3.28.84 \(\int \frac {(1+x^2) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1-x^2)^2 \sqrt {x+\sqrt {1+x^2}}} \, dx\) [2784]

Optimal. Leaf size=267 \[ -\frac {x \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (-1+x^2\right ) \sqrt {x+\sqrt {1+x^2}}}-\frac {1}{8} \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\& ,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2}{2 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\& \right ]-\frac {1}{8} \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\& ,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-2 \text {$\#$1}+4 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\& \right ] \]

[Out]

Unintegrable

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Rubi [F]
time = 2.43, 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 {\left (1+x^2\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {\left (1+x^2\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx &=\int \left (\frac {2 \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}}-\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right ) \sqrt {x+\sqrt {1+x^2}}}\right ) \, dx\\ &=2 \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx-\int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx\\ &=2 \int \left (\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{4 (1-x)^2 \sqrt {x+\sqrt {1+x^2}}}+\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{4 (1+x)^2 \sqrt {x+\sqrt {1+x^2}}}+\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 \left (1-x^2\right ) \sqrt {x+\sqrt {1+x^2}}}\right ) \, dx-\int \left (\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 (1-x) \sqrt {x+\sqrt {1+x^2}}}+\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 (1+x) \sqrt {x+\sqrt {1+x^2}}}\right ) \, dx\\ &=\frac {1}{2} \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1-x)^2 \sqrt {x+\sqrt {1+x^2}}} \, dx-\frac {1}{2} \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1-x) \sqrt {x+\sqrt {1+x^2}}} \, dx+\frac {1}{2} \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1+x)^2 \sqrt {x+\sqrt {1+x^2}}} \, dx-\frac {1}{2} \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1+x) \sqrt {x+\sqrt {1+x^2}}} \, dx+\int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx\\ &=\frac {1}{2} \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1-x)^2 \sqrt {x+\sqrt {1+x^2}}} \, dx-\frac {1}{2} \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1-x) \sqrt {x+\sqrt {1+x^2}}} \, dx+\frac {1}{2} \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1+x)^2 \sqrt {x+\sqrt {1+x^2}}} \, dx-\frac {1}{2} \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1+x) \sqrt {x+\sqrt {1+x^2}}} \, dx+\int \left (\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 (1-x) \sqrt {x+\sqrt {1+x^2}}}+\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 (1+x) \sqrt {x+\sqrt {1+x^2}}}\right ) \, dx\\ &=\frac {1}{2} \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1-x)^2 \sqrt {x+\sqrt {1+x^2}}} \, dx+\frac {1}{2} \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1+x)^2 \sqrt {x+\sqrt {1+x^2}}} \, dx\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 416, normalized size = 1.56 \begin {gather*} \frac {1}{8} \left (-\frac {8 x \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (-1+x^2\right ) \sqrt {x+\sqrt {1+x^2}}}+4 \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )}{2 \text {$\#$1}-3 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ]-\text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+5 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2}{2 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]+4 \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}}{-2+4 \text {$\#$1}^2-3 \text {$\#$1}^4+\text {$\#$1}^6}\&\right ]-\text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+5 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-2 \text {$\#$1}+4 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 1.70, size = 6469, normalized size = 24.23 \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+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^2+1)^2/(x+(x^2+1)^(1/2))^(1/2),x, algorithm="fricas")

[Out]

-1/16*(sqrt(2)*(x^2 - 1)*sqrt(sqrt(2)*sqrt(-3/32*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) + 7*sqrt(2) - 8)^2 - 3/32*(2*s
qrt(1/2)*sqrt(sqrt(2) + 1) - 7*sqrt(2) + 8)^2 + 1/16*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) + 7*sqrt(2) - 8)*(2*sqrt(1
/2)*sqrt(sqrt(2) + 1) - 7*sqrt(2) - 24) + 4*sqrt(1/2)*sqrt(sqrt(2) + 1) - 14*sqrt(2) + 25) - 7/2*sqrt(2) - 4)*
log(1/4*((6*sqrt(2)*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) - 7*sqrt(2) + 8) - 721*sqrt(2))*(2*sqrt(1/2)*sqrt(sqrt(2) +
1) + 7*sqrt(2) - 8)^2 - 721*sqrt(2)*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) - 7*sqrt(2) + 8)^2 - 3*(2*sqrt(2)*(2*sqrt(
1/2)*sqrt(sqrt(2) + 1) - 7*sqrt(2) + 8)^2 - 64*sqrt(2)*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) - 7*sqrt(2) + 8) + 2597*
sqrt(2))*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) + 7*sqrt(2) - 8) + 8*((12*sqrt(1/2)*sqrt(sqrt(2) + 1) - 42*sqrt(2) - 6
73)*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) + 7*sqrt(2) - 8) + 1442*sqrt(1/2)*sqrt(sqrt(2) + 1) - 5047*sqrt(2) - 9513)*
sqrt(-3/32*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) + 7*sqrt(2) - 8)^2 - 3/32*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) - 7*sqrt(2)
+ 8)^2 + 1/16*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) + 7*sqrt(2) - 8)*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) - 7*sqrt(2) - 24
) + 4*sqrt(1/2)*sqrt(sqrt(2) + 1) - 14*sqrt(2) + 25) + 7791*sqrt(2)*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) - 7*sqrt(2)
+ 8) + 32760*sqrt(2))*sqrt(sqrt(2)*sqrt(-3/32*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) + 7*sqrt(2) - 8)^2 - 3/32*(2*sqr
t(1/2)*sqrt(sqrt(2) + 1) - 7*sqrt(2) + 8)^2 + 1/16*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) + 7*sqrt(2) - 8)*(2*sqrt(1/2
)*sqrt(sqrt(2) + 1) - 7*sqrt(2) - 24) + 4*sqrt(1/2)*sqrt(sqrt(2) + 1) - 14*sqrt(2) + 25) - 7/2*sqrt(2) - 4) +
32935*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - sqrt(2)*(x^2 - 1)*sqrt(sqrt(2)*sqrt(-3/32*(2*sqrt(1/2)*sqrt(sqrt(2)
+ 1) + 7*sqrt(2) - 8)^2 - 3/32*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) - 7*sqrt(2) + 8)^2 + 1/16*(2*sqrt(1/2)*sqrt(sqr
t(2) + 1) + 7*sqrt(2) - 8)*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) - 7*sqrt(2) - 24) + 4*sqrt(1/2)*sqrt(sqrt(2) + 1) -
14*sqrt(2) + 25) - 7/2*sqrt(2) - 4)*log(-1/4*((6*sqrt(2)*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) - 7*sqrt(2) + 8) - 721
*sqrt(2))*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) + 7*sqrt(2) - 8)^2 - 721*sqrt(2)*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) - 7*s
qrt(2) + 8)^2 - 3*(2*sqrt(2)*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) - 7*sqrt(2) + 8)^2 - 64*sqrt(2)*(2*sqrt(1/2)*sqrt(
sqrt(2) + 1) - 7*sqrt(2) + 8) + 2597*sqrt(2))*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) + 7*sqrt(2) - 8) + 8*((12*sqrt(1/
2)*sqrt(sqrt(2) + 1) - 42*sqrt(2) - 673)*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) + 7*sqrt(2) - 8) + 1442*sqrt(1/2)*sqrt
(sqrt(2) + 1) - 5047*sqrt(2) - 9513)*sqrt(-3/32*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) + 7*sqrt(2) - 8)^2 - 3/32*(2*sq
rt(1/2)*sqrt(sqrt(2) + 1) - 7*sqrt(2) + 8)^2 + 1/16*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) + 7*sqrt(2) - 8)*(2*sqrt(1/
2)*sqrt(sqrt(2) + 1) - 7*sqrt(2) - 24) + 4*sqrt(1/2)*sqrt(sqrt(2) + 1) - 14*sqrt(2) + 25) + 7791*sqrt(2)*(2*sq
rt(1/2)*sqrt(sqrt(2) + 1) - 7*sqrt(2) + 8) + 32760*sqrt(2))*sqrt(sqrt(2)*sqrt(-3/32*(2*sqrt(1/2)*sqrt(sqrt(2)
+ 1) + 7*sqrt(2) - 8)^2 - 3/32*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) - 7*sqrt(2) + 8)^2 + 1/16*(2*sqrt(1/2)*sqrt(sqrt
(2) + 1) + 7*sqrt(2) - 8)*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) - 7*sqrt(2) - 24) + 4*sqrt(1/2)*sqrt(sqrt(2) + 1) - 1
4*sqrt(2) + 25) - 7/2*sqrt(2) - 4) + 32935*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + sqrt(2)*(x^2 - 1)*sqrt(-sqrt(2
)*sqrt(-3/32*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) + 7*sqrt(2) - 8)^2 - 3/32*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) - 7*sqrt(
2) + 8)^2 + 1/16*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) + 7*sqrt(2) - 8)*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) - 7*sqrt(2) -
24) + 4*sqrt(1/2)*sqrt(sqrt(2) + 1) - 14*sqrt(2) + 25) - 7/2*sqrt(2) - 4)*log(1/4*((6*sqrt(2)*(2*sqrt(1/2)*sqr
t(sqrt(2) + 1) - 7*sqrt(2) + 8) - 721*sqrt(2))*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) + 7*sqrt(2) - 8)^2 - 721*sqrt(2)
*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) - 7*sqrt(2) + 8)^2 - 3*(2*sqrt(2)*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) - 7*sqrt(2) +
8)^2 - 64*sqrt(2)*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) - 7*sqrt(2) + 8) + 2597*sqrt(2))*(2*sqrt(1/2)*sqrt(sqrt(2) +
1) + 7*sqrt(2) - 8) - 8*((12*sqrt(1/2)*sqrt(sqrt(2) + 1) - 42*sqrt(2) - 673)*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) +
7*sqrt(2) - 8) + 1442*sqrt(1/2)*sqrt(sqrt(2) + 1) - 5047*sqrt(2) - 9513)*sqrt(-3/32*(2*sqrt(1/2)*sqrt(sqrt(2)
+ 1) + 7*sqrt(2) - 8)^2 - 3/32*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) - 7*sqrt(2) + 8)^2 + 1/16*(2*sqrt(1/2)*sqrt(sqr
t(2) + 1) + 7*sqrt(2) - 8)*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) - 7*sqrt(2) - 24) + 4*sqrt(1/2)*sqrt(sqrt(2) + 1) -
14*sqrt(2) + 25) + 7791*sqrt(2)*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) - 7*sqrt(2) + 8) + 32760*sqrt(2))*sqrt(-sqrt(2)
*sqrt(-3/32*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) + 7*sqrt(2) - 8)^2 - 3/32*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) - 7*sqrt(2
) + 8)^2 + 1/16*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) + 7*sqrt(2) - 8)*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) - 7*sqrt(2) - 2
4) + 4*sqrt(1/2)*sqrt(sqrt(2) + 1) - 14*sqrt(2) + 25) - 7/2*sqrt(2) - 4) + 32935*sqrt(sqrt(x + sqrt(x^2 + 1))
+ 1)) - sqrt(2)*(x^2 - 1)*sqrt(-sqrt(2)*sqrt(-3/32*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) + 7*sqrt(2) - 8)^2 - 3/32*(2
*sqrt(1/2)*sqrt(sqrt(2) + 1) - 7*sqrt(2) + 8)^2 + 1/16*(2*sqrt(1/2)*sqrt(sqrt(2) + 1) + 7*sqrt(2) - 8)*(2*sqrt
(1/2)*sqrt(sqrt(2) + 1) - 7*sqrt(2) - 24) + 4*s...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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