3.28.53 \(\int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^2} \, dx\) [2753]
Optimal. Leaf size=257 \[ -\frac {1}{2} \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\& ,\frac {2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{2 \text {$\#$1}-3 \text {$\#$1}^3+\text {$\#$1}^5}\& \right ]-\frac {1}{2} \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\& ,\frac {-2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^3+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^5}{-2+4 \text {$\#$1}^2-3 \text {$\#$1}^4+\text {$\#$1}^6}\& \right ] \]
[Out]
Unintegrable
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Rubi [F]
time = 0.82, 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]
Defer[Int][(Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x), x]/2 + Defer[Int][(Sqrt[x + Sq
rt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 + x), x]/2
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 {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 (1-x)}+\frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 (1+x)}\right ) \, dx\\ &=\frac {1}{2} \int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x} \, dx+\frac {1}{2} \int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1+x} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 257, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{2 \text {$\#$1}-3 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ]-\frac {1}{2} \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^3+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^5}{-2+4 \text {$\#$1}^2-3 \text {$\#$1}^4+\text {$\#$1}^6}\&\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]
-1/2*RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt[1 + S
qrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(2*#1 - 3*#1^3 + #1^5)
& ] - RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (-2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^3 + L
og[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^5)/(-2 + 4*#1^2 - 3*#1^4 + #1^6) & ]/2
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Maple [F]
time = 0.04, 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 = 1.47, size = 5201, normalized size = 20.24 \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]
1/2*sqrt(sqrt(2) + sqrt(5*sqrt(2) + 7) + 1)*log(((4*sqrt(2) - 4*sqrt(5*sqrt(2) + 7) + 5)*(sqrt(2) + sqrt(5*sqr
t(2) + 7) + 1)^2 + 4*(sqrt(2) - sqrt(5*sqrt(2) + 7) + 1)^3 + (4*(sqrt(2) - sqrt(5*sqrt(2) + 7) + 1)^2 - 16*sqr
t(2) + 16*sqrt(5*sqrt(2) + 7) - 19)*(sqrt(2) + sqrt(5*sqrt(2) + 7) + 1) - 16*(sqrt(2) - sqrt(5*sqrt(2) + 7) +
1)^2 - 48*sqrt(2) + 48*sqrt(5*sqrt(2) + 7) - 63)*sqrt(sqrt(2) + sqrt(5*sqrt(2) + 7) + 1) + 5*sqrt(sqrt(x + sqr
t(x^2 + 1)) + 1)) - 1/2*sqrt(sqrt(2) + sqrt(5*sqrt(2) + 7) + 1)*log(-((4*sqrt(2) - 4*sqrt(5*sqrt(2) + 7) + 5)*
(sqrt(2) + sqrt(5*sqrt(2) + 7) + 1)^2 + 4*(sqrt(2) - sqrt(5*sqrt(2) + 7) + 1)^3 + (4*(sqrt(2) - sqrt(5*sqrt(2)
+ 7) + 1)^2 - 16*sqrt(2) + 16*sqrt(5*sqrt(2) + 7) - 19)*(sqrt(2) + sqrt(5*sqrt(2) + 7) + 1) - 16*(sqrt(2) - s
qrt(5*sqrt(2) + 7) + 1)^2 - 48*sqrt(2) + 48*sqrt(5*sqrt(2) + 7) - 63)*sqrt(sqrt(2) + sqrt(5*sqrt(2) + 7) + 1)
+ 5*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 1/2*sqrt(sqrt(2) - sqrt(5*sqrt(2) + 7) + 1)*log((4*(sqrt(2) - sqrt(5*
sqrt(2) + 7) + 1)^3 - 17*(sqrt(2) - sqrt(5*sqrt(2) + 7) + 1)^2 - 45*sqrt(2) + 45*sqrt(5*sqrt(2) + 7) - 62)*sqr
t(sqrt(2) - sqrt(5*sqrt(2) + 7) + 1) + 5*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + 1/2*sqrt(sqrt(2) - sqrt(5*sqrt(2
) + 7) + 1)*log(-(4*(sqrt(2) - sqrt(5*sqrt(2) + 7) + 1)^3 - 17*(sqrt(2) - sqrt(5*sqrt(2) + 7) + 1)^2 - 45*sqrt
(2) + 45*sqrt(5*sqrt(2) + 7) - 62)*sqrt(sqrt(2) - sqrt(5*sqrt(2) + 7) + 1) + 5*sqrt(sqrt(x + sqrt(x^2 + 1)) +
1)) + 1/2*sqrt(sqrt(2) + sqrt(5*sqrt(2) - 7) - 1)*log(((16*sqrt(2) - 16*sqrt(5*sqrt(2) - 7) - 21)*(sqrt(2) + s
qrt(5*sqrt(2) - 7) - 1)^2 + 16*(sqrt(2) - sqrt(5*sqrt(2) - 7) - 1)^3 + (16*(sqrt(2) - sqrt(5*sqrt(2) - 7) - 1)
^2 + 64*sqrt(2) - 64*sqrt(5*sqrt(2) - 7) - 97)*(sqrt(2) + sqrt(5*sqrt(2) - 7) - 1) + 64*(sqrt(2) - sqrt(5*sqrt
(2) - 7) - 1)^2 + 256*sqrt(2) - 256*sqrt(5*sqrt(2) - 7) - 373)*sqrt(sqrt(2) + sqrt(5*sqrt(2) - 7) - 1) + 61*sq
rt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 1/2*sqrt(sqrt(2) + sqrt(5*sqrt(2) - 7) - 1)*log(-((16*sqrt(2) - 16*sqrt(5*s
qrt(2) - 7) - 21)*(sqrt(2) + sqrt(5*sqrt(2) - 7) - 1)^2 + 16*(sqrt(2) - sqrt(5*sqrt(2) - 7) - 1)^3 + (16*(sqrt
(2) - sqrt(5*sqrt(2) - 7) - 1)^2 + 64*sqrt(2) - 64*sqrt(5*sqrt(2) - 7) - 97)*(sqrt(2) + sqrt(5*sqrt(2) - 7) -
1) + 64*(sqrt(2) - sqrt(5*sqrt(2) - 7) - 1)^2 + 256*sqrt(2) - 256*sqrt(5*sqrt(2) - 7) - 373)*sqrt(sqrt(2) + sq
rt(5*sqrt(2) - 7) - 1) + 61*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 1/2*sqrt(sqrt(2) - sqrt(5*sqrt(2) - 7) - 1)*l
og((16*(sqrt(2) - sqrt(5*sqrt(2) - 7) - 1)^3 + 69*(sqrt(2) - sqrt(5*sqrt(2) - 7) - 1)^2 + 289*sqrt(2) - 289*sq
rt(5*sqrt(2) - 7) - 428)*sqrt(sqrt(2) - sqrt(5*sqrt(2) - 7) - 1) + 61*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + 1/2
*sqrt(sqrt(2) - sqrt(5*sqrt(2) - 7) - 1)*log(-(16*(sqrt(2) - sqrt(5*sqrt(2) - 7) - 1)^3 + 69*(sqrt(2) - sqrt(5
*sqrt(2) - 7) - 1)^2 + 289*sqrt(2) - 289*sqrt(5*sqrt(2) - 7) - 428)*sqrt(sqrt(2) - sqrt(5*sqrt(2) - 7) - 1) +
61*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 1/2*sqrt(-sqrt(2) + 2*sqrt(-3/16*(sqrt(2) + sqrt(5*sqrt(2) + 7) + 1)^2
- 3/16*(sqrt(2) - sqrt(5*sqrt(2) + 7) + 1)^2 - 1/8*(sqrt(2) + sqrt(5*sqrt(2) + 7) + 1)*(sqrt(2) - sqrt(5*sqrt
(2) + 7) - 3) + 1/2*sqrt(2) - 1/2*sqrt(5*sqrt(2) + 7) + 9/2) + 1)*log(1/2*((4*sqrt(2) - 4*sqrt(5*sqrt(2) + 7)
+ 5)*(sqrt(2) + sqrt(5*sqrt(2) + 7) + 1)^2 + (4*(sqrt(2) - sqrt(5*sqrt(2) + 7) + 1)^2 - 16*sqrt(2) + 16*sqrt(5
*sqrt(2) + 7) - 19)*(sqrt(2) + sqrt(5*sqrt(2) + 7) + 1) + (sqrt(2) - sqrt(5*sqrt(2) + 7) + 1)^2 + 4*((4*sqrt(2
) - 4*sqrt(5*sqrt(2) + 7) + 5)*(sqrt(2) + sqrt(5*sqrt(2) + 7) + 1) + sqrt(2) - sqrt(5*sqrt(2) + 7))*sqrt(-3/16
*(sqrt(2) + sqrt(5*sqrt(2) + 7) + 1)^2 - 3/16*(sqrt(2) - sqrt(5*sqrt(2) + 7) + 1)^2 - 1/8*(sqrt(2) + sqrt(5*sq
rt(2) + 7) + 1)*(sqrt(2) - sqrt(5*sqrt(2) + 7) - 3) + 1/2*sqrt(2) - 1/2*sqrt(5*sqrt(2) + 7) + 9/2) - 3*sqrt(2)
+ 3*sqrt(5*sqrt(2) + 7) - 1)*sqrt(-sqrt(2) + 2*sqrt(-3/16*(sqrt(2) + sqrt(5*sqrt(2) + 7) + 1)^2 - 3/16*(sqrt(
2) - sqrt(5*sqrt(2) + 7) + 1)^2 - 1/8*(sqrt(2) + sqrt(5*sqrt(2) + 7) + 1)*(sqrt(2) - sqrt(5*sqrt(2) + 7) - 3)
+ 1/2*sqrt(2) - 1/2*sqrt(5*sqrt(2) + 7) + 9/2) + 1) + 5*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + 1/2*sqrt(-sqrt(2)
+ 2*sqrt(-3/16*(sqrt(2) + sqrt(5*sqrt(2) + 7) + 1)^2 - 3/16*(sqrt(2) - sqrt(5*sqrt(2) + 7) + 1)^2 - 1/8*(sqrt
(2) + sqrt(5*sqrt(2) + 7) + 1)*(sqrt(2) - sqrt(5*sqrt(2) + 7) - 3) + 1/2*sqrt(2) - 1/2*sqrt(5*sqrt(2) + 7) + 9
/2) + 1)*log(-1/2*((4*sqrt(2) - 4*sqrt(5*sqrt(2) + 7) + 5)*(sqrt(2) + sqrt(5*sqrt(2) + 7) + 1)^2 + (4*(sqrt(2)
- sqrt(5*sqrt(2) + 7) + 1)^2 - 16*sqrt(2) + 16*sqrt(5*sqrt(2) + 7) - 19)*(sqrt(2) + sqrt(5*sqrt(2) + 7) + 1)
+ (sqrt(2) - sqrt(5*sqrt(2) + 7) + 1)^2 + 4*((4*sqrt(2) - 4*sqrt(5*sqrt(2) + 7) + 5)*(sqrt(2) + sqrt(5*sqrt(2)
+ 7) + 1) + sqrt(2) - sqrt(5*sqrt(2) + 7))*sqrt(-3/16*(sqrt(2) + sqrt(5*sqrt(2) + 7) + 1)^2 - 3/16*(sqrt(2) -
sqrt(5*sqrt(2) + 7) + 1)^2 - 1/8*(sqrt(2) + sqrt(5*sqrt(2) + 7) + 1)*(sqrt(2) - sqrt(5*sqrt(2) + 7) - 3) + 1/
2*sqrt(2) - 1/2*sqrt(5*sqrt(2) + 7) + 9/2) - 3*sqrt(2) + 3*sqrt(5*sqrt(2) + 7) - 1)*sqrt(-sqrt(2) + 2*sqrt(-3/
16*(sqrt(2) + sqrt(5*sqrt(2) + 7) + 1)^2 - 3/16...
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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 = -0.00 \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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