Integrand size = 33, antiderivative size = 130 \[ \int \frac {\sqrt {1+x} \left (-1+x^2\right )}{\left (1+x^2\right ) \sqrt {1+\sqrt {1+x}}} \, dx=-\frac {16}{15} \sqrt {1+x} \sqrt {1+\sqrt {1+x}}+\frac {4}{15} (11+3 x) \sqrt {1+\sqrt {1+x}}-\text {RootSum}\left [1+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right )+\log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-2 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ] \] Output:
Unintegrable
Time = 0.00 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {1+x} \left (-1+x^2\right )}{\left (1+x^2\right ) \sqrt {1+\sqrt {1+x}}} \, dx=\frac {4}{15} \sqrt {1+\sqrt {1+x}} \left (8-4 \sqrt {1+x}+3 (1+x)\right )-\text {RootSum}\left [1+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right )+\log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-2 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ] \] Input:
Integrate[(Sqrt[1 + x]*(-1 + x^2))/((1 + x^2)*Sqrt[1 + Sqrt[1 + x]]),x]
Output:
(4*Sqrt[1 + Sqrt[1 + x]]*(8 - 4*Sqrt[1 + x] + 3*(1 + x)))/15 - RootSum[1 + 4*#1^4 - 4*#1^6 + #1^8 & , (-Log[Sqrt[1 + Sqrt[1 + x]] - #1] + Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^2)/(-2*#1^3 + #1^5) & ]
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.
\(\displaystyle \int \frac {\sqrt {x+1} \left (x^2-1\right )}{\left (x^2+1\right ) \sqrt {\sqrt {x+1}+1}} \, dx\) |
\(\Big \downarrow \) 2003 |
\(\displaystyle \int \frac {(x-1) (x+1)^{3/2}}{\left (x^2+1\right ) \sqrt {\sqrt {x+1}+1}}dx\) |
\(\Big \downarrow \) 7267 |
\(\displaystyle 2 \int -\frac {(1-x) (x+1)^2}{\left (x^2+1\right ) \sqrt {\sqrt {x+1}+1}}d\sqrt {x+1}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -2 \int \frac {(1-x) (x+1)^2}{\left (x^2+1\right ) \sqrt {\sqrt {x+1}+1}}d\sqrt {x+1}\) |
\(\Big \downarrow \) 2091 |
\(\displaystyle -2 \int \frac {(1-x) (x+1)^2}{\sqrt {\sqrt {x+1}+1} \left ((x+1)^2-2 (x+1)+2\right )}d\sqrt {x+1}\) |
\(\Big \downarrow \) 7267 |
\(\displaystyle 4 \int -\frac {x^4 \left (-(x+1)^2+2 (x+1)+1\right )}{(x+1)^4-4 (x+1)^3+4 (x+1)^2+1}d\sqrt {\sqrt {x+1}+1}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -4 \int \frac {x^4 \left (-(x+1)^2+2 (x+1)+1\right )}{(x+1)^4-4 (x+1)^3+4 (x+1)^2+1}d\sqrt {\sqrt {x+1}+1}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -4 \int \left (-(x+1)^2+2 (x+1)+\frac {2 \left ((x+1)^2-2 (x+1)+1\right )}{(x+1)^4-4 (x+1)^3+4 (x+1)^2+1}-1\right )d\sqrt {\sqrt {x+1}+1}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 4 \left (-2 \int \frac {1}{(x+1)^4-4 (x+1)^3+4 (x+1)^2+1}d\sqrt {\sqrt {x+1}+1}+4 \int \frac {x+1}{(x+1)^4-4 (x+1)^3+4 (x+1)^2+1}d\sqrt {\sqrt {x+1}+1}-2 \int \frac {(x+1)^2}{(x+1)^4-4 (x+1)^3+4 (x+1)^2+1}d\sqrt {\sqrt {x+1}+1}+\frac {1}{5} (x+1)^{5/2}-\frac {2}{3} (x+1)^{3/2}+\sqrt {\sqrt {x+1}+1}\right )\) |
Input:
Int[(Sqrt[1 + x]*(-1 + x^2))/((1 + x^2)*Sqrt[1 + Sqrt[1 + x]]),x]
Output:
$Aborted
Time = 0.09 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.75
method | result | size |
derivativedivides | \(\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {5}{2}}}{5}-\frac {8 \left (1+\sqrt {1+x}\right )^{\frac {3}{2}}}{3}+4 \sqrt {1+\sqrt {1+x}}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+4 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (-\textit {\_R}^{4}+2 \textit {\_R}^{2}-1\right ) \ln \left (\sqrt {1+\sqrt {1+x}}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5}+2 \textit {\_R}^{3}}\right )\) | \(97\) |
default | \(\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {5}{2}}}{5}-\frac {8 \left (1+\sqrt {1+x}\right )^{\frac {3}{2}}}{3}+4 \sqrt {1+\sqrt {1+x}}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+4 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (-\textit {\_R}^{4}+2 \textit {\_R}^{2}-1\right ) \ln \left (\sqrt {1+\sqrt {1+x}}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5}+2 \textit {\_R}^{3}}\right )\) | \(97\) |
Input:
int((1+x)^(1/2)*(x^2-1)/(x^2+1)/(1+(1+x)^(1/2))^(1/2),x,method=_RETURNVERB OSE)
Output:
4/5*(1+(1+x)^(1/2))^(5/2)-8/3*(1+(1+x)^(1/2))^(3/2)+4*(1+(1+x)^(1/2))^(1/2 )+sum((-_R^4+2*_R^2-1)/(_R^7-3*_R^5+2*_R^3)*ln((1+(1+x)^(1/2))^(1/2)-_R),_ R=RootOf(_Z^8-4*_Z^6+4*_Z^4+1))
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.90 (sec) , antiderivative size = 1268, normalized size of antiderivative = 9.75 \[ \int \frac {\sqrt {1+x} \left (-1+x^2\right )}{\left (1+x^2\right ) \sqrt {1+\sqrt {1+x}}} \, dx=\text {Too large to display} \] Input:
integrate((1+x)^(1/2)*(x^2-1)/(x^2+1)/(1+(1+x)^(1/2))^(1/2),x, algorithm=" fricas")
Output:
4/15*(3*x - 4*sqrt(x + 1) + 11)*sqrt(sqrt(x + 1) + 1) + 1/2*sqrt(sqrt(8*I + 8) + sqrt(-8*I + 8) + 2*sqrt(-3/4*(sqrt(8*I + 8) - 2*I - 2)^2 - 1/2*(sqr t(8*I + 8) - 2*I - 2)*(sqrt(-8*I + 8) + 2*I + 6) - 3/4*(sqrt(-8*I + 8) + 2 *I - 2)^2 - 4*sqrt(-8*I + 8) - 8*I + 8) + 4)*log(1/8*((sqrt(8*I + 8) - 2*I - 2)^2*(3*sqrt(-8*I + 8) + 6*I - 10) + (3*(sqrt(-8*I + 8) + 2*I - 2)^2 + 24*sqrt(-8*I + 8) + 48*I - 88)*(sqrt(8*I + 8) - 2*I - 2) - 4*(sqrt(-8*I + 8) + 2*I - 2)^2 - 2*sqrt(-3/4*(sqrt(8*I + 8) - 2*I - 2)^2 - 1/2*(sqrt(8*I + 8) - 2*I - 2)*(sqrt(-8*I + 8) + 2*I + 6) - 3/4*(sqrt(-8*I + 8) + 2*I - 2 )^2 - 4*sqrt(-8*I + 8) - 8*I + 8)*((sqrt(8*I + 8) - 2*I - 2)*(3*sqrt(-8*I + 8) + 6*I - 10) - 4*sqrt(-8*I + 8) - 8*I + 16) - 40*sqrt(-8*I + 8) - 80*I + 80)*sqrt(sqrt(8*I + 8) + sqrt(-8*I + 8) + 2*sqrt(-3/4*(sqrt(8*I + 8) - 2*I - 2)^2 - 1/2*(sqrt(8*I + 8) - 2*I - 2)*(sqrt(-8*I + 8) + 2*I + 6) - 3/ 4*(sqrt(-8*I + 8) + 2*I - 2)^2 - 4*sqrt(-8*I + 8) - 8*I + 8) + 4) + 32*sqr t(sqrt(x + 1) + 1)) - 1/2*sqrt(sqrt(8*I + 8) + sqrt(-8*I + 8) + 2*sqrt(-3/ 4*(sqrt(8*I + 8) - 2*I - 2)^2 - 1/2*(sqrt(8*I + 8) - 2*I - 2)*(sqrt(-8*I + 8) + 2*I + 6) - 3/4*(sqrt(-8*I + 8) + 2*I - 2)^2 - 4*sqrt(-8*I + 8) - 8*I + 8) + 4)*log(-1/8*((sqrt(8*I + 8) - 2*I - 2)^2*(3*sqrt(-8*I + 8) + 6*I - 10) + (3*(sqrt(-8*I + 8) + 2*I - 2)^2 + 24*sqrt(-8*I + 8) + 48*I - 88)*(s qrt(8*I + 8) - 2*I - 2) - 4*(sqrt(-8*I + 8) + 2*I - 2)^2 - 2*sqrt(-3/4*(sq rt(8*I + 8) - 2*I - 2)^2 - 1/2*(sqrt(8*I + 8) - 2*I - 2)*(sqrt(-8*I + 8...
Timed out. \[ \int \frac {\sqrt {1+x} \left (-1+x^2\right )}{\left (1+x^2\right ) \sqrt {1+\sqrt {1+x}}} \, dx=\text {Timed out} \] Input:
integrate((1+x)**(1/2)*(x**2-1)/(x**2+1)/(1+(1+x)**(1/2))**(1/2),x)
Output:
Timed out
Not integrable
Time = 0.33 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.22 \[ \int \frac {\sqrt {1+x} \left (-1+x^2\right )}{\left (1+x^2\right ) \sqrt {1+\sqrt {1+x}}} \, dx=\int { \frac {{\left (x^{2} - 1\right )} \sqrt {x + 1}}{{\left (x^{2} + 1\right )} \sqrt {\sqrt {x + 1} + 1}} \,d x } \] Input:
integrate((1+x)^(1/2)*(x^2-1)/(x^2+1)/(1+(1+x)^(1/2))^(1/2),x, algorithm=" maxima")
Output:
integrate((x^2 - 1)*sqrt(x + 1)/((x^2 + 1)*sqrt(sqrt(x + 1) + 1)), x)
Exception generated. \[ \int \frac {\sqrt {1+x} \left (-1+x^2\right )}{\left (1+x^2\right ) \sqrt {1+\sqrt {1+x}}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((1+x)^(1/2)*(x^2-1)/(x^2+1)/(1+(1+x)^(1/2))^(1/2),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 ValueInvalid _EXT in replace_ext Error: Bad Argument ValueDon e
Not integrable
Time = 0.00 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.22 \[ \int \frac {\sqrt {1+x} \left (-1+x^2\right )}{\left (1+x^2\right ) \sqrt {1+\sqrt {1+x}}} \, dx=\int \frac {\left (x^2-1\right )\,\sqrt {x+1}}{\left (x^2+1\right )\,\sqrt {\sqrt {x+1}+1}} \,d x \] Input:
int(((x^2 - 1)*(x + 1)^(1/2))/((x^2 + 1)*((x + 1)^(1/2) + 1)^(1/2)),x)
Output:
int(((x^2 - 1)*(x + 1)^(1/2))/((x^2 + 1)*((x + 1)^(1/2) + 1)^(1/2)), x)
Not integrable
Time = 0.43 (sec) , antiderivative size = 280, normalized size of antiderivative = 2.15 \[ \int \frac {\sqrt {1+x} \left (-1+x^2\right )}{\left (1+x^2\right ) \sqrt {1+\sqrt {1+x}}} \, dx=-4 \sqrt {x +1}\, \sqrt {\sqrt {x +1}+1}-3 \left (\int \frac {\sqrt {\sqrt {x +1}+1}}{x^{4}+x^{3}+x^{2}+x}d x \right )-3 \left (\int \frac {\sqrt {\sqrt {x +1}+1}}{x^{3}+x^{2}+x +1}d x \right )+\int \frac {\sqrt {\sqrt {x +1}+1}\, x^{3}}{x^{3}+x^{2}+x +1}d x +\int \frac {\sqrt {\sqrt {x +1}+1}\, x^{2}}{x^{3}+x^{2}+x +1}d x -2 \left (\int \frac {\sqrt {\sqrt {x +1}+1}\, x}{x^{3}+x^{2}+x +1}d x \right )+2 \left (\int \frac {\sqrt {x +1}\, \sqrt {\sqrt {x +1}+1}\, x^{2}}{x^{3}+x^{2}+x +1}d x \right )+\int \frac {\sqrt {x +1}\, \sqrt {\sqrt {x +1}+1}\, x}{x^{3}+x^{2}+x +1}d x +3 \left (\int \frac {\sqrt {x +1}\, \sqrt {\sqrt {x +1}+1}}{x^{4}+x^{3}+x^{2}+x}d x \right )+4 \left (\int \frac {\sqrt {x +1}\, \sqrt {\sqrt {x +1}+1}}{x^{3}+x^{2}+x +1}d x \right )-\mathrm {log}\left (-2 \sqrt {\sqrt {x +1}+1}+\sqrt {x +1}+2\right )+\mathrm {log}\left (2 \sqrt {\sqrt {x +1}+1}+\sqrt {x +1}+2\right ) \] Input:
int((1+x)^(1/2)*(x^2-1)/(x^2+1)/(1+(1+x)^(1/2))^(1/2),x)
Output:
- 4*sqrt(x + 1)*sqrt(sqrt(x + 1) + 1) - 3*int(sqrt(sqrt(x + 1) + 1)/(x**4 + x**3 + x**2 + x),x) - 3*int(sqrt(sqrt(x + 1) + 1)/(x**3 + x**2 + x + 1) ,x) + int((sqrt(sqrt(x + 1) + 1)*x**3)/(x**3 + x**2 + x + 1),x) + int((sqr t(sqrt(x + 1) + 1)*x**2)/(x**3 + x**2 + x + 1),x) - 2*int((sqrt(sqrt(x + 1 ) + 1)*x)/(x**3 + x**2 + x + 1),x) + 2*int((sqrt(x + 1)*sqrt(sqrt(x + 1) + 1)*x**2)/(x**3 + x**2 + x + 1),x) + int((sqrt(x + 1)*sqrt(sqrt(x + 1) + 1 )*x)/(x**3 + x**2 + x + 1),x) + 3*int((sqrt(x + 1)*sqrt(sqrt(x + 1) + 1))/ (x**4 + x**3 + x**2 + x),x) + 4*int((sqrt(x + 1)*sqrt(sqrt(x + 1) + 1))/(x **3 + x**2 + x + 1),x) - log( - 2*sqrt(sqrt(x + 1) + 1) + sqrt(x + 1) + 2) + log(2*sqrt(sqrt(x + 1) + 1) + sqrt(x + 1) + 2)