Integrand size = 31, antiderivative size = 97 \[ \int \frac {x+\sqrt {1+x^2}}{1+\sqrt {x+\sqrt {1+x^2}}} \, dx=-\frac {x}{2}+\frac {1}{3} (3+x) \sqrt {x+\sqrt {1+x^2}}+\sqrt {1+x^2} \left (-\frac {1}{2}+\frac {1}{3} \sqrt {x+\sqrt {1+x^2}}\right )+\frac {1}{2} \log \left (x+\sqrt {1+x^2}\right )-2 \log \left (1+\sqrt {x+\sqrt {1+x^2}}\right ) \] Output:
-1/2*x+1/3*(3+x)*(x+(x^2+1)^(1/2))^(1/2)+(x^2+1)^(1/2)*(-1/2+1/3*(x+(x^2+1 )^(1/2))^(1/2))+1/2*ln(x+(x^2+1)^(1/2))-2*ln(1+(x+(x^2+1)^(1/2))^(1/2))
Time = 0.12 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.94 \[ \int \frac {x+\sqrt {1+x^2}}{1+\sqrt {x+\sqrt {1+x^2}}} \, dx=\frac {1}{6} \left (-3 x+2 (3+x) \sqrt {x+\sqrt {1+x^2}}+\sqrt {1+x^2} \left (-3+2 \sqrt {x+\sqrt {1+x^2}}\right )+3 \log \left (x+\sqrt {1+x^2}\right )-12 \log \left (1+\sqrt {x+\sqrt {1+x^2}}\right )\right ) \] Input:
Integrate[(x + Sqrt[1 + x^2])/(1 + Sqrt[x + Sqrt[1 + x^2]]),x]
Output:
(-3*x + 2*(3 + x)*Sqrt[x + Sqrt[1 + x^2]] + Sqrt[1 + x^2]*(-3 + 2*Sqrt[x + Sqrt[1 + x^2]]) + 3*Log[x + Sqrt[1 + x^2]] - 12*Log[1 + Sqrt[x + Sqrt[1 + x^2]]])/6
Time = 0.90 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.94, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {7293, 2009}
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.
\(\displaystyle \int \frac {\sqrt {x^2+1}+x}{\sqrt {\sqrt {x^2+1}+x}+1} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {x}{\sqrt {\sqrt {x^2+1}+x}+1}+\frac {\sqrt {x^2+1}}{\sqrt {\sqrt {x^2+1}+x}+1}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \text {arctanh}\left (\sqrt {x^2+1}\right )-2 \text {arctanh}\left (\sqrt {\sqrt {x^2+1}+x}\right )+\frac {1}{3} \left (\sqrt {x^2+1}+x\right )^{3/2}+\sqrt {\sqrt {x^2+1}+x}-\frac {\sqrt {x^2+1}}{2}-\frac {x}{2}-\frac {\log (x)}{2}\) |
Input:
Int[(x + Sqrt[1 + x^2])/(1 + Sqrt[x + Sqrt[1 + x^2]]),x]
Output:
-1/2*x - Sqrt[1 + x^2]/2 + Sqrt[x + Sqrt[1 + x^2]] + (x + Sqrt[1 + x^2])^( 3/2)/3 + ArcTanh[Sqrt[1 + x^2]]/2 - 2*ArcTanh[Sqrt[x + Sqrt[1 + x^2]]] - L og[x]/2
\[\int \frac {x +\sqrt {x^{2}+1}}{1+\sqrt {x +\sqrt {x^{2}+1}}}d x\]
Input:
int((x+(x^2+1)^(1/2))/(1+(x+(x^2+1)^(1/2))^(1/2)),x)
Output:
int((x+(x^2+1)^(1/2))/(1+(x+(x^2+1)^(1/2))^(1/2)),x)
Time = 0.08 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.66 \[ \int \frac {x+\sqrt {1+x^2}}{1+\sqrt {x+\sqrt {1+x^2}}} \, dx=\frac {1}{3} \, {\left (x + \sqrt {x^{2} + 1} + 3\right )} \sqrt {x + \sqrt {x^{2} + 1}} - \frac {1}{2} \, x - \frac {1}{2} \, \sqrt {x^{2} + 1} - 2 \, \log \left (\sqrt {x + \sqrt {x^{2} + 1}} + 1\right ) + \log \left (\sqrt {x + \sqrt {x^{2} + 1}}\right ) \] Input:
integrate((x+(x^2+1)^(1/2))/(1+(x+(x^2+1)^(1/2))^(1/2)),x, algorithm="fric as")
Output:
1/3*(x + sqrt(x^2 + 1) + 3)*sqrt(x + sqrt(x^2 + 1)) - 1/2*x - 1/2*sqrt(x^2 + 1) - 2*log(sqrt(x + sqrt(x^2 + 1)) + 1) + log(sqrt(x + sqrt(x^2 + 1)))
\[ \int \frac {x+\sqrt {1+x^2}}{1+\sqrt {x+\sqrt {1+x^2}}} \, dx=\int \frac {x + \sqrt {x^{2} + 1}}{\sqrt {x + \sqrt {x^{2} + 1}} + 1}\, dx \] Input:
integrate((x+(x**2+1)**(1/2))/(1+(x+(x**2+1)**(1/2))**(1/2)),x)
Output:
Integral((x + sqrt(x**2 + 1))/(sqrt(x + sqrt(x**2 + 1)) + 1), x)
\[ \int \frac {x+\sqrt {1+x^2}}{1+\sqrt {x+\sqrt {1+x^2}}} \, dx=\int { \frac {x + \sqrt {x^{2} + 1}}{\sqrt {x + \sqrt {x^{2} + 1}} + 1} \,d x } \] Input:
integrate((x+(x^2+1)^(1/2))/(1+(x+(x^2+1)^(1/2))^(1/2)),x, algorithm="maxi ma")
Output:
1/4*x^2 + 1/2*integrate(sqrt(x^2 + 1), x) - integrate(1/2*(2*x^2 + sqrt(x^ 2 + 1)*(2*x - 1) - x + 1)/(x + sqrt(x^2 + 1) + 2*sqrt(x + sqrt(x^2 + 1)) + 1), x)
\[ \int \frac {x+\sqrt {1+x^2}}{1+\sqrt {x+\sqrt {1+x^2}}} \, dx=\int { \frac {x + \sqrt {x^{2} + 1}}{\sqrt {x + \sqrt {x^{2} + 1}} + 1} \,d x } \] Input:
integrate((x+(x^2+1)^(1/2))/(1+(x+(x^2+1)^(1/2))^(1/2)),x, algorithm="giac ")
Output:
integrate((x + sqrt(x^2 + 1))/(sqrt(x + sqrt(x^2 + 1)) + 1), x)
Timed out. \[ \int \frac {x+\sqrt {1+x^2}}{1+\sqrt {x+\sqrt {1+x^2}}} \, dx=\int \frac {x+\sqrt {x^2+1}}{\sqrt {x+\sqrt {x^2+1}}+1} \,d x \] Input:
int((x + (x^2 + 1)^(1/2))/((x + (x^2 + 1)^(1/2))^(1/2) + 1),x)
Output:
int((x + (x^2 + 1)^(1/2))/((x + (x^2 + 1)^(1/2))^(1/2) + 1), x)
\[ \int \frac {x+\sqrt {1+x^2}}{1+\sqrt {x+\sqrt {1+x^2}}} \, dx=\frac {\sqrt {\sqrt {x^{2}+1}+x}\, \sqrt {x^{2}+1}}{2}-\frac {\sqrt {x^{2}+1}}{2}+\frac {\left (\int \frac {\sqrt {\sqrt {x^{2}+1}+x}}{x^{2}+1}d x \right )}{4}+\frac {\left (\int \frac {\sqrt {\sqrt {x^{2}+1}+x}\, x^{2}}{x^{2}+1}d x \right )}{4}+\frac {\left (\int \frac {\sqrt {\sqrt {x^{2}+1}+x}\, \sqrt {x^{2}+1}}{x^{2}+1}d x \right )}{2}-\frac {\mathrm {log}\left (\sqrt {x^{2}+1}+x -1\right )}{2}+\frac {\mathrm {log}\left (\sqrt {x^{2}+1}+x +1\right )}{2}+\mathrm {log}\left (\sqrt {\sqrt {x^{2}+1}+x}-1\right )-\mathrm {log}\left (\sqrt {\sqrt {x^{2}+1}+x}+1\right )-\frac {\mathrm {log}\left (x \right )}{2}-\frac {x}{2} \] Input:
int((x+(x^2+1)^(1/2))/(1+(x+(x^2+1)^(1/2))^(1/2)),x)
Output:
(2*sqrt(sqrt(x**2 + 1) + x)*sqrt(x**2 + 1) - 2*sqrt(x**2 + 1) + int(sqrt(s qrt(x**2 + 1) + x)/(x**2 + 1),x) + int((sqrt(sqrt(x**2 + 1) + x)*x**2)/(x* *2 + 1),x) + 2*int((sqrt(sqrt(x**2 + 1) + x)*sqrt(x**2 + 1))/(x**2 + 1),x) - 2*log(sqrt(x**2 + 1) + x - 1) + 2*log(sqrt(x**2 + 1) + x + 1) + 4*log(s qrt(sqrt(x**2 + 1) + x) - 1) - 4*log(sqrt(sqrt(x**2 + 1) + x) + 1) - 2*log (x) - 2*x)/4