3.30.56 \(\int \frac {\sqrt {1+x^2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1-x^2)^2} \, dx\) [2956]

Optimal. Leaf size=357 \[ \frac {\left (-1-3 x^2\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-3 x \sqrt {1+x^2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{4 x \left (-1+x^2\right ) \sqrt {1+x^2}+2 \left (-1+x^2\right ) \left (1+2 x^2\right )}-\frac {1}{16} \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\& ,\frac {-2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2+3 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{2 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\& \right ]+\frac {1}{16} \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\& ,\frac {-2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}+3 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^3}{-2+4 \text {$\#$1}^2-3 \text {$\#$1}^4+\text {$\#$1}^6}\& \right ] \]

[Out]

Unintegrable

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Rubi [F]
time = 1.38, 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}}}}{\left (1-x^2\right )^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)^2,x]

[Out]

Defer[Int][(Sqrt[1 + x^2]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x)^2, x]/4 + Defer[Int][(Sqrt[1 + x^2]*Sqrt[
1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x), x]/4 + Defer[Int][(Sqrt[1 + x^2]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1
+ x)^2, x]/4 + Defer[Int][(Sqrt[1 + x^2]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 + x), x]/4

Rubi steps

\begin {align*} \int \frac {\sqrt {1+x^2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx &=\int \left (\frac {\sqrt {1+x^2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{4 (1-x)^2}+\frac {\sqrt {1+x^2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{4 (1+x)^2}+\frac {\sqrt {1+x^2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 \left (1-x^2\right )}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\sqrt {1+x^2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1-x)^2} \, dx+\frac {1}{4} \int \frac {\sqrt {1+x^2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1+x)^2} \, dx+\frac {1}{2} \int \frac {\sqrt {1+x^2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^2} \, dx\\ &=\frac {1}{4} \int \frac {\sqrt {1+x^2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1-x)^2} \, dx+\frac {1}{4} \int \frac {\sqrt {1+x^2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1+x)^2} \, dx+\frac {1}{2} \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}{4} \int \frac {\sqrt {1+x^2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1-x)^2} \, dx+\frac {1}{4} \int \frac {\sqrt {1+x^2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x} \, dx+\frac {1}{4} \int \frac {\sqrt {1+x^2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1+x)^2} \, dx+\frac {1}{4} \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 = 0.43, size = 511, normalized size = 1.43 \begin {gather*} -\frac {\left (1+3 x^2+3 x \sqrt {1+x^2}\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 \left (-1+x^2\right ) \left (1+2 x^2+2 x \sqrt {1+x^2}\right )}+\text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )}{2 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]-\frac {1}{16} \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {14 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2+3 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{2 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]+\text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )}{-2 \text {$\#$1}+4 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]+\frac {1}{16} \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-16 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2+3 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{-2 \text {$\#$1}+4 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((1 + 3*x^2 + 3*x*Sqrt[1 + x^2])*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/((-1 + x^2)*(1 + 2*x^2 + 2*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*#1^3 - 3*#1^5
+ #1^7) & ] - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (14*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log
[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + 3*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(2*#1^3 -
3*#1^5 + #1^7) & ]/16 + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]
- #1]/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ] + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-16*Log[Sqrt[1 +
Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + 3*Log[Sqrt[1 + Sqrt[x +
Sqrt[1 + x^2]]] - #1]*#1^4)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ]/16

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x^{2}+1}\, \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{\left (-x^{2}+1\right )^{2}}\, dx\]

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)^2,x)

[Out]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 1.52, size = 6606, normalized size = 18.50 \begin {gather*} \text {Too large to display} \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)^2,x, algorithm="fricas")

[Out]

-1/16*(sqrt(1/2)*(x^2 - 1)*sqrt(2*sqrt(1/2)*sqrt(85*sqrt(2) - 41) - sqrt(2) + 14)*log(1/4*sqrt(1/2)*(10203*(2*
sqrt(1/2)*sqrt(85*sqrt(2) - 41) + sqrt(2) - 14)^3 + (20406*sqrt(1/2)*sqrt(85*sqrt(2) - 41) + 10203*sqrt(2) - 1
48696)*(2*sqrt(1/2)*sqrt(85*sqrt(2) - 41) - sqrt(2) + 14)^2 + 571368*(2*sqrt(1/2)*sqrt(85*sqrt(2) - 41) + sqrt
(2) - 14)^2 - 3*(3401*(2*sqrt(1/2)*sqrt(85*sqrt(2) - 41) + sqrt(2) - 14)^2 + 380912*sqrt(1/2)*sqrt(85*sqrt(2)
- 41) + 190456*sqrt(2) - 2723784)*(2*sqrt(1/2)*sqrt(85*sqrt(2) - 41) - sqrt(2) + 14) + 27262416*sqrt(1/2)*sqrt
(85*sqrt(2) - 41) + 13631208*sqrt(2) - 192953624)*sqrt(2*sqrt(1/2)*sqrt(85*sqrt(2) - 41) - sqrt(2) + 14) + 825
917*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - sqrt(1/2)*(x^2 - 1)*sqrt(2*sqrt(1/2)*sqrt(85*sqrt(2) - 41) - sqrt(2)
+ 14)*log(-1/4*sqrt(1/2)*(10203*(2*sqrt(1/2)*sqrt(85*sqrt(2) - 41) + sqrt(2) - 14)^3 + (20406*sqrt(1/2)*sqrt(8
5*sqrt(2) - 41) + 10203*sqrt(2) - 148696)*(2*sqrt(1/2)*sqrt(85*sqrt(2) - 41) - sqrt(2) + 14)^2 + 571368*(2*sqr
t(1/2)*sqrt(85*sqrt(2) - 41) + sqrt(2) - 14)^2 - 3*(3401*(2*sqrt(1/2)*sqrt(85*sqrt(2) - 41) + sqrt(2) - 14)^2
+ 380912*sqrt(1/2)*sqrt(85*sqrt(2) - 41) + 190456*sqrt(2) - 2723784)*(2*sqrt(1/2)*sqrt(85*sqrt(2) - 41) - sqrt
(2) + 14) + 27262416*sqrt(1/2)*sqrt(85*sqrt(2) - 41) + 13631208*sqrt(2) - 192953624)*sqrt(2*sqrt(1/2)*sqrt(85*
sqrt(2) - 41) - sqrt(2) + 14) + 825917*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - sqrt(1/2)*(x^2 - 1)*sqrt(2*sqrt(1/
2)*sqrt(65*sqrt(2) + 47) + sqrt(2) + 16)*log(1/4*sqrt(1/2)*((6150*sqrt(1/2)*sqrt(65*sqrt(2) + 47) - 3075*sqrt(
2) - 129614)*(2*sqrt(1/2)*sqrt(65*sqrt(2) + 47) + sqrt(2) + 16)^2 + 3075*(2*sqrt(1/2)*sqrt(65*sqrt(2) + 47) -
sqrt(2) - 16)^3 - 3*(1025*(2*sqrt(1/2)*sqrt(65*sqrt(2) + 47) - sqrt(2) - 16)^2 + 131200*sqrt(1/2)*sqrt(65*sqrt
(2) + 47) - 65600*sqrt(2) - 1943144)*(2*sqrt(1/2)*sqrt(65*sqrt(2) + 47) + sqrt(2) + 16) + 196800*(2*sqrt(1/2)*
sqrt(65*sqrt(2) + 47) - sqrt(2) - 16)^2 + 8265600*sqrt(1/2)*sqrt(65*sqrt(2) + 47) - 4132800*sqrt(2) - 75955768
)*sqrt(2*sqrt(1/2)*sqrt(65*sqrt(2) + 47) + sqrt(2) + 16) + 10121717*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + sqrt(
1/2)*(x^2 - 1)*sqrt(2*sqrt(1/2)*sqrt(65*sqrt(2) + 47) + sqrt(2) + 16)*log(-1/4*sqrt(1/2)*((6150*sqrt(1/2)*sqrt
(65*sqrt(2) + 47) - 3075*sqrt(2) - 129614)*(2*sqrt(1/2)*sqrt(65*sqrt(2) + 47) + sqrt(2) + 16)^2 + 3075*(2*sqrt
(1/2)*sqrt(65*sqrt(2) + 47) - sqrt(2) - 16)^3 - 3*(1025*(2*sqrt(1/2)*sqrt(65*sqrt(2) + 47) - sqrt(2) - 16)^2 +
131200*sqrt(1/2)*sqrt(65*sqrt(2) + 47) - 65600*sqrt(2) - 1943144)*(2*sqrt(1/2)*sqrt(65*sqrt(2) + 47) + sqrt(2
) + 16) + 196800*(2*sqrt(1/2)*sqrt(65*sqrt(2) + 47) - sqrt(2) - 16)^2 + 8265600*sqrt(1/2)*sqrt(65*sqrt(2) + 47
) - 4132800*sqrt(2) - 75955768)*sqrt(2*sqrt(1/2)*sqrt(65*sqrt(2) + 47) + sqrt(2) + 16) + 10121717*sqrt(sqrt(x
+ sqrt(x^2 + 1)) + 1)) - (x^2 - 1)*sqrt(sqrt(2)*sqrt(-3/32*(2*sqrt(1/2)*sqrt(85*sqrt(2) - 41) + sqrt(2) - 14)^
2 + 1/16*(2*sqrt(1/2)*sqrt(85*sqrt(2) - 41) + sqrt(2) + 42)*(2*sqrt(1/2)*sqrt(85*sqrt(2) - 41) - sqrt(2) + 14)
- 3/32*(2*sqrt(1/2)*sqrt(85*sqrt(2) - 41) - sqrt(2) + 14)^2 - 7*sqrt(1/2)*sqrt(85*sqrt(2) - 41) - 7/2*sqrt(2)
- 20) + 1/2*sqrt(2) + 7)*log(1/8*((20406*sqrt(1/2)*sqrt(85*sqrt(2) - 41) + 10203*sqrt(2) - 148696)*(2*sqrt(1/
2)*sqrt(85*sqrt(2) - 41) - sqrt(2) + 14)^2 - 5854*(2*sqrt(1/2)*sqrt(85*sqrt(2) - 41) + sqrt(2) - 14)^2 - 3*(34
01*(2*sqrt(1/2)*sqrt(85*sqrt(2) - 41) + sqrt(2) - 14)^2 + 380912*sqrt(1/2)*sqrt(85*sqrt(2) - 41) + 190456*sqrt
(2) - 2723784)*(2*sqrt(1/2)*sqrt(85*sqrt(2) - 41) - sqrt(2) + 14) + 4*sqrt(-3/32*(2*sqrt(1/2)*sqrt(85*sqrt(2)
- 41) + sqrt(2) - 14)^2 + 1/16*(2*sqrt(1/2)*sqrt(85*sqrt(2) - 41) + sqrt(2) + 42)*(2*sqrt(1/2)*sqrt(85*sqrt(2)
- 41) - sqrt(2) + 14) - 3/32*(2*sqrt(1/2)*sqrt(85*sqrt(2) - 41) - sqrt(2) + 14)^2 - 7*sqrt(1/2)*sqrt(85*sqrt(
2) - 41) - 7/2*sqrt(2) - 20)*((10203*sqrt(2)*(2*sqrt(1/2)*sqrt(85*sqrt(2) - 41) + sqrt(2) - 14) - 5854*sqrt(2)
)*(2*sqrt(1/2)*sqrt(85*sqrt(2) - 41) - sqrt(2) + 14) + 5854*sqrt(2)*(2*sqrt(1/2)*sqrt(85*sqrt(2) - 41) + sqrt(
2) - 14) + 155624*sqrt(2)) - 344400*sqrt(1/2)*sqrt(85*sqrt(2) - 41) - 172200*sqrt(2) - 282720)*sqrt(sqrt(2)*sq
rt(-3/32*(2*sqrt(1/2)*sqrt(85*sqrt(2) - 41) + sqrt(2) - 14)^2 + 1/16*(2*sqrt(1/2)*sqrt(85*sqrt(2) - 41) + sqrt
(2) + 42)*(2*sqrt(1/2)*sqrt(85*sqrt(2) - 41) - sqrt(2) + 14) - 3/32*(2*sqrt(1/2)*sqrt(85*sqrt(2) - 41) - sqrt(
2) + 14)^2 - 7*sqrt(1/2)*sqrt(85*sqrt(2) - 41) - 7/2*sqrt(2) - 20) + 1/2*sqrt(2) + 7) + 825917*sqrt(sqrt(x + s
qrt(x^2 + 1)) + 1)) + (x^2 - 1)*sqrt(sqrt(2)*sqrt(-3/32*(2*sqrt(1/2)*sqrt(85*sqrt(2) - 41) + sqrt(2) - 14)^2 +
1/16*(2*sqrt(1/2)*sqrt(85*sqrt(2) - 41) + sqrt(2) + 42)*(2*sqrt(1/2)*sqrt(85*sqrt(2) - 41) - sqrt(2) + 14) -
3/32*(2*sqrt(1/2)*sqrt(85*sqrt(2) - 41) - sqrt(2) + 14)^2 - 7*sqrt(1/2)*sqrt(85*sqrt(2) - 41) - 7/2*sqrt(2) -
20) + 1/2*sqrt(2) + 7)*log(-1/8*((20406*sqrt(1/2)*sqrt(85*sqrt(2) - 41) + 10203*sqrt(2) - 148696)*(2*sqrt(1/2)
*sqrt(85*sqrt(2) - 41) - sqrt(2) + 14)^2 - 5854*(2*sqrt(1/2)*sqrt(85*sqrt(2) - 41) + sqrt(2) - 14)^2 - 3*(3401
*(2*sqrt(1/2)*sqrt(85*sqrt(2) - 41) + sqrt(2) -...

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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}}{\left (x - 1\right )^{2} \left (x + 1\right )^{2}}\, 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)**2,x)

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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)^2,x, algorithm="giac")

[Out]

integrate(sqrt(x^2 + 1)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)/(x^2 - 1)^2, 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}}{{\left (x^2-1\right )}^2} \,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)^2,x)

[Out]

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

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