3.31.30 \(\int \frac {1}{(-1+x^2)^2 \sqrt {x+\sqrt {1+x^2}}} \, dx\) [3030]

3.31.30.1 Optimal result

Integrand size = 23, antiderivative size = 434 \[ \int \frac {1}{\left (-1+x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx=\frac {x^2 \sqrt {x+\sqrt {1+x^2}}}{2 \left (-1+x^2\right )}-\frac {x \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{2 \left (-1+x^2\right )}+\frac {\left (\sqrt {2 \left (1+5 \sqrt {2}\right )}-4 \sqrt {1+\sqrt {2}} x^2+\sqrt {2 \left (1+\sqrt {2}\right )} x^2\right ) \arctan \left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {-1+\sqrt {2}}}\right )}{8 (-1+x) (1+x)}+\frac {\left (-\sqrt {2 \left (-1+5 \sqrt {2}\right )}+4 \sqrt {-1+\sqrt {2}} x^2+\sqrt {2 \left (-1+\sqrt {2}\right )} x^2\right ) \arctan \left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {1+\sqrt {2}}}\right )}{8 (-1+x) (1+x)}+\frac {\left (\sqrt {2 \left (1+5 \sqrt {2}\right )}-4 \sqrt {1+\sqrt {2}} x^2+\sqrt {2 \left (1+\sqrt {2}\right )} x^2\right ) \text {arctanh}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {-1+\sqrt {2}}}\right )}{8 (-1+x) (1+x)}+\frac {\left (-\sqrt {2 \left (-1+5 \sqrt {2}\right )}+4 \sqrt {-1+\sqrt {2}} x^2+\sqrt {2 \left (-1+\sqrt {2}\right )} x^2\right ) \text {arctanh}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {1+\sqrt {2}}}\right )}{8 (-1+x) (1+x)} \]

output

x^2*(x+(x^2+1)^(1/2))^(1/2)/(2*x^2-2)-x*(x^2+1)^(1/2)*(x+(x^2+1)^(1/2))^(1 
/2)/(2*x^2-2)+1/8*((2+10*2^(1/2))^(1/2)-4*(1+2^(1/2))^(1/2)*x^2+(2+2*2^(1/ 
2))^(1/2)*x^2)*arctan((x+(x^2+1)^(1/2))^(1/2)/(2^(1/2)-1)^(1/2))/(-1+x)/(1 
+x)+1/8*(-(-2+10*2^(1/2))^(1/2)+4*(2^(1/2)-1)^(1/2)*x^2+(-2+2*2^(1/2))^(1/ 
2)*x^2)*arctan((x+(x^2+1)^(1/2))^(1/2)/(1+2^(1/2))^(1/2))/(-1+x)/(1+x)+1/8 
*((2+10*2^(1/2))^(1/2)-4*(1+2^(1/2))^(1/2)*x^2+(2+2*2^(1/2))^(1/2)*x^2)*ar 
ctanh((x+(x^2+1)^(1/2))^(1/2)/(2^(1/2)-1)^(1/2))/(-1+x)/(1+x)+1/8*(-(-2+10 
*2^(1/2))^(1/2)+4*(2^(1/2)-1)^(1/2)*x^2+(-2+2*2^(1/2))^(1/2)*x^2)*arctanh( 
(x+(x^2+1)^(1/2))^(1/2)/(1+2^(1/2))^(1/2))/(-1+x)/(1+x)
 

3.31.30.2 Mathematica [A] (verified)

Time = 2.64 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.54 \[ \int \frac {1}{\left (-1+x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx=\frac {1}{8} \left (\frac {4 x^2 \sqrt {x+\sqrt {1+x^2}}}{-1+x^2}-\frac {4 x \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1+x^2}+\sqrt {-2+10 \sqrt {2}} \arctan \left (\sqrt {-1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right )-\sqrt {2+10 \sqrt {2}} \arctan \left (\sqrt {1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right )+\sqrt {-2+10 \sqrt {2}} \text {arctanh}\left (\sqrt {-1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right )-\sqrt {2+10 \sqrt {2}} \text {arctanh}\left (\sqrt {1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right )\right ) \]

input

Integrate[1/((-1 + x^2)^2*Sqrt[x + Sqrt[1 + x^2]]),x]
 

output

((4*x^2*Sqrt[x + Sqrt[1 + x^2]])/(-1 + x^2) - (4*x*Sqrt[1 + x^2]*Sqrt[x + 
Sqrt[1 + x^2]])/(-1 + x^2) + Sqrt[-2 + 10*Sqrt[2]]*ArcTan[Sqrt[-1 + Sqrt[2 
]]*Sqrt[x + Sqrt[1 + x^2]]] - Sqrt[2 + 10*Sqrt[2]]*ArcTan[Sqrt[1 + Sqrt[2] 
]*Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[-2 + 10*Sqrt[2]]*ArcTanh[Sqrt[-1 + Sqrt[ 
2]]*Sqrt[x + Sqrt[1 + x^2]]] - Sqrt[2 + 10*Sqrt[2]]*ArcTanh[Sqrt[1 + Sqrt[ 
2]]*Sqrt[x + Sqrt[1 + x^2]]])/8
 

3.31.30.3 Rubi [A] (verified)

Time = 1.08 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.07, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, 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.

\(\displaystyle \int \frac {1}{\left (x^2-1\right )^2 \sqrt {\sqrt {x^2+1}+x}} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {1}{2 \left (1-x^2\right ) \sqrt {\sqrt {x^2+1}+x}}+\frac {1}{4 (1-x)^2 \sqrt {\sqrt {x^2+1}+x}}+\frac {1}{4 (x+1)^2 \sqrt {\sqrt {x^2+1}+x}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\arctan \left (\sqrt {\sqrt {2}-1} \sqrt {\sqrt {x^2+1}+x}\right )}{2 \sqrt {1+\sqrt {2}}}+\frac {1}{4} \sqrt {\frac {1}{2} \left (\sqrt {2}-1\right )} \arctan \left (\sqrt {\sqrt {2}-1} \sqrt {\sqrt {x^2+1}+x}\right )+\frac {1}{4} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \arctan \left (\sqrt {1+\sqrt {2}} \sqrt {\sqrt {x^2+1}+x}\right )-\frac {\arctan \left (\sqrt {1+\sqrt {2}} \sqrt {\sqrt {x^2+1}+x}\right )}{2 \sqrt {\sqrt {2}-1}}+\frac {\text {arctanh}\left (\sqrt {\sqrt {2}-1} \sqrt {\sqrt {x^2+1}+x}\right )}{2 \sqrt {1+\sqrt {2}}}+\frac {1}{4} \sqrt {\frac {1}{2} \left (\sqrt {2}-1\right )} \text {arctanh}\left (\sqrt {\sqrt {2}-1} \sqrt {\sqrt {x^2+1}+x}\right )+\frac {1}{4} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \text {arctanh}\left (\sqrt {1+\sqrt {2}} \sqrt {\sqrt {x^2+1}+x}\right )-\frac {\text {arctanh}\left (\sqrt {1+\sqrt {2}} \sqrt {\sqrt {x^2+1}+x}\right )}{2 \sqrt {\sqrt {2}-1}}+\frac {\sqrt {\sqrt {x^2+1}+x}}{2 \left (-\left (\sqrt {x^2+1}+x\right )^2-2 \left (\sqrt {x^2+1}+x\right )+1\right )}+\frac {\sqrt {\sqrt {x^2+1}+x}}{2 \left (-\left (\sqrt {x^2+1}+x\right )^2+2 \left (\sqrt {x^2+1}+x\right )+1\right )}\)

input

Int[1/((-1 + x^2)^2*Sqrt[x + Sqrt[1 + x^2]]),x]
 

output

Sqrt[x + Sqrt[1 + x^2]]/(2*(1 - 2*(x + Sqrt[1 + x^2]) - (x + Sqrt[1 + x^2] 
)^2)) + Sqrt[x + Sqrt[1 + x^2]]/(2*(1 + 2*(x + Sqrt[1 + x^2]) - (x + Sqrt[ 
1 + x^2])^2)) + (Sqrt[(-1 + Sqrt[2])/2]*ArcTan[Sqrt[-1 + Sqrt[2]]*Sqrt[x + 
 Sqrt[1 + x^2]]])/4 + ArcTan[Sqrt[-1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]]/( 
2*Sqrt[1 + Sqrt[2]]) - ArcTan[Sqrt[1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]]/( 
2*Sqrt[-1 + Sqrt[2]]) + (Sqrt[(1 + Sqrt[2])/2]*ArcTan[Sqrt[1 + Sqrt[2]]*Sq 
rt[x + Sqrt[1 + x^2]]])/4 + (Sqrt[(-1 + Sqrt[2])/2]*ArcTanh[Sqrt[-1 + Sqrt 
[2]]*Sqrt[x + Sqrt[1 + x^2]]])/4 + ArcTanh[Sqrt[-1 + Sqrt[2]]*Sqrt[x + Sqr 
t[1 + x^2]]]/(2*Sqrt[1 + Sqrt[2]]) - ArcTanh[Sqrt[1 + Sqrt[2]]*Sqrt[x + Sq 
rt[1 + x^2]]]/(2*Sqrt[-1 + Sqrt[2]]) + (Sqrt[(1 + Sqrt[2])/2]*ArcTanh[Sqrt 
[1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]])/4
 

3.31.30.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.31.30.4 Maple [F]

\[\int \frac {1}{\left (x^{2}-1\right )^{2} \sqrt {x +\sqrt {x^{2}+1}}}d x\]

input

int(1/(x^2-1)^2/(x+(x^2+1)^(1/2))^(1/2),x)
 

output

int(1/(x^2-1)^2/(x+(x^2+1)^(1/2))^(1/2),x)
 

3.31.30.5 Fricas [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 429, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\left (-1+x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx=\frac {\sqrt {2} {\left (x^{2} - 1\right )} \sqrt {5 \, \sqrt {2} - 1} \log \left (\sqrt {5 \, \sqrt {2} - 1} {\left (\sqrt {2} + 3\right )} + 7 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) - \sqrt {2} {\left (x^{2} - 1\right )} \sqrt {5 \, \sqrt {2} - 1} \log \left (-\sqrt {5 \, \sqrt {2} - 1} {\left (\sqrt {2} + 3\right )} + 7 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) + \sqrt {2} {\left (x^{2} - 1\right )} \sqrt {5 \, \sqrt {2} + 1} \log \left (\sqrt {5 \, \sqrt {2} + 1} {\left (\sqrt {2} - 3\right )} + 7 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) - \sqrt {2} {\left (x^{2} - 1\right )} \sqrt {5 \, \sqrt {2} + 1} \log \left (-\sqrt {5 \, \sqrt {2} + 1} {\left (\sqrt {2} - 3\right )} + 7 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) + \sqrt {2} {\left (x^{2} - 1\right )} \sqrt {-5 \, \sqrt {2} + 1} \log \left ({\left (\sqrt {2} + 3\right )} \sqrt {-5 \, \sqrt {2} + 1} + 7 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) - \sqrt {2} {\left (x^{2} - 1\right )} \sqrt {-5 \, \sqrt {2} + 1} \log \left (-{\left (\sqrt {2} + 3\right )} \sqrt {-5 \, \sqrt {2} + 1} + 7 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) + \sqrt {2} {\left (x^{2} - 1\right )} \sqrt {-5 \, \sqrt {2} - 1} \log \left ({\left (\sqrt {2} - 3\right )} \sqrt {-5 \, \sqrt {2} - 1} + 7 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) - \sqrt {2} {\left (x^{2} - 1\right )} \sqrt {-5 \, \sqrt {2} - 1} \log \left (-{\left (\sqrt {2} - 3\right )} \sqrt {-5 \, \sqrt {2} - 1} + 7 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) + 8 \, {\left (x^{2} - \sqrt {x^{2} + 1} x\right )} \sqrt {x + \sqrt {x^{2} + 1}}}{16 \, {\left (x^{2} - 1\right )}} \]

input

integrate(1/(x^2-1)^2/(x+(x^2+1)^(1/2))^(1/2),x, algorithm="fricas")
 

output

1/16*(sqrt(2)*(x^2 - 1)*sqrt(5*sqrt(2) - 1)*log(sqrt(5*sqrt(2) - 1)*(sqrt( 
2) + 3) + 7*sqrt(x + sqrt(x^2 + 1))) - sqrt(2)*(x^2 - 1)*sqrt(5*sqrt(2) - 
1)*log(-sqrt(5*sqrt(2) - 1)*(sqrt(2) + 3) + 7*sqrt(x + sqrt(x^2 + 1))) + s 
qrt(2)*(x^2 - 1)*sqrt(5*sqrt(2) + 1)*log(sqrt(5*sqrt(2) + 1)*(sqrt(2) - 3) 
 + 7*sqrt(x + sqrt(x^2 + 1))) - sqrt(2)*(x^2 - 1)*sqrt(5*sqrt(2) + 1)*log( 
-sqrt(5*sqrt(2) + 1)*(sqrt(2) - 3) + 7*sqrt(x + sqrt(x^2 + 1))) + sqrt(2)* 
(x^2 - 1)*sqrt(-5*sqrt(2) + 1)*log((sqrt(2) + 3)*sqrt(-5*sqrt(2) + 1) + 7* 
sqrt(x + sqrt(x^2 + 1))) - sqrt(2)*(x^2 - 1)*sqrt(-5*sqrt(2) + 1)*log(-(sq 
rt(2) + 3)*sqrt(-5*sqrt(2) + 1) + 7*sqrt(x + sqrt(x^2 + 1))) + sqrt(2)*(x^ 
2 - 1)*sqrt(-5*sqrt(2) - 1)*log((sqrt(2) - 3)*sqrt(-5*sqrt(2) - 1) + 7*sqr 
t(x + sqrt(x^2 + 1))) - sqrt(2)*(x^2 - 1)*sqrt(-5*sqrt(2) - 1)*log(-(sqrt( 
2) - 3)*sqrt(-5*sqrt(2) - 1) + 7*sqrt(x + sqrt(x^2 + 1))) + 8*(x^2 - sqrt( 
x^2 + 1)*x)*sqrt(x + sqrt(x^2 + 1)))/(x^2 - 1)
 

3.31.30.6 Sympy [F]

\[ \int \frac {1}{\left (-1+x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx=\int \frac {1}{\left (x - 1\right )^{2} \left (x + 1\right )^{2} \sqrt {x + \sqrt {x^{2} + 1}}}\, dx \]

input

integrate(1/(x**2-1)**2/(x+(x**2+1)**(1/2))**(1/2),x)
 

output

Integral(1/((x - 1)**2*(x + 1)**2*sqrt(x + sqrt(x**2 + 1))), x)
 

3.31.30.7 Maxima [F]

\[ \int \frac {1}{\left (-1+x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx=\int { \frac {1}{{\left (x^{2} - 1\right )}^{2} \sqrt {x + \sqrt {x^{2} + 1}}} \,d x } \]

input

integrate(1/(x^2-1)^2/(x+(x^2+1)^(1/2))^(1/2),x, algorithm="maxima")
 

output

integrate(1/((x^2 - 1)^2*sqrt(x + sqrt(x^2 + 1))), x)
 

3.31.30.8 Giac [F]

\[ \int \frac {1}{\left (-1+x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx=\int { \frac {1}{{\left (x^{2} - 1\right )}^{2} \sqrt {x + \sqrt {x^{2} + 1}}} \,d x } \]

input

integrate(1/(x^2-1)^2/(x+(x^2+1)^(1/2))^(1/2),x, algorithm="giac")
 

output

integrate(1/((x^2 - 1)^2*sqrt(x + sqrt(x^2 + 1))), x)
 

3.31.30.9 Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (-1+x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx=\int \frac {1}{{\left (x^2-1\right )}^2\,\sqrt {x+\sqrt {x^2+1}}} \,d x \]

input

int(1/((x^2 - 1)^2*(x + (x^2 + 1)^(1/2))^(1/2)),x)
 

output

int(1/((x^2 - 1)^2*(x + (x^2 + 1)^(1/2))^(1/2)), x)