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

Optimal. Leaf size=266 \[ -\text {RootSum}\left [\text {$\#$1}^8+8 \text {$\#$1}^7+24 \text {$\#$1}^6+32 \text {$\#$1}^5+14 \text {$\#$1}^4-8 \text {$\#$1}^3-8 \text {$\#$1}^2-1\& ,\frac {\log \left (-\text {$\#$1}+\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-1\right )}{\text {$\#$1}^3+3 \text {$\#$1}^2+\text {$\#$1}-1}\& \right ]+\text {RootSum}\left [\text {$\#$1}^8+8 \text {$\#$1}^7+24 \text {$\#$1}^6+32 \text {$\#$1}^5+18 \text {$\#$1}^4+8 \text {$\#$1}^3+8 \text {$\#$1}^2-1\& ,\frac {\text {$\#$1} \log \left (-\text {$\#$1}+\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-1\right )+\log \left (-\text {$\#$1}+\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-1\right )}{\text {$\#$1}^4+4 \text {$\#$1}^3+4 \text {$\#$1}^2+1}\& \right ]-4 \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}+4 \tanh ^{-1}\left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right ) \]

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Rubi [F]  time = 0.84, 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 {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[1 + x^2]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^2),x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.15, size = 211, normalized size = 0.79 \begin {gather*} -\text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+4 \text {$\#$1}^4-2\&,\frac {\log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )}{\text {$\#$1}^3-2 \text {$\#$1}}\&\right ]+\text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+8 \text {$\#$1}^4-8 \text {$\#$1}^2+2\&,\frac {\text {$\#$1} \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )}{\text {$\#$1}^4-2 \text {$\#$1}^2+2}\&\right ]-4 \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-2 \log \left (1-\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right )+2 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.00, size = 181, normalized size = 0.68 \begin {gather*} -4 \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+4 \tanh ^{-1}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )-\text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )}{-2 \text {$\#$1}+\text {$\#$1}^3}\&\right ]+\text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}}{2-2 \text {$\#$1}^2+\text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 1.44, size = 2152, normalized size = 8.09

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2*sqrt(2))*(4*sqrt(2) + 8)^(1/4)*sqrt(sqrt(2) + 2)*sqrt(sqrt(2)
+ 1)*(sqrt(2) - 2)*arctan(1/8*sqrt(2*(sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + sqrt(2))*sqrt(-2*sqrt(sqrt(2) + 2)*(s
qrt(2) - 1) + 2*sqrt(2))*(4*sqrt(2) + 8)^(1/4)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 4*sqrt(x + sqrt(x^2 + 1)) +
4*sqrt(sqrt(2) + 2) + 4)*((2*sqrt(2) - 3)*sqrt(sqrt(2) + 2)*sqrt(sqrt(2) + 1) + (2*sqrt(2) - 3)*sqrt(sqrt(2)
+ 1))*sqrt(-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2*sqrt(2))*(4*sqrt(2) + 8)^(3/4) - 1/4*((2*sqrt(2) - 3)*sqrt(s
qrt(2) + 2)*sqrt(sqrt(2) + 1) + (2*sqrt(2) - 3)*sqrt(sqrt(2) + 1))*sqrt(-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2
*sqrt(2))*(4*sqrt(2) + 8)^(3/4)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + sqrt(sqrt(2) + 2)*sqrt(sqrt(2) + 1)*(sqrt(
2) - 1) + sqrt(sqrt(2) + 1)*(sqrt(2) - 1)) - 1/2*sqrt(-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2*sqrt(2))*(4*sqrt(
2) + 8)^(1/4)*sqrt(sqrt(2) + 2)*sqrt(sqrt(2) + 1)*(sqrt(2) - 2)*arctan(1/8*sqrt(-2*(sqrt(sqrt(2) + 2)*(sqrt(2)
- 1) + sqrt(2))*sqrt(-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2*sqrt(2))*(4*sqrt(2) + 8)^(1/4)*sqrt(sqrt(x + sqrt
(x^2 + 1)) + 1) + 4*sqrt(x + sqrt(x^2 + 1)) + 4*sqrt(sqrt(2) + 2) + 4)*((2*sqrt(2) - 3)*sqrt(sqrt(2) + 2)*sqrt
(sqrt(2) + 1) + (2*sqrt(2) - 3)*sqrt(sqrt(2) + 1))*sqrt(-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2*sqrt(2))*(4*sqr
t(2) + 8)^(3/4) - 1/4*((2*sqrt(2) - 3)*sqrt(sqrt(2) + 2)*sqrt(sqrt(2) + 1) + (2*sqrt(2) - 3)*sqrt(sqrt(2) + 1)
)*sqrt(-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2*sqrt(2))*(4*sqrt(2) + 8)^(3/4)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)
- sqrt(sqrt(2) + 2)*sqrt(sqrt(2) + 1)*(sqrt(2) - 1) - sqrt(sqrt(2) + 1)*(sqrt(2) - 1)) + 1/2*4^(1/4)*2^(7/8)*
sqrt(-2*2^(1/4)*(sqrt(2) + 1) + 2*sqrt(2) + 4)*sqrt(sqrt(2) - 1)*arctan(1/8*4^(3/4)*2^(3/8)*sqrt(2*4^(1/4)*2^(
1/8)*sqrt(-2*2^(1/4)*(sqrt(2) + 1) + 2*sqrt(2) + 4)*(sqrt(2) + 2^(1/4))*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 4*
sqrt(x + sqrt(x^2 + 1)) + 4*2^(1/4) + 4)*(2^(1/4)*(sqrt(2) + 1)*sqrt(sqrt(2) - 1) + (sqrt(2) + 1)*sqrt(sqrt(2)
- 1))*sqrt(-2*2^(1/4)*(sqrt(2) + 1) + 2*sqrt(2) + 4) - 1/4*4^(3/4)*2^(3/8)*(2^(1/4)*(sqrt(2) + 1)*sqrt(sqrt(2
) - 1) + (sqrt(2) + 1)*sqrt(sqrt(2) - 1))*sqrt(-2*2^(1/4)*(sqrt(2) + 1) + 2*sqrt(2) + 4)*sqrt(sqrt(x + sqrt(x^
2 + 1)) + 1) - 2^(1/4)*(sqrt(2) + 1)*sqrt(sqrt(2) - 1) - (sqrt(2) + 1)*sqrt(sqrt(2) - 1)) + 1/2*4^(1/4)*2^(7/8
)*sqrt(-2*2^(1/4)*(sqrt(2) + 1) + 2*sqrt(2) + 4)*sqrt(sqrt(2) - 1)*arctan(1/8*4^(3/4)*2^(3/8)*sqrt(-2*4^(1/4)*
2^(1/8)*sqrt(-2*2^(1/4)*(sqrt(2) + 1) + 2*sqrt(2) + 4)*(sqrt(2) + 2^(1/4))*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) +
4*sqrt(x + sqrt(x^2 + 1)) + 4*2^(1/4) + 4)*(2^(1/4)*(sqrt(2) + 1)*sqrt(sqrt(2) - 1) + (sqrt(2) + 1)*sqrt(sqrt
(2) - 1))*sqrt(-2*2^(1/4)*(sqrt(2) + 1) + 2*sqrt(2) + 4) - 1/4*4^(3/4)*2^(3/8)*(2^(1/4)*(sqrt(2) + 1)*sqrt(sqr
t(2) - 1) + (sqrt(2) + 1)*sqrt(sqrt(2) - 1))*sqrt(-2*2^(1/4)*(sqrt(2) + 1) + 2*sqrt(2) + 4)*sqrt(sqrt(x + sqrt
(x^2 + 1)) + 1) + 2^(1/4)*(sqrt(2) + 1)*sqrt(sqrt(2) - 1) + (sqrt(2) + 1)*sqrt(sqrt(2) - 1)) - 1/8*4^(1/4)*2^(
1/8)*sqrt(-2*2^(1/4)*(sqrt(2) + 1) + 2*sqrt(2) + 4)*(2^(3/4) + 2)*log(2*4^(1/4)*2^(1/8)*sqrt(-2*2^(1/4)*(sqrt(
2) + 1) + 2*sqrt(2) + 4)*(sqrt(2) + 2^(1/4))*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 4*sqrt(x + sqrt(x^2 + 1)) + 4
*2^(1/4) + 4) + 1/8*4^(1/4)*2^(1/8)*sqrt(-2*2^(1/4)*(sqrt(2) + 1) + 2*sqrt(2) + 4)*(2^(3/4) + 2)*log(-2*4^(1/4
)*2^(1/8)*sqrt(-2*2^(1/4)*(sqrt(2) + 1) + 2*sqrt(2) + 4)*(sqrt(2) + 2^(1/4))*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)
+ 4*sqrt(x + sqrt(x^2 + 1)) + 4*2^(1/4) + 4) - 1/8*sqrt(-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2*sqrt(2))*(sqrt
(sqrt(2) + 2)*(sqrt(2) - 2) - 2)*(4*sqrt(2) + 8)^(1/4)*log(2*(sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + sqrt(2))*sqrt(
-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2*sqrt(2))*(4*sqrt(2) + 8)^(1/4)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 4*sq
rt(x + sqrt(x^2 + 1)) + 4*sqrt(sqrt(2) + 2) + 4) + 1/8*sqrt(-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2*sqrt(2))*(s
qrt(sqrt(2) + 2)*(sqrt(2) - 2) - 2)*(4*sqrt(2) + 8)^(1/4)*log(-2*(sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + sqrt(2))*s
qrt(-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2*sqrt(2))*(4*sqrt(2) + 8)^(1/4)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) +
4*sqrt(x + sqrt(x^2 + 1)) + 4*sqrt(sqrt(2) + 2) + 4) - 2*sqrt(2*sqrt(sqrt(2) + 1) - 2)*arctan(1/2*sqrt(sqrt(x
+ sqrt(x^2 + 1)) + sqrt(sqrt(2) + 1))*sqrt(2*sqrt(sqrt(2) + 1) - 2)*(sqrt(sqrt(2) + 1) + 1) - 1/2*sqrt(sqrt(x
+ sqrt(x^2 + 1)) + 1)*sqrt(2*sqrt(sqrt(2) + 1) - 2)*(sqrt(sqrt(2) + 1) + 1)) + 1/2*sqrt(2*sqrt(sqrt(2) + 1) +
2)*log(sqrt(2)*sqrt(2*sqrt(sqrt(2) + 1) + 2) + 2*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 1/2*sqrt(2*sqrt(sqrt(2)
+ 1) + 2)*log(-sqrt(2)*sqrt(2*sqrt(sqrt(2) + 1) + 2) + 2*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 1/2*sqrt(2*sqrt(
sqrt(2) - 1) + 2)*log(sqrt(2)*sqrt(2*sqrt(sqrt(2) - 1) + 2) + 2*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + 1/2*sqrt(
2*sqrt(sqrt(2) - 1) + 2)*log(-sqrt(2)*sqrt(2*sqrt(sqrt(2) - 1) + 2) + 2*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 1
/2*sqrt(-2*sqrt(sqrt(2) - 1) + 2)*log(sqrt(2)*sqrt(-2*sqrt(sqrt(2) - 1) + 2) + 2*sqrt(sqrt(x + sqrt(x^2 + 1))
+ 1)) + 1/2*sqrt(-2*sqrt(sqrt(2) - 1) + 2)*log(-sqrt(2)*sqrt(-2*sqrt(sqrt(2) - 1) + 2) + 2*sqrt(sqrt(x + sqrt(
x^2 + 1)) + 1)) - 4*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 2*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 1) - 2*log(s
qrt(sqrt(x + sqrt(x^2 + 1)) + 1) - 1)

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giac [F(-1)]  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^2+1)^(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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maple [F]  time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x^{2}+1}\, \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{-x^{2}+1}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((x^2+1)^(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*} -\int \frac {\sqrt {x^{2} + 1} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{x^{2} - 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}\,\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^2 + 1)^(1/2))/(x^2 - 1),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {\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**2+1)**(1/2)*(1+(x+(x**2+1)**(1/2))**(1/2))**(1/2)/(-x**2+1),x)

[Out]

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

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