Integrand size = 30, antiderivative size = 86 \[ \int \frac {\sqrt {x+\sqrt {1+x^2}}}{(1+x) \sqrt {1+x^2}} \, dx=\sqrt {2 \left (1+\sqrt {2}\right )} \arctan \left (\sqrt {-1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right )-\sqrt {2 \left (-1+\sqrt {2}\right )} \text {arctanh}\left (\sqrt {1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right ) \] Output:
(2+2*2^(1/2))^(1/2)*arctan((2^(1/2)-1)^(1/2)*(x+(x^2+1)^(1/2))^(1/2))-(-2+ 2*2^(1/2))^(1/2)*arctanh((1+2^(1/2))^(1/2)*(x+(x^2+1)^(1/2))^(1/2))
Time = 0.25 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {x+\sqrt {1+x^2}}}{(1+x) \sqrt {1+x^2}} \, dx=\sqrt {2 \left (1+\sqrt {2}\right )} \arctan \left (\sqrt {-1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right )-\sqrt {2 \left (-1+\sqrt {2}\right )} \text {arctanh}\left (\sqrt {1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right ) \] Input:
Integrate[Sqrt[x + Sqrt[1 + x^2]]/((1 + x)*Sqrt[1 + x^2]),x]
Output:
Sqrt[2*(1 + Sqrt[2])]*ArcTan[Sqrt[-1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]] - Sqrt[2*(-1 + Sqrt[2])]*ArcTanh[Sqrt[1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^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.
\(\displaystyle \int \frac {\sqrt {\sqrt {x^2+1}+x}}{(x+1) \sqrt {x^2+1}} \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \frac {\sqrt {\sqrt {x^2+1}+x}}{(x+1) \sqrt {x^2+1}}dx\) |
Input:
Int[Sqrt[x + Sqrt[1 + x^2]]/((1 + x)*Sqrt[1 + x^2]),x]
Output:
$Aborted
\[\int \frac {\sqrt {x +\sqrt {x^{2}+1}}}{\left (1+x \right ) \sqrt {x^{2}+1}}d x\]
Input:
int((x+(x^2+1)^(1/2))^(1/2)/(1+x)/(x^2+1)^(1/2),x)
Output:
int((x+(x^2+1)^(1/2))^(1/2)/(1+x)/(x^2+1)^(1/2),x)
Time = 0.09 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.31 \[ \int \frac {\sqrt {x+\sqrt {1+x^2}}}{(1+x) \sqrt {1+x^2}} \, dx=-2 \, \sqrt {\frac {1}{2} \, \sqrt {2} + \frac {1}{2}} \arctan \left (\sqrt {x + \sqrt {x^{2} + 1}} {\left (\sqrt {2} - 2\right )} \sqrt {\frac {1}{2} \, \sqrt {2} + \frac {1}{2}}\right ) - \sqrt {\frac {1}{2} \, \sqrt {2} - \frac {1}{2}} \log \left (\sqrt {2} \sqrt {\frac {1}{2} \, \sqrt {2} - \frac {1}{2}} + \sqrt {x + \sqrt {x^{2} + 1}}\right ) + \sqrt {\frac {1}{2} \, \sqrt {2} - \frac {1}{2}} \log \left (-\sqrt {2} \sqrt {\frac {1}{2} \, \sqrt {2} - \frac {1}{2}} + \sqrt {x + \sqrt {x^{2} + 1}}\right ) \] Input:
integrate((x+(x^2+1)^(1/2))^(1/2)/(1+x)/(x^2+1)^(1/2),x, algorithm="fricas ")
Output:
-2*sqrt(1/2*sqrt(2) + 1/2)*arctan(sqrt(x + sqrt(x^2 + 1))*(sqrt(2) - 2)*sq rt(1/2*sqrt(2) + 1/2)) - sqrt(1/2*sqrt(2) - 1/2)*log(sqrt(2)*sqrt(1/2*sqrt (2) - 1/2) + sqrt(x + sqrt(x^2 + 1))) + sqrt(1/2*sqrt(2) - 1/2)*log(-sqrt( 2)*sqrt(1/2*sqrt(2) - 1/2) + sqrt(x + sqrt(x^2 + 1)))
\[ \int \frac {\sqrt {x+\sqrt {1+x^2}}}{(1+x) \sqrt {1+x^2}} \, dx=\int \frac {\sqrt {x + \sqrt {x^{2} + 1}}}{\left (x + 1\right ) \sqrt {x^{2} + 1}}\, dx \] Input:
integrate((x+(x**2+1)**(1/2))**(1/2)/(1+x)/(x**2+1)**(1/2),x)
Output:
Integral(sqrt(x + sqrt(x**2 + 1))/((x + 1)*sqrt(x**2 + 1)), x)
\[ \int \frac {\sqrt {x+\sqrt {1+x^2}}}{(1+x) \sqrt {1+x^2}} \, dx=\int { \frac {\sqrt {x + \sqrt {x^{2} + 1}}}{\sqrt {x^{2} + 1} {\left (x + 1\right )}} \,d x } \] Input:
integrate((x+(x^2+1)^(1/2))^(1/2)/(1+x)/(x^2+1)^(1/2),x, algorithm="maxima ")
Output:
integrate(sqrt(x + sqrt(x^2 + 1))/(sqrt(x^2 + 1)*(x + 1)), x)
\[ \int \frac {\sqrt {x+\sqrt {1+x^2}}}{(1+x) \sqrt {1+x^2}} \, dx=\int { \frac {\sqrt {x + \sqrt {x^{2} + 1}}}{\sqrt {x^{2} + 1} {\left (x + 1\right )}} \,d x } \] Input:
integrate((x+(x^2+1)^(1/2))^(1/2)/(1+x)/(x^2+1)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(x + sqrt(x^2 + 1))/(sqrt(x^2 + 1)*(x + 1)), x)
Timed out. \[ \int \frac {\sqrt {x+\sqrt {1+x^2}}}{(1+x) \sqrt {1+x^2}} \, dx=\int \frac {\sqrt {x+\sqrt {x^2+1}}}{\sqrt {x^2+1}\,\left (x+1\right )} \,d x \] Input:
int((x + (x^2 + 1)^(1/2))^(1/2)/((x^2 + 1)^(1/2)*(x + 1)),x)
Output:
int((x + (x^2 + 1)^(1/2))^(1/2)/((x^2 + 1)^(1/2)*(x + 1)), x)
Time = 0.18 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.56 \[ \int \frac {\sqrt {x+\sqrt {1+x^2}}}{(1+x) \sqrt {1+x^2}} \, dx=\frac {\sqrt {2}\, \left (-\sqrt {\sqrt {2}+1}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {2}+1}\, \sqrt {\sqrt {x^{2}+1}+x}\, \sqrt {x^{2}+1}}{2}-\frac {\sqrt {\sqrt {2}+1}\, \sqrt {\sqrt {x^{2}+1}+x}\, \sqrt {2}}{2}-\frac {\sqrt {\sqrt {2}+1}\, \sqrt {\sqrt {x^{2}+1}+x}\, x}{2}+\frac {\sqrt {\sqrt {2}+1}\, \sqrt {\sqrt {x^{2}+1}+x}}{2}\right )+\sqrt {\sqrt {2}-1}\, \mathrm {log}\left (\sqrt {\sqrt {x^{2}+1}+x}-\sqrt {\sqrt {2}-1}\right )-\sqrt {\sqrt {2}-1}\, \mathrm {log}\left (\sqrt {\sqrt {x^{2}+1}+x}+\sqrt {\sqrt {2}-1}\right )\right )}{2} \] Input:
int((x+(x^2+1)^(1/2))^(1/2)/(1+x)/(x^2+1)^(1/2),x)
Output:
(sqrt(2)*( - sqrt(sqrt(2) + 1)*atan((sqrt(sqrt(2) + 1)*sqrt(sqrt(x**2 + 1) + x)*sqrt(x**2 + 1) - sqrt(sqrt(2) + 1)*sqrt(sqrt(x**2 + 1) + x)*sqrt(2) - sqrt(sqrt(2) + 1)*sqrt(sqrt(x**2 + 1) + x)*x + sqrt(sqrt(2) + 1)*sqrt(sq rt(x**2 + 1) + x))/2) + sqrt(sqrt(2) - 1)*log(sqrt(sqrt(x**2 + 1) + x) - s qrt(sqrt(2) - 1)) - sqrt(sqrt(2) - 1)*log(sqrt(sqrt(x**2 + 1) + x) + sqrt( sqrt(2) - 1))))/2