Integrand size = 24, antiderivative size = 127 \[ \int \frac {(-1+x) \sqrt {x+\sqrt {1+x^2}}}{1+x} \, dx=\frac {2 (-6+x) \sqrt {1+x^2}+2 \left (-1-6 x+x^2\right )}{3 \sqrt {x+\sqrt {1+x^2}}}+4 \sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {1+\sqrt {2}}}\right )+4 \sqrt {-1+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {-1+\sqrt {2}}}\right ) \] Output:
1/3*(2*(-6+x)*(x^2+1)^(1/2)+2*x^2-12*x-2)/(x+(x^2+1)^(1/2))^(1/2)+4*(1+2^( 1/2))^(1/2)*arctan((x+(x^2+1)^(1/2))^(1/2)/(1+2^(1/2))^(1/2))+4*(2^(1/2)-1 )^(1/2)*arctanh((x+(x^2+1)^(1/2))^(1/2)/(2^(1/2)-1)^(1/2))
Time = 0.25 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.97 \[ \int \frac {(-1+x) \sqrt {x+\sqrt {1+x^2}}}{1+x} \, dx=\frac {2 \left (-1-6 x+x^2+(-6+x) \sqrt {1+x^2}\right )}{3 \sqrt {x+\sqrt {1+x^2}}}+4 \sqrt {1+\sqrt {2}} \arctan \left (\sqrt {-1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right )+4 \sqrt {-1+\sqrt {2}} \text {arctanh}\left (\sqrt {1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right ) \] Input:
Integrate[((-1 + x)*Sqrt[x + Sqrt[1 + x^2]])/(1 + x),x]
Output:
(2*(-1 - 6*x + x^2 + (-6 + x)*Sqrt[1 + x^2]))/(3*Sqrt[x + Sqrt[1 + x^2]]) + 4*Sqrt[1 + Sqrt[2]]*ArcTan[Sqrt[-1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]] + 4*Sqrt[-1 + Sqrt[2]]*ArcTanh[Sqrt[1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]]
Time = 0.54 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.07, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, 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.
\(\displaystyle \int \frac {(x-1) \sqrt {\sqrt {x^2+1}+x}}{x+1} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\sqrt {\sqrt {x^2+1}+x}-\frac {2 \sqrt {\sqrt {x^2+1}+x}}{x+1}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 4 \sqrt {1+\sqrt {2}} \arctan \left (\sqrt {\sqrt {2}-1} \sqrt {\sqrt {x^2+1}+x}\right )+4 \sqrt {\sqrt {2}-1} \text {arctanh}\left (\sqrt {1+\sqrt {2}} \sqrt {\sqrt {x^2+1}+x}\right )+\frac {1}{3} \left (\sqrt {x^2+1}+x\right )^{3/2}-4 \sqrt {\sqrt {x^2+1}+x}-\frac {1}{\sqrt {\sqrt {x^2+1}+x}}\) |
Input:
Int[((-1 + x)*Sqrt[x + Sqrt[1 + x^2]])/(1 + x),x]
Output:
-(1/Sqrt[x + Sqrt[1 + x^2]]) - 4*Sqrt[x + Sqrt[1 + x^2]] + (x + Sqrt[1 + x ^2])^(3/2)/3 + 4*Sqrt[1 + Sqrt[2]]*ArcTan[Sqrt[-1 + Sqrt[2]]*Sqrt[x + Sqrt [1 + x^2]]] + 4*Sqrt[-1 + Sqrt[2]]*ArcTanh[Sqrt[1 + Sqrt[2]]*Sqrt[x + Sqrt [1 + x^2]]]
\[\int \frac {\left (-1+x \right ) \sqrt {x +\sqrt {x^{2}+1}}}{1+x}d x\]
Input:
int((-1+x)*(x+(x^2+1)^(1/2))^(1/2)/(1+x),x)
Output:
int((-1+x)*(x+(x^2+1)^(1/2))^(1/2)/(1+x),x)
Time = 0.09 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.01 \[ \int \frac {(-1+x) \sqrt {x+\sqrt {1+x^2}}}{1+x} \, dx=\frac {2}{3} \, {\left (2 \, x - \sqrt {x^{2} + 1} - 6\right )} \sqrt {x + \sqrt {x^{2} + 1}} + 4 \, \sqrt {\sqrt {2} + 1} \arctan \left (\sqrt {x + \sqrt {x^{2} + 1}} \sqrt {\sqrt {2} + 1} {\left (\sqrt {2} - 1\right )}\right ) + 2 \, \sqrt {\sqrt {2} - 1} \log \left (2 \, \sqrt {x + \sqrt {x^{2} + 1}} + 2 \, \sqrt {\sqrt {2} - 1}\right ) - 2 \, \sqrt {\sqrt {2} - 1} \log \left (2 \, \sqrt {x + \sqrt {x^{2} + 1}} - 2 \, \sqrt {\sqrt {2} - 1}\right ) \] Input:
integrate((-1+x)*(x+(x^2+1)^(1/2))^(1/2)/(1+x),x, algorithm="fricas")
Output:
2/3*(2*x - sqrt(x^2 + 1) - 6)*sqrt(x + sqrt(x^2 + 1)) + 4*sqrt(sqrt(2) + 1 )*arctan(sqrt(x + sqrt(x^2 + 1))*sqrt(sqrt(2) + 1)*(sqrt(2) - 1)) + 2*sqrt (sqrt(2) - 1)*log(2*sqrt(x + sqrt(x^2 + 1)) + 2*sqrt(sqrt(2) - 1)) - 2*sqr t(sqrt(2) - 1)*log(2*sqrt(x + sqrt(x^2 + 1)) - 2*sqrt(sqrt(2) - 1))
\[ \int \frac {(-1+x) \sqrt {x+\sqrt {1+x^2}}}{1+x} \, dx=\int \frac {\left (x - 1\right ) \sqrt {x + \sqrt {x^{2} + 1}}}{x + 1}\, dx \] Input:
integrate((-1+x)*(x+(x**2+1)**(1/2))**(1/2)/(1+x),x)
Output:
Integral((x - 1)*sqrt(x + sqrt(x**2 + 1))/(x + 1), x)
\[ \int \frac {(-1+x) \sqrt {x+\sqrt {1+x^2}}}{1+x} \, dx=\int { \frac {\sqrt {x + \sqrt {x^{2} + 1}} {\left (x - 1\right )}}{x + 1} \,d x } \] Input:
integrate((-1+x)*(x+(x^2+1)^(1/2))^(1/2)/(1+x),x, algorithm="maxima")
Output:
integrate(sqrt(x + sqrt(x^2 + 1))*(x - 1)/(x + 1), x)
\[ \int \frac {(-1+x) \sqrt {x+\sqrt {1+x^2}}}{1+x} \, dx=\int { \frac {\sqrt {x + \sqrt {x^{2} + 1}} {\left (x - 1\right )}}{x + 1} \,d x } \] Input:
integrate((-1+x)*(x+(x^2+1)^(1/2))^(1/2)/(1+x),x, algorithm="giac")
Output:
integrate(sqrt(x + sqrt(x^2 + 1))*(x - 1)/(x + 1), x)
Timed out. \[ \int \frac {(-1+x) \sqrt {x+\sqrt {1+x^2}}}{1+x} \, dx=\int \frac {\sqrt {x+\sqrt {x^2+1}}\,\left (x-1\right )}{x+1} \,d x \] Input:
int(((x + (x^2 + 1)^(1/2))^(1/2)*(x - 1))/(x + 1),x)
Output:
int(((x + (x^2 + 1)^(1/2))^(1/2)*(x - 1))/(x + 1), x)
Time = 0.17 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.35 \[ \int \frac {(-1+x) \sqrt {x+\sqrt {1+x^2}}}{1+x} \, dx=-2 \sqrt {\sqrt {2}+1}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {2}+1}\, \sqrt {\sqrt {x^{2}+1}+x}\, \sqrt {x^{2}+1}}{2}-\frac {\sqrt {\sqrt {2}+1}\, \sqrt {\sqrt {x^{2}+1}+x}\, \sqrt {2}}{2}-\frac {\sqrt {\sqrt {2}+1}\, \sqrt {\sqrt {x^{2}+1}+x}\, x}{2}+\frac {\sqrt {\sqrt {2}+1}\, \sqrt {\sqrt {x^{2}+1}+x}}{2}\right )-\frac {2 \sqrt {\sqrt {x^{2}+1}+x}\, \sqrt {x^{2}+1}}{3}+\frac {4 \sqrt {\sqrt {x^{2}+1}+x}\, x}{3}-4 \sqrt {\sqrt {x^{2}+1}+x}-2 \sqrt {\sqrt {2}-1}\, \mathrm {log}\left (\sqrt {\sqrt {x^{2}+1}+x}-\sqrt {\sqrt {2}-1}\right )+2 \sqrt {\sqrt {2}-1}\, \mathrm {log}\left (\sqrt {\sqrt {x^{2}+1}+x}+\sqrt {\sqrt {2}-1}\right ) \] Input:
int((-1+x)*(x+(x^2+1)^(1/2))^(1/2)/(1+x),x)
Output:
(2*( - 3*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) - sq rt(sqrt(2) + 1)*sqrt(sqrt(x**2 + 1) + x)*x + sqrt(sqrt(2) + 1)*sqrt(sqrt(x **2 + 1) + x))/2) - sqrt(sqrt(x**2 + 1) + x)*sqrt(x**2 + 1) + 2*sqrt(sqrt( x**2 + 1) + x)*x - 6*sqrt(sqrt(x**2 + 1) + x) - 3*sqrt(sqrt(2) - 1)*log(sq rt(sqrt(x**2 + 1) + x) - sqrt(sqrt(2) - 1)) + 3*sqrt(sqrt(2) - 1)*log(sqrt (sqrt(x**2 + 1) + x) + sqrt(sqrt(2) - 1))))/3