\(\int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1+x^2)^{3/2}} \, dx\) [1829]

Optimal result

Integrand size = 31, antiderivative size = 124 \[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^{3/2}} \, dx=-\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1+x^2+x \sqrt {1+x^2}}+\frac {1}{4} \text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )}{-\text {$\#$1}+3 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ] \]

[Out]

Unintegrable

Rubi [F]

\[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^{3/2}} \, dx \]

[In]

Int[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/(1 + x^2)^(3/2),x]

[Out]

Defer[Int][Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/(1 + x^2)^(3/2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^{3/2}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^{3/2}} \, dx=-\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1+x^2+x \sqrt {1+x^2}}+\frac {1}{4} \text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )}{-\text {$\#$1}+3 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ] \]

[In]

Integrate[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/(1 + x^2)^(3/2),x]

[Out]

-(Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/(1 + x^2 + x*Sqrt[1 + x^2])) + RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8
& , Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]/(-#1 + 3*#1^3 - 3*#1^5 + #1^7) & ]/4

Maple [N/A] (verified)

Not integrable

Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.19

\[\int \frac {\sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{\left (x^{2}+1\right )^{\frac {3}{2}}}d x\]

[In]

int((1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(x^2+1)^(3/2),x)

[Out]

int((1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(x^2+1)^(3/2),x)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.29 (sec) , antiderivative size = 682, normalized size of antiderivative = 5.50 \[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^{3/2}} \, dx=\frac {\sqrt {2} {\left (x^{2} + 1\right )} \sqrt {\sqrt {2 i \, \sqrt {2} + 1} - 1} \log \left ({\left ({\left (\sqrt {2} - i\right )} \sqrt {2 i \, \sqrt {2} + 1} + 3 \, \sqrt {2} - 3 i\right )} \sqrt {\sqrt {2 i \, \sqrt {2} + 1} - 1} + 6 \, \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}\right ) - \sqrt {2} {\left (x^{2} + 1\right )} \sqrt {\sqrt {2 i \, \sqrt {2} + 1} - 1} \log \left (-{\left ({\left (\sqrt {2} - i\right )} \sqrt {2 i \, \sqrt {2} + 1} + 3 \, \sqrt {2} - 3 i\right )} \sqrt {\sqrt {2 i \, \sqrt {2} + 1} - 1} + 6 \, \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}\right ) - \sqrt {2} {\left (x^{2} + 1\right )} \sqrt {-\sqrt {2 i \, \sqrt {2} + 1} - 1} \log \left ({\left ({\left (\sqrt {2} - i\right )} \sqrt {2 i \, \sqrt {2} + 1} - 3 \, \sqrt {2} + 3 i\right )} \sqrt {-\sqrt {2 i \, \sqrt {2} + 1} - 1} + 6 \, \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}\right ) + \sqrt {2} {\left (x^{2} + 1\right )} \sqrt {-\sqrt {2 i \, \sqrt {2} + 1} - 1} \log \left (-{\left ({\left (\sqrt {2} - i\right )} \sqrt {2 i \, \sqrt {2} + 1} - 3 \, \sqrt {2} + 3 i\right )} \sqrt {-\sqrt {2 i \, \sqrt {2} + 1} - 1} + 6 \, \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}\right ) + \sqrt {2} {\left (x^{2} + 1\right )} \sqrt {\sqrt {-2 i \, \sqrt {2} + 1} - 1} \log \left ({\left ({\left (\sqrt {2} + i\right )} \sqrt {-2 i \, \sqrt {2} + 1} + 3 \, \sqrt {2} + 3 i\right )} \sqrt {\sqrt {-2 i \, \sqrt {2} + 1} - 1} + 6 \, \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}\right ) - \sqrt {2} {\left (x^{2} + 1\right )} \sqrt {\sqrt {-2 i \, \sqrt {2} + 1} - 1} \log \left (-{\left ({\left (\sqrt {2} + i\right )} \sqrt {-2 i \, \sqrt {2} + 1} + 3 \, \sqrt {2} + 3 i\right )} \sqrt {\sqrt {-2 i \, \sqrt {2} + 1} - 1} + 6 \, \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}\right ) - \sqrt {2} {\left (x^{2} + 1\right )} \sqrt {-\sqrt {-2 i \, \sqrt {2} + 1} - 1} \log \left ({\left ({\left (\sqrt {2} + i\right )} \sqrt {-2 i \, \sqrt {2} + 1} - 3 \, \sqrt {2} - 3 i\right )} \sqrt {-\sqrt {-2 i \, \sqrt {2} + 1} - 1} + 6 \, \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}\right ) + \sqrt {2} {\left (x^{2} + 1\right )} \sqrt {-\sqrt {-2 i \, \sqrt {2} + 1} - 1} \log \left (-{\left ({\left (\sqrt {2} + i\right )} \sqrt {-2 i \, \sqrt {2} + 1} - 3 \, \sqrt {2} - 3 i\right )} \sqrt {-\sqrt {-2 i \, \sqrt {2} + 1} - 1} + 6 \, \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}\right ) - 8 \, {\left (x^{2} - \sqrt {x^{2} + 1} x + 1\right )} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{8 \, {\left (x^{2} + 1\right )}} \]

[In]

integrate((1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(x^2+1)^(3/2),x, algorithm="fricas")

[Out]

1/8*(sqrt(2)*(x^2 + 1)*sqrt(sqrt(2*I*sqrt(2) + 1) - 1)*log(((sqrt(2) - I)*sqrt(2*I*sqrt(2) + 1) + 3*sqrt(2) -
3*I)*sqrt(sqrt(2*I*sqrt(2) + 1) - 1) + 6*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - sqrt(2)*(x^2 + 1)*sqrt(sqrt(2*I*
sqrt(2) + 1) - 1)*log(-((sqrt(2) - I)*sqrt(2*I*sqrt(2) + 1) + 3*sqrt(2) - 3*I)*sqrt(sqrt(2*I*sqrt(2) + 1) - 1)
+ 6*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - sqrt(2)*(x^2 + 1)*sqrt(-sqrt(2*I*sqrt(2) + 1) - 1)*log(((sqrt(2) - I
)*sqrt(2*I*sqrt(2) + 1) - 3*sqrt(2) + 3*I)*sqrt(-sqrt(2*I*sqrt(2) + 1) - 1) + 6*sqrt(sqrt(x + sqrt(x^2 + 1)) +
1)) + sqrt(2)*(x^2 + 1)*sqrt(-sqrt(2*I*sqrt(2) + 1) - 1)*log(-((sqrt(2) - I)*sqrt(2*I*sqrt(2) + 1) - 3*sqrt(2
) + 3*I)*sqrt(-sqrt(2*I*sqrt(2) + 1) - 1) + 6*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + sqrt(2)*(x^2 + 1)*sqrt(sqrt
(-2*I*sqrt(2) + 1) - 1)*log(((sqrt(2) + I)*sqrt(-2*I*sqrt(2) + 1) + 3*sqrt(2) + 3*I)*sqrt(sqrt(-2*I*sqrt(2) +
1) - 1) + 6*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - sqrt(2)*(x^2 + 1)*sqrt(sqrt(-2*I*sqrt(2) + 1) - 1)*log(-((sqr
t(2) + I)*sqrt(-2*I*sqrt(2) + 1) + 3*sqrt(2) + 3*I)*sqrt(sqrt(-2*I*sqrt(2) + 1) - 1) + 6*sqrt(sqrt(x + sqrt(x^
2 + 1)) + 1)) - sqrt(2)*(x^2 + 1)*sqrt(-sqrt(-2*I*sqrt(2) + 1) - 1)*log(((sqrt(2) + I)*sqrt(-2*I*sqrt(2) + 1)
- 3*sqrt(2) - 3*I)*sqrt(-sqrt(-2*I*sqrt(2) + 1) - 1) + 6*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + sqrt(2)*(x^2 + 1
)*sqrt(-sqrt(-2*I*sqrt(2) + 1) - 1)*log(-((sqrt(2) + I)*sqrt(-2*I*sqrt(2) + 1) - 3*sqrt(2) - 3*I)*sqrt(-sqrt(-
2*I*sqrt(2) + 1) - 1) + 6*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 8*(x^2 - sqrt(x^2 + 1)*x + 1)*sqrt(sqrt(x + sqr
t(x^2 + 1)) + 1))/(x^2 + 1)

Sympy [N/A]

Not integrable

Time = 1.84 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.22 \[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{\left (x^{2} + 1\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate((1+(x+(x**2+1)**(1/2))**(1/2))**(1/2)/(x**2+1)**(3/2),x)

[Out]

Integral(sqrt(sqrt(x + sqrt(x**2 + 1)) + 1)/(x**2 + 1)**(3/2), x)

Maxima [N/A]

Not integrable

Time = 0.60 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.20 \[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{{\left (x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(x^2+1)^(3/2),x, algorithm="maxima")

[Out]

integrate(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)/(x^2 + 1)^(3/2), x)

Giac [N/A]

Not integrable

Time = 124.68 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.20 \[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{{\left (x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(x^2+1)^(3/2),x, algorithm="giac")

[Out]

integrate(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)/(x^2 + 1)^(3/2), x)

Mupad [N/A]

Not integrable

Time = 7.81 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.20 \[ \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{{\left (x^2+1\right )}^{3/2}} \,d x \]

[In]

int(((x + (x^2 + 1)^(1/2))^(1/2) + 1)^(1/2)/(x^2 + 1)^(3/2),x)

[Out]

int(((x + (x^2 + 1)^(1/2))^(1/2) + 1)^(1/2)/(x^2 + 1)^(3/2), x)