^{4}+x +1}d x \right )}{7}+\frac {5 \left (\int \frac {\sqrt {\sqrt {x +1}+1}\, x^{4}}{x^{5}+x^{4}+x +1}d x \right )}{21}+\frac {\left (\int \frac {\sqrt {\sqrt {x +1}+1}\, x^{3}}{x^{5}+x^{4}+x +1}d x \right )}{7}+\frac {\left (\int \frac {\sqrt {\sqrt {x +1}+1}\, x}{x^{5}+x^{4}+x +1}d x \right )}{7}-\frac {\left (\int \frac {\sqrt {x +1}\, \sqrt {\sqrt {x +1}+1}\, x^{3}}{x^{5}+x^{4}+x +1}d x \right )}{7}-2 \left (\int \frac {\sqrt {x +1}\, \sqrt {\sqrt {x +1}+1}\, x}{x^{5}+x^{4}+x +1}d x \right )-\frac {\left (\int \frac {\sqrt {x +1}\, \sqrt {\sqrt {x +1}+1}}{x^{6}+x^{5}+x^{2}+x}d x \right )}{7}-2 \left (\int \frac {\sqrt {x +1}\, \sqrt {\sqrt {x +1}+1}}{x^{5}+x^{4}+x +1}d x \right )+\frac {4 \,\mathrm {log}\left (-2 \sqrt {\sqrt {x +1}+1}-\sqrt {x +1}-2\right )}{21}+\frac {5 \,\mathrm {log}\left (-2 \sqrt {\sqrt {x +1}+1}+\sqrt {x +1}+2\right )}{21}-\frac {4 \,\mathrm {log}\left (2 \sqrt {\sqrt {x +1}+1}-\sqrt {x +1}-2\right )}{21}-\frac {5 \,\mathrm {log}\left (2 \sqrt {\sqrt {x +1}+1}+\sqrt {x +1}+2\right )}{21} \] Input:
int((1+x)^(1/2)*(x^4-1)*(1+(1+x)^(1/2))^(1/2)/(x^4+1),x)
Output:
(12*sqrt(x + 1)*sqrt(sqrt(x + 1) + 1)*x + 8*sqrt(x + 1)*sqrt(sqrt(x + 1) + 1) + 3*int(sqrt(sqrt(x + 1) + 1)/(x**6 + x**5 + x**2 + x),x) + 5*int(sqrt (sqrt(x + 1) + 1)/(x**5 + x**4 + x + 1),x) + 3*int((sqrt(sqrt(x + 1) + 1)* x**5)/(x**5 + x**4 + x + 1),x) + 5*int((sqrt(sqrt(x + 1) + 1)*x**4)/(x**5 + x**4 + x + 1),x) + 3*int((sqrt(sqrt(x + 1) + 1)*x**3)/(x**5 + x**4 + x + 1),x) + 3*int((sqrt(sqrt(x + 1) + 1)*x)/(x**5 + x**4 + x + 1),x) - 3*int( (sqrt(x + 1)*sqrt(sqrt(x + 1) + 1)*x**3)/(x**5 + x**4 + x + 1),x) - 42*int ((sqrt(x + 1)*sqrt(sqrt(x + 1) + 1)*x)/(x**5 + x**4 + x + 1),x) - 3*int((s qrt(x + 1)*sqrt(sqrt(x + 1) + 1))/(x**6 + x**5 + x**2 + x),x) - 42*int((sq rt(x + 1)*sqrt(sqrt(x + 1) + 1))/(x**5 + x**4 + x + 1),x) + 4*log( - 2*sqr t(sqrt(x + 1) + 1) - sqrt(x + 1) - 2) + 5*log( - 2*sqrt(sqrt(x + 1) + 1) + sqrt(x + 1) + 2) - 4*log(2*sqrt(sqrt(x + 1) + 1) - sqrt(x + 1) - 2) - 5*l og(2*sqrt(sqrt(x + 1) + 1) + sqrt(x + 1) + 2))/21