Integrand size = 28, antiderivative size = 199 \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx=-\frac {1}{3 \left (x+\sqrt {1+x^2}\right )^{3/2}}+\sqrt {x+\sqrt {1+x^2}}+2 \sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {-1+\sqrt {2}}}\right )-2 \sqrt {-1+\sqrt {2}} \arctan \left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {1+\sqrt {2}}}\right )+2 \sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {-1+\sqrt {2}}}\right )-2 \sqrt {-1+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {1+\sqrt {2}}}\right ) \]
-1/3/(x+(x^2+1)^(1/2))^(3/2)+(x+(x^2+1)^(1/2))^(1/2)+2*(1+2^(1/2))^(1/2)*a rctan((x+(x^2+1)^(1/2))^(1/2)/(2^(1/2)-1)^(1/2))-2*(2^(1/2)-1)^(1/2)*arcta n((x+(x^2+1)^(1/2))^(1/2)/(1+2^(1/2))^(1/2))+2*(1+2^(1/2))^(1/2)*arctanh(( x+(x^2+1)^(1/2))^(1/2)/(2^(1/2)-1)^(1/2))-2*(2^(1/2)-1)^(1/2)*arctanh((x+( x^2+1)^(1/2))^(1/2)/(1+2^(1/2))^(1/2))
Time = 0.25 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.00 \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx=-\frac {1}{3 \left (x+\sqrt {1+x^2}\right )^{3/2}}+\sqrt {x+\sqrt {1+x^2}}-2 \sqrt {-1+\sqrt {2}} \arctan \left (\sqrt {-1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right )+2 \sqrt {1+\sqrt {2}} \arctan \left (\sqrt {1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right )-2 \sqrt {-1+\sqrt {2}} \text {arctanh}\left (\sqrt {-1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right )+2 \sqrt {1+\sqrt {2}} \text {arctanh}\left (\sqrt {1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right ) \]
-1/3*1/(x + Sqrt[1 + x^2])^(3/2) + Sqrt[x + Sqrt[1 + x^2]] - 2*Sqrt[-1 + S qrt[2]]*ArcTan[Sqrt[-1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]] + 2*Sqrt[1 + Sq rt[2]]*ArcTan[Sqrt[1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]] - 2*Sqrt[-1 + Sqr t[2]]*ArcTanh[Sqrt[-1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]] + 2*Sqrt[1 + Sqr t[2]]*ArcTanh[Sqrt[1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]]
Time = 0.80 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {7276, 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.
\(\displaystyle \int \frac {x^2+1}{\left (x^2-1\right ) \sqrt {\sqrt {x^2+1}+x}} \, dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (\frac {2}{\left (x^2-1\right ) \sqrt {\sqrt {x^2+1}+x}}+\frac {1}{\sqrt {\sqrt {x^2+1}+x}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 \arctan \left (\sqrt {\sqrt {2}-1} \sqrt {\sqrt {x^2+1}+x}\right )}{\sqrt {1+\sqrt {2}}}+\frac {2 \arctan \left (\sqrt {1+\sqrt {2}} \sqrt {\sqrt {x^2+1}+x}\right )}{\sqrt {\sqrt {2}-1}}-\frac {2 \text {arctanh}\left (\sqrt {\sqrt {2}-1} \sqrt {\sqrt {x^2+1}+x}\right )}{\sqrt {1+\sqrt {2}}}+\frac {2 \text {arctanh}\left (\sqrt {1+\sqrt {2}} \sqrt {\sqrt {x^2+1}+x}\right )}{\sqrt {\sqrt {2}-1}}+\sqrt {\sqrt {x^2+1}+x}-\frac {1}{3 \left (\sqrt {x^2+1}+x\right )^{3/2}}\) |
-1/3*1/(x + Sqrt[1 + x^2])^(3/2) + Sqrt[x + Sqrt[1 + x^2]] - (2*ArcTan[Sqr t[-1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]])/Sqrt[1 + Sqrt[2]] + (2*ArcTan[Sq rt[1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]])/Sqrt[-1 + Sqrt[2]] - (2*ArcTanh[ Sqrt[-1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]])/Sqrt[1 + Sqrt[2]] + (2*ArcTan h[Sqrt[1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]])/Sqrt[-1 + Sqrt[2]]
3.25.57.3.1 Defintions of rubi rules used
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
\[\int \frac {x^{2}+1}{\left (x^{2}-1\right ) \sqrt {x +\sqrt {x^{2}+1}}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 345 vs. \(2 (141) = 282\).
Time = 0.27 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.73 \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx=-\frac {2}{3} \, {\left (x^{2} - \sqrt {x^{2} + 1} x - 1\right )} \sqrt {x + \sqrt {x^{2} + 1}} - \sqrt {\sqrt {2} - 1} \log \left (2 \, {\left (\sqrt {2} + 1\right )} \sqrt {\sqrt {2} - 1} + 2 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) + \sqrt {\sqrt {2} - 1} \log \left (-2 \, {\left (\sqrt {2} + 1\right )} \sqrt {\sqrt {2} - 1} + 2 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) + \sqrt {\sqrt {2} + 1} \log \left (2 \, \sqrt {\sqrt {2} + 1} {\left (\sqrt {2} - 1\right )} + 2 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) - \sqrt {\sqrt {2} + 1} \log \left (-2 \, \sqrt {\sqrt {2} + 1} {\left (\sqrt {2} - 1\right )} + 2 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) - \frac {1}{2} \, \sqrt {-4 \, \sqrt {2} + 4} \log \left ({\left (\sqrt {2} + 1\right )} \sqrt {-4 \, \sqrt {2} + 4} + 2 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) + \frac {1}{2} \, \sqrt {-4 \, \sqrt {2} + 4} \log \left (-{\left (\sqrt {2} + 1\right )} \sqrt {-4 \, \sqrt {2} + 4} + 2 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) + \frac {1}{2} \, \sqrt {-4 \, \sqrt {2} - 4} \log \left ({\left (\sqrt {2} - 1\right )} \sqrt {-4 \, \sqrt {2} - 4} + 2 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) - \frac {1}{2} \, \sqrt {-4 \, \sqrt {2} - 4} \log \left (-{\left (\sqrt {2} - 1\right )} \sqrt {-4 \, \sqrt {2} - 4} + 2 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) \]
-2/3*(x^2 - sqrt(x^2 + 1)*x - 1)*sqrt(x + sqrt(x^2 + 1)) - sqrt(sqrt(2) - 1)*log(2*(sqrt(2) + 1)*sqrt(sqrt(2) - 1) + 2*sqrt(x + sqrt(x^2 + 1))) + sq rt(sqrt(2) - 1)*log(-2*(sqrt(2) + 1)*sqrt(sqrt(2) - 1) + 2*sqrt(x + sqrt(x ^2 + 1))) + sqrt(sqrt(2) + 1)*log(2*sqrt(sqrt(2) + 1)*(sqrt(2) - 1) + 2*sq rt(x + sqrt(x^2 + 1))) - sqrt(sqrt(2) + 1)*log(-2*sqrt(sqrt(2) + 1)*(sqrt( 2) - 1) + 2*sqrt(x + sqrt(x^2 + 1))) - 1/2*sqrt(-4*sqrt(2) + 4)*log((sqrt( 2) + 1)*sqrt(-4*sqrt(2) + 4) + 2*sqrt(x + sqrt(x^2 + 1))) + 1/2*sqrt(-4*sq rt(2) + 4)*log(-(sqrt(2) + 1)*sqrt(-4*sqrt(2) + 4) + 2*sqrt(x + sqrt(x^2 + 1))) + 1/2*sqrt(-4*sqrt(2) - 4)*log((sqrt(2) - 1)*sqrt(-4*sqrt(2) - 4) + 2*sqrt(x + sqrt(x^2 + 1))) - 1/2*sqrt(-4*sqrt(2) - 4)*log(-(sqrt(2) - 1)*s qrt(-4*sqrt(2) - 4) + 2*sqrt(x + sqrt(x^2 + 1)))
\[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx=\int \frac {x^{2} + 1}{\left (x - 1\right ) \left (x + 1\right ) \sqrt {x + \sqrt {x^{2} + 1}}}\, dx \]
\[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx=\int { \frac {x^{2} + 1}{{\left (x^{2} - 1\right )} \sqrt {x + \sqrt {x^{2} + 1}}} \,d x } \]
\[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx=\int { \frac {x^{2} + 1}{{\left (x^{2} - 1\right )} \sqrt {x + \sqrt {x^{2} + 1}}} \,d x } \]
Timed out. \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx=\int \frac {x^2+1}{\left (x^2-1\right )\,\sqrt {x+\sqrt {x^2+1}}} \,d x \]