3.31.82 \(\int \frac {(1+x^4) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^4} \, dx\) [3082]

Optimal. Leaf size=514 \[ \frac {\left (75+8 x-60 x^2\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+(-16-6 x) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\sqrt {1+x^2} \left ((8-60 x) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-6 \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )}{105 \sqrt {x+\sqrt {1+x^2}}}+\tanh ^{-1}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )-\frac {1}{2} \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\& ,\frac {2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{2 \text {$\#$1}-3 \text {$\#$1}^3+\text {$\#$1}^5}\& \right ]+\text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\& ,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^2}\& \right ]-\frac {1}{2} \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\& ,\frac {-2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^3+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^5}{-2+4 \text {$\#$1}^2-3 \text {$\#$1}^4+\text {$\#$1}^6}\& \right ] \]

[Out]

Unintegrable

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Rubi [F]
time = 2.05, 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^4\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 477, normalized size = 0.93 \begin {gather*} -\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \left (-75+60 x^2-8 \sqrt {1+x^2}+16 \sqrt {x+\sqrt {1+x^2}}+6 \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}+x \left (-8+60 \sqrt {1+x^2}+6 \sqrt {x+\sqrt {1+x^2}}\right )\right )}{105 \sqrt {x+\sqrt {1+x^2}}}+\tanh ^{-1}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )-\frac {1}{2} \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{2 \text {$\#$1}-3 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ]+\text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^2}\&\right ]-\frac {1}{2} \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^3+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^5}{-2+4 \text {$\#$1}^2-3 \text {$\#$1}^4+\text {$\#$1}^6}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/105*(Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]*(-75 + 60*x^2 - 8*Sqrt[1 + x^2] + 16*Sqrt[x + Sqrt[1 + x^2]] + 6*Sqr
t[1 + x^2]*Sqrt[x + Sqrt[1 + x^2]] + x*(-8 + 60*Sqrt[1 + x^2] + 6*Sqrt[x + Sqrt[1 + x^2]])))/Sqrt[x + Sqrt[1 +
x^2]] + ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^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 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + Log[Sqrt[1 + Sqrt[x + Sq
rt[1 + x^2]]] - #1]*#1^4)/(2*#1 - 3*#1^3 + #1^5) & ]/2 + 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) & ] - 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 + Log[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 = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{4}+1\right ) \sqrt {x +\sqrt {x^{2}+1}}\, \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{-x^{4}+1}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4+1)*(x+(x^2+1)^(1/2))^(1/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^4+1),x)

[Out]

int((x^4+1)*(x+(x^2+1)^(1/2))^(1/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^4+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^4+1)*(x+(x^2+1)^(1/2))^(1/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^4+1),x, algorithm="maxima")

[Out]

-integrate((x^4 + 1)*sqrt(x + sqrt(x^2 + 1))*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)/(x^4 - 1), x)

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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 1.95, size = 7760, normalized size = 15.10 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)*(x+(x^2+1)^(1/2))^(1/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^4+1),x, algorithm="fricas")

[Out]

-1/105*((135*x - 75*sqrt(x^2 + 1) - 8)*sqrt(x + sqrt(x^2 + 1)) + 6*x + 6*sqrt(x^2 + 1) + 16)*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)*(s
qrt(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) - sqr
t(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) - 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) - 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 + 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*sqr
t(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(-(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) + 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*(sq
rt(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*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 1/2*sqrt(sqrt(2) + sqrt(5*sqrt(2) - 7) - 1)*log(-((16*sq
rt(2) - 16*sqrt(5*sqrt(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) + sqr
t(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*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 + 2
89*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) - sqrt(5*sqrt(2) - 7) - 1)*log(-(16*(sqrt(2) - sqrt(5*sqrt(2) - 7) - 1)^3 + 6
9*(sqrt(2) - sqrt(5*sqrt(2) - 7) - 1)^2 + 289*sqrt(2) - 289*sqrt(5*sqrt(2) - 7) - 428)*sqrt(sqrt(2) - sqrt(5*s
qrt(2) - 7) - 1) + 61*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 + 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 + 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*sqr
t(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*s
qrt(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(-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*s
qrt(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*sqr
t(5*sqrt(2) + 7) - 19)*(sqrt(2) + sqrt(5*sqrt(2) + 7) + 1) + (sqrt(2) - sqrt(5*sqrt(2) + 7) + 1)^2 + 4*((4*sqr
t(2) - 4*sqrt(5*sqrt(2) + 7) + 5)*(sqrt(2) + sq...

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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^{4} - 1}\, dx - \int \frac {x^{4} \sqrt {x + \sqrt {x^{2} + 1}} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{x^{4} - 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4+1)*(x+(x**2+1)**(1/2))**(1/2)*(1+(x+(x**2+1)**(1/2))**(1/2))**(1/2)/(-x**4+1),x)

[Out]

-Integral(sqrt(x + sqrt(x**2 + 1))*sqrt(sqrt(x + sqrt(x**2 + 1)) + 1)/(x**4 - 1), x) - Integral(x**4*sqrt(x +
sqrt(x**2 + 1))*sqrt(sqrt(x + sqrt(x**2 + 1)) + 1)/(x**4 - 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^4+1)*(x+(x^2+1)^(1/2))^(1/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^4+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}\,\left (x^4+1\right )\,\sqrt {x+\sqrt {x^2+1}}}{x^4-1} \,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^4 + 1)*(x + (x^2 + 1)^(1/2))^(1/2))/(x^4 - 1),x)

[Out]

int(-(((x + (x^2 + 1)^(1/2))^(1/2) + 1)^(1/2)*(x^4 + 1)*(x + (x^2 + 1)^(1/2))^(1/2))/(x^4 - 1), x)

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