3.9.64 \(\int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1+x^2} \, dx\) [864]

Optimal. Leaf size=65 \[ \text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\& ,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^2}\& \right ] \]

[Out]

Unintegrable

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Rubi [F]
time = 0.79, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(I/2)*Defer[Int][(Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(I - x), x] + (I/2)*Defer[Int][(S
qrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(I + x), x]

Rubi steps

\begin {align*} \int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1+x^2} \, dx &=\int \left (\frac {i \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 (i-x)}+\frac {i \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 (i+x)}\right ) \, dx\\ &=\frac {1}{2} i \int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{i-x} \, dx+\frac {1}{2} i \int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{i+x} \, dx\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 65, normalized size = 1.00 \begin {gather*} \text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^2}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.17, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x +\sqrt {x^{2}+1}}\, \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{x^{2}+1}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 1.21, size = 2463, normalized size = 37.89 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))*log((2*(1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))^3 - 4*(1/2
*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))^2 + 3*I*sqrt(2) - 3*sqrt(4*I*sqrt(2) - 2) + 2)*sqrt(1/2*I*sqrt(2) - 1/
2*sqrt(4*I*sqrt(2) - 2)) + 6*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + sqrt(1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) -
2))*log(-(2*(1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))^3 - 4*(1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))^2 +
3*I*sqrt(2) - 3*sqrt(4*I*sqrt(2) - 2) + 2)*sqrt(1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2)) + 6*sqrt(sqrt(x + s
qrt(x^2 + 1)) + 1)) + sqrt(-1/2*I*sqrt(2) - 1/2*sqrt(-4*I*sqrt(2) - 2))*log((2*(1/2*I*sqrt(2) - 1/2*sqrt(4*I*s
qrt(2) - 2))^3 - (-1/2*I*sqrt(2) - 1/2*sqrt(-4*I*sqrt(2) - 2))^2*(-I*sqrt(2) + sqrt(4*I*sqrt(2) - 2) - 4) - ((
1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))^2 - 1)*(I*sqrt(2) + sqrt(-4*I*sqrt(2) - 2)) + 2*I*sqrt(2) - 2*sqrt(
4*I*sqrt(2) - 2) + 6)*sqrt(-1/2*I*sqrt(2) - 1/2*sqrt(-4*I*sqrt(2) - 2)) + 6*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1))
- sqrt(-1/2*I*sqrt(2) - 1/2*sqrt(-4*I*sqrt(2) - 2))*log(-(2*(1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))^3 - (
-1/2*I*sqrt(2) - 1/2*sqrt(-4*I*sqrt(2) - 2))^2*(-I*sqrt(2) + sqrt(4*I*sqrt(2) - 2) - 4) - ((1/2*I*sqrt(2) - 1/
2*sqrt(4*I*sqrt(2) - 2))^2 - 1)*(I*sqrt(2) + sqrt(-4*I*sqrt(2) - 2)) + 2*I*sqrt(2) - 2*sqrt(4*I*sqrt(2) - 2) +
6)*sqrt(-1/2*I*sqrt(2) - 1/2*sqrt(-4*I*sqrt(2) - 2)) + 6*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + 1/2*sqrt(2*sqrt
(-3*(1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))^2 - 3*(-1/2*I*sqrt(2) - 1/2*sqrt(-4*I*sqrt(2) - 2))^2 - 1/2*(I
*sqrt(2) + sqrt(-4*I*sqrt(2) - 2))*(-I*sqrt(2) + sqrt(4*I*sqrt(2) - 2)) - 8) + sqrt(4*I*sqrt(2) - 2) + sqrt(-4
*I*sqrt(2) - 2))*log(1/4*(2*(-1/2*I*sqrt(2) - 1/2*sqrt(-4*I*sqrt(2) - 2))^2*(-I*sqrt(2) + sqrt(4*I*sqrt(2) - 2
) - 4) + 2*((1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))^2 - 1)*(I*sqrt(2) + sqrt(-4*I*sqrt(2) - 2)) - 8*(1/2*I
*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))^2 - sqrt(-3*(1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))^2 - 3*(-1/2*I*sq
rt(2) - 1/2*sqrt(-4*I*sqrt(2) - 2))^2 - 1/2*(I*sqrt(2) + sqrt(-4*I*sqrt(2) - 2))*(-I*sqrt(2) + sqrt(4*I*sqrt(2
) - 2)) - 8)*((I*sqrt(2) + sqrt(-4*I*sqrt(2) - 2))*(-I*sqrt(2) + sqrt(4*I*sqrt(2) - 2) - 4) + 4*I*sqrt(2) - 4*
sqrt(4*I*sqrt(2) - 2) + 4) + 2*I*sqrt(2) - 2*sqrt(4*I*sqrt(2) - 2) - 8)*sqrt(2*sqrt(-3*(1/2*I*sqrt(2) - 1/2*sq
rt(4*I*sqrt(2) - 2))^2 - 3*(-1/2*I*sqrt(2) - 1/2*sqrt(-4*I*sqrt(2) - 2))^2 - 1/2*(I*sqrt(2) + sqrt(-4*I*sqrt(2
) - 2))*(-I*sqrt(2) + sqrt(4*I*sqrt(2) - 2)) - 8) + sqrt(4*I*sqrt(2) - 2) + sqrt(-4*I*sqrt(2) - 2)) + 12*sqrt(
sqrt(x + sqrt(x^2 + 1)) + 1)) - 1/2*sqrt(2*sqrt(-3*(1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))^2 - 3*(-1/2*I*s
qrt(2) - 1/2*sqrt(-4*I*sqrt(2) - 2))^2 - 1/2*(I*sqrt(2) + sqrt(-4*I*sqrt(2) - 2))*(-I*sqrt(2) + sqrt(4*I*sqrt(
2) - 2)) - 8) + sqrt(4*I*sqrt(2) - 2) + sqrt(-4*I*sqrt(2) - 2))*log(-1/4*(2*(-1/2*I*sqrt(2) - 1/2*sqrt(-4*I*sq
rt(2) - 2))^2*(-I*sqrt(2) + sqrt(4*I*sqrt(2) - 2) - 4) + 2*((1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))^2 - 1)
*(I*sqrt(2) + sqrt(-4*I*sqrt(2) - 2)) - 8*(1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))^2 - sqrt(-3*(1/2*I*sqrt(
2) - 1/2*sqrt(4*I*sqrt(2) - 2))^2 - 3*(-1/2*I*sqrt(2) - 1/2*sqrt(-4*I*sqrt(2) - 2))^2 - 1/2*(I*sqrt(2) + sqrt(
-4*I*sqrt(2) - 2))*(-I*sqrt(2) + sqrt(4*I*sqrt(2) - 2)) - 8)*((I*sqrt(2) + sqrt(-4*I*sqrt(2) - 2))*(-I*sqrt(2)
+ sqrt(4*I*sqrt(2) - 2) - 4) + 4*I*sqrt(2) - 4*sqrt(4*I*sqrt(2) - 2) + 4) + 2*I*sqrt(2) - 2*sqrt(4*I*sqrt(2)
- 2) - 8)*sqrt(2*sqrt(-3*(1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))^2 - 3*(-1/2*I*sqrt(2) - 1/2*sqrt(-4*I*sqr
t(2) - 2))^2 - 1/2*(I*sqrt(2) + sqrt(-4*I*sqrt(2) - 2))*(-I*sqrt(2) + sqrt(4*I*sqrt(2) - 2)) - 8) + sqrt(4*I*s
qrt(2) - 2) + sqrt(-4*I*sqrt(2) - 2)) + 12*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + 1/2*sqrt(-2*sqrt(-3*(1/2*I*sqr
t(2) - 1/2*sqrt(4*I*sqrt(2) - 2))^2 - 3*(-1/2*I*sqrt(2) - 1/2*sqrt(-4*I*sqrt(2) - 2))^2 - 1/2*(I*sqrt(2) + sqr
t(-4*I*sqrt(2) - 2))*(-I*sqrt(2) + sqrt(4*I*sqrt(2) - 2)) - 8) + sqrt(4*I*sqrt(2) - 2) + sqrt(-4*I*sqrt(2) - 2
))*log(1/4*(2*(-1/2*I*sqrt(2) - 1/2*sqrt(-4*I*sqrt(2) - 2))^2*(-I*sqrt(2) + sqrt(4*I*sqrt(2) - 2) - 4) + 2*((1
/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))^2 - 1)*(I*sqrt(2) + sqrt(-4*I*sqrt(2) - 2)) - 8*(1/2*I*sqrt(2) - 1/2
*sqrt(4*I*sqrt(2) - 2))^2 + sqrt(-3*(1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))^2 - 3*(-1/2*I*sqrt(2) - 1/2*sq
rt(-4*I*sqrt(2) - 2))^2 - 1/2*(I*sqrt(2) + sqrt(-4*I*sqrt(2) - 2))*(-I*sqrt(2) + sqrt(4*I*sqrt(2) - 2)) - 8)*(
(I*sqrt(2) + sqrt(-4*I*sqrt(2) - 2))*(-I*sqrt(2) + sqrt(4*I*sqrt(2) - 2) - 4) + 4*I*sqrt(2) - 4*sqrt(4*I*sqrt(
2) - 2) + 4) + 2*I*sqrt(2) - 2*sqrt(4*I*sqrt(2) - 2) - 8)*sqrt(-2*sqrt(-3*(1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2
) - 2))^2 - 3*(-1/2*I*sqrt(2) - 1/2*sqrt(-4*I*sqrt(2) - 2))^2 - 1/2*(I*sqrt(2) + sqrt(-4*I*sqrt(2) - 2))*(-I*s
qrt(2) + sqrt(4*I*sqrt(2) - 2)) - 8) + sqrt(4*I*sqrt(2) - 2) + sqrt(-4*I*sqrt(2) - 2)) + 12*sqrt(sqrt(x + sqrt
(x^2 + 1)) + 1)) - 1/2*sqrt(-2*sqrt(-3*(1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))^2 - 3*(-1/2*I*sqrt(2) - 1/2
*sqrt(-4*I*sqrt(2) - 2))^2 - 1/2*(I*sqrt(2) + s...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x + \sqrt {x^{2} + 1}} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{x^{2} + 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}\,\sqrt {x+\sqrt {x^2+1}}}{x^2+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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