3.31.24 \(\int \frac {(1+x^4) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {1+x^2} (1-x^4)} \, dx\)

Optimal. Leaf size=428 \[ \frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8+8 \text {$\#$1}^7+24 \text {$\#$1}^6+32 \text {$\#$1}^5+16 \text {$\#$1}^4+1\& ,\frac {\log \left (-\text {$\#$1}+\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-1\right )}{\text {$\#$1}^7+7 \text {$\#$1}^6+18 \text {$\#$1}^5+20 \text {$\#$1}^4+8 \text {$\#$1}^3}\& \right ]-\frac {1}{2} \text {RootSum}\left [\text {$\#$1}^8+8 \text {$\#$1}^7+24 \text {$\#$1}^6+32 \text {$\#$1}^5+14 \text {$\#$1}^4-8 \text {$\#$1}^3-8 \text {$\#$1}^2-1\& ,\frac {\log \left (-\text {$\#$1}+\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-1\right )}{\text {$\#$1}^3+3 \text {$\#$1}^2+\text {$\#$1}-1}\& \right ]+\frac {1}{2} \text {RootSum}\left [\text {$\#$1}^8+8 \text {$\#$1}^7+24 \text {$\#$1}^6+32 \text {$\#$1}^5+18 \text {$\#$1}^4+8 \text {$\#$1}^3+8 \text {$\#$1}^2-1\& ,\frac {\text {$\#$1} \log \left (-\text {$\#$1}+\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-1\right )+\log \left (-\text {$\#$1}+\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-1\right )}{\text {$\#$1}^4+4 \text {$\#$1}^3+4 \text {$\#$1}^2+1}\& \right ]+\frac {\left (-4 x^2-5\right ) \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-4 x \sqrt {x^2+1} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{x^2+\sqrt {x^2+1} x+1}+4 \tanh ^{-1}\left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right ) \]

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Rubi [F]  time = 1.65, 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 {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {1+x^2} \left (1-x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Defer[Int][Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/(1 + x^2)^(3/2), x] - Defer[Int][Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]
]/Sqrt[1 + x^2], x] + Defer[Int][Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/((1 - x)*Sqrt[1 + x^2]), x]/2 + Defer[Int][
Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/((1 + x)*Sqrt[1 + x^2]), x]/2

Rubi steps

\begin {align*} \int \frac {\left (1+x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {1+x^2} \left (1-x^4\right )} \, dx &=\int \left (-\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {1+x^2}}+\frac {2 \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {1+x^2} \left (1-x^4\right )}\right ) \, dx\\ &=2 \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {1+x^2} \left (1-x^4\right )} \, dx-\int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {1+x^2}} \, dx\\ &=2 \int \left (\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 \left (1+x^2\right )^{3/2}}+\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 \left (1-x^2\right ) \sqrt {1+x^2}}\right ) \, dx-\int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {1+x^2}} \, dx\\ &=\int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^{3/2}} \, dx-\int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {1+x^2}} \, dx+\int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right ) \sqrt {1+x^2}} \, dx\\ &=\int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^{3/2}} \, dx-\int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {1+x^2}} \, dx+\int \left (\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 (1-x) \sqrt {1+x^2}}+\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 (1+x) \sqrt {1+x^2}}\right ) \, dx\\ &=\frac {1}{2} \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1-x) \sqrt {1+x^2}} \, dx+\frac {1}{2} \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1+x) \sqrt {1+x^2}} \, dx+\int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^{3/2}} \, dx-\int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {1+x^2}} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.29, size = 556, normalized size = 1.30 \begin {gather*} -\frac {2 \left (x^2+\sqrt {x^2+1} x+1\right ) \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+4 \text {$\#$1}^4-2\&,\frac {\log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )}{\text {$\#$1}^3-2 \text {$\#$1}}\&\right ]-2 \left (x^2+\sqrt {x^2+1} x+1\right ) \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+8 \text {$\#$1}^4-8 \text {$\#$1}^2+2\&,\frac {\text {$\#$1} \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )}{\text {$\#$1}^4-2 \text {$\#$1}^2+2}\&\right ]-\left (x^2+\sqrt {x^2+1} x+1\right ) \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+6 \text {$\#$1}^4-4 \text {$\#$1}^2+2\&,\frac {\log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )}{\text {$\#$1}^7-3 \text {$\#$1}^5+3 \text {$\#$1}^3-\text {$\#$1}}\&\right ]+16 \sqrt {\sqrt {\sqrt {x^2+1}+x}+1} x^2+16 \sqrt {x^2+1} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1} x+20 \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}+8 x^2 \log \left (1-\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right )-8 x^2 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}+1\right )+8 \sqrt {x^2+1} x \log \left (1-\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right )-8 \sqrt {x^2+1} x \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}+1\right )+8 \log \left (1-\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right )-8 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}+1\right )}{4 \left (x^2+\sqrt {x^2+1} x+1\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(20*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + 16*x^2*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + 16*x*Sqrt[1 + x^2]*Sqr
t[1 + Sqrt[x + Sqrt[1 + x^2]]] + 8*Log[1 - Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]] + 8*x^2*Log[1 - Sqrt[1 + Sqrt[x
+ Sqrt[1 + x^2]]]] + 8*x*Sqrt[1 + x^2]*Log[1 - Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]] - 8*Log[1 + Sqrt[1 + Sqrt[x
+ Sqrt[1 + x^2]]]] - 8*x^2*Log[1 + Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]] - 8*x*Sqrt[1 + x^2]*Log[1 + Sqrt[1 + Sqr
t[x + Sqrt[1 + x^2]]]] + 2*(1 + x^2 + x*Sqrt[1 + x^2])*RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , Log[Sqrt[1 + Sq
rt[x + Sqrt[1 + x^2]]] - #1]/(-2*#1 + #1^3) & ] - (1 + x^2 + x*Sqrt[1 + x^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 + 3*#1^3 - 3*#1^5 + #1^7) & ] - 2*(1 + x^2 +
x*Sqrt[1 + x^2])*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) & ])/(1 + x^2 + x*Sqrt[1 + x^2])

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IntegrateAlgebraic [A]  time = 0.00, size = 328, normalized size = 0.77 \begin {gather*} \frac {\left (-5-4 x^2\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-4 x \sqrt {1+x^2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1+x^2+x \sqrt {1+x^2}}+4 \tanh ^{-1}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )-\frac {1}{2} \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )}{-2 \text {$\#$1}+\text {$\#$1}^3}\&\right ]+\frac {1}{4} \text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )}{-\text {$\#$1}+3 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]+\frac {1}{2} \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}}{2-2 \text {$\#$1}^2+\text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-5 - 4*x^2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - 4*x*Sqrt[1 + x^2]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 + x^
2 + 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) & ]/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 + 3*#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 + Sqrt[1 + x^2]]] - #1]*#1)/(2 - 2*#1^2 + #1^4) & ]/2

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fricas [B]  time = 0.90, size = 4111, normalized size = 9.61

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32*(16*(x^2 - sqrt(2)*(x^2 + 1) + 1)*sqrt(-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2*sqrt(2))*(sqrt(2) + 2)^(3/
4)*sqrt(sqrt(2) + 1)*arctan(1/2*sqrt(sqrt(-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2*sqrt(2))*(sqrt(sqrt(2) + 2)*(
sqrt(2) - 2) - 2)*(sqrt(2) + 2)^(1/4)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 2*sqrt(x + sqrt(x^2 + 1)) + 2*sqrt(s
qrt(2) + 2) + 2)*((2*sqrt(2) - 3)*sqrt(sqrt(2) + 2)*sqrt(sqrt(2) + 1) + (2*sqrt(2) - 3)*sqrt(sqrt(2) + 1))*sqr
t(-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2*sqrt(2))*(sqrt(2) + 2)^(3/4) + 1/2*((3*sqrt(2) - 4)*sqrt(sqrt(2) + 2)
*sqrt(sqrt(2) + 1) + (3*sqrt(2) - 4)*sqrt(sqrt(2) + 1))*sqrt(-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2*sqrt(2))*(
sqrt(2) + 2)^(3/4)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) - sqrt(sqrt(2) + 2)*sqrt(sqrt(2) + 1)*(sqrt(2) - 1) - sqr
t(sqrt(2) + 1)*(sqrt(2) - 1)) + 16*(x^2 - sqrt(2)*(x^2 + 1) + 1)*sqrt(-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2*s
qrt(2))*(sqrt(2) + 2)^(3/4)*sqrt(sqrt(2) + 1)*arctan(1/2*sqrt(-sqrt(-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2*sqr
t(2))*(sqrt(sqrt(2) + 2)*(sqrt(2) - 2) - 2)*(sqrt(2) + 2)^(1/4)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 2*sqrt(x +
sqrt(x^2 + 1)) + 2*sqrt(sqrt(2) + 2) + 2)*((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))*(sqrt(2) + 2)^(3/4) + 1/2*((3*sqrt
(2) - 4)*sqrt(sqrt(2) + 2)*sqrt(sqrt(2) + 1) + (3*sqrt(2) - 4)*sqrt(sqrt(2) + 1))*sqrt(-2*sqrt(sqrt(2) + 2)*(s
qrt(2) - 1) + 2*sqrt(2))*(sqrt(2) + 2)^(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)) - 16*2^(3/8)*(x^2 + 1)*sqrt(-2*2^(1/4)*(sqrt(2) + 1) +
2*sqrt(2) + 4)*sqrt(sqrt(2) - 1)*arctan(1/2*2^(3/8)*sqrt(2^(1/8)*sqrt(-2*2^(1/4)*(sqrt(2) + 1) + 2*sqrt(2) +
4)*(2^(3/4) + 2)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 2*sqrt(x + sqrt(x^2 + 1)) + 2*2^(1/4) + 2)*(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) -
1/2*2^(3/8)*(2^(1/4)*(sqrt(2) + 2)*sqrt(sqrt(2) - 1) + (sqrt(2) + 2)*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)) - 16*2^(3/8)*(x^2 + 1)*sqrt(-2*2^(1/4)*(sqrt(2) + 1) + 2*sqrt(2) + 4)*sqrt(sqrt(2)
- 1)*arctan(1/2*2^(3/8)*sqrt(-2^(1/8)*sqrt(-2*2^(1/4)*(sqrt(2) + 1) + 2*sqrt(2) + 4)*(2^(3/4) + 2)*sqrt(sqrt(x
+ sqrt(x^2 + 1)) + 1) + 2*sqrt(x + sqrt(x^2 + 1)) + 2*2^(1/4) + 2)*(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) - 1/2*2^(3/8)*(2^(1/4)*(sqrt(
2) + 2)*sqrt(sqrt(2) - 1) + (sqrt(2) + 2)*sqrt(sqrt(2) - 1))*sqrt(-2*2^(1/4)*(sqrt(2) + 1) + 2*sqrt(2) + 4)*sq
rt(sqrt(x + sqrt(x^2 + 1)) + 1) + 2^(1/4)*(sqrt(2) + 1)*sqrt(sqrt(2) - 1) + (sqrt(2) + 1)*sqrt(sqrt(2) - 1)) -
4*(x^2 - sqrt(2)*(x^2 + 1) + 1)*sqrt(2*sqrt(2*sqrt(2) + 4)*(sqrt(2) + 1) + 4*sqrt(2) + 8)*(2*sqrt(2) + 4)^(3/
4)*arctan(1/8*sqrt(sqrt(2*sqrt(2*sqrt(2) + 4)*(sqrt(2) + 1) + 4*sqrt(2) + 8)*(sqrt(2*sqrt(2) + 4)*(sqrt(2) - 2
) + 2*sqrt(2) - 2)*(2*sqrt(2) + 4)^(1/4)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) - 2*sqrt(2*sqrt(2) + 4)*(sqrt(2) -
2) + 4*sqrt(x + sqrt(x^2 + 1)) + 4)*(sqrt(2*sqrt(2) + 4)*(2*sqrt(2) - 3) + 2*sqrt(2) - 2)*sqrt(2*sqrt(2*sqrt(2
) + 4)*(sqrt(2) + 1) + 4*sqrt(2) + 8)*(2*sqrt(2) + 4)^(3/4) - 1/4*(sqrt(2*sqrt(2) + 4)*(2*sqrt(2) - 3) + 2*sqr
t(2) - 2)*sqrt(2*sqrt(2*sqrt(2) + 4)*(sqrt(2) + 1) + 4*sqrt(2) + 8)*(2*sqrt(2) + 4)^(3/4)*sqrt(sqrt(x + sqrt(x
^2 + 1)) + 1) + sqrt(2*sqrt(2) + 4)*(sqrt(2) - 1) + sqrt(2) - 1) - 4*(x^2 - sqrt(2)*(x^2 + 1) + 1)*sqrt(2*sqrt
(2*sqrt(2) + 4)*(sqrt(2) + 1) + 4*sqrt(2) + 8)*(2*sqrt(2) + 4)^(3/4)*arctan(1/24*sqrt(-9*sqrt(2*sqrt(2*sqrt(2)
+ 4)*(sqrt(2) + 1) + 4*sqrt(2) + 8)*(sqrt(2*sqrt(2) + 4)*(sqrt(2) - 2) + 2*sqrt(2) - 2)*(2*sqrt(2) + 4)^(1/4)
*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) - 18*sqrt(2*sqrt(2) + 4)*(sqrt(2) - 2) + 36*sqrt(x + sqrt(x^2 + 1)) + 36)*(
sqrt(2*sqrt(2) + 4)*(2*sqrt(2) - 3) + 2*sqrt(2) - 2)*sqrt(2*sqrt(2*sqrt(2) + 4)*(sqrt(2) + 1) + 4*sqrt(2) + 8)
*(2*sqrt(2) + 4)^(3/4) - 1/4*(sqrt(2*sqrt(2) + 4)*(2*sqrt(2) - 3) + 2*sqrt(2) - 2)*sqrt(2*sqrt(2*sqrt(2) + 4)*
(sqrt(2) + 1) + 4*sqrt(2) + 8)*(2*sqrt(2) + 4)^(3/4)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) - sqrt(2*sqrt(2) + 4)*(
sqrt(2) - 1) - sqrt(2) + 1) + 4*(x^2 + sqrt(2)*(x^2 + 1) + 1)*sqrt(-2*(sqrt(2) - 1)*sqrt(-2*sqrt(2) + 4) - 4*s
qrt(2) + 8)*(-2*sqrt(2) + 4)^(3/4)*arctan(1/8*sqrt(((sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2) + 2)*sqrt(-
2*(sqrt(2) - 1)*sqrt(-2*sqrt(2) + 4) - 4*sqrt(2) + 8)*(-2*sqrt(2) + 4)^(1/4)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)
+ 2*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 4*sqrt(x + sqrt(x^2 + 1)) + 4)*((2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4)
+ 2*sqrt(2) + 2)*sqrt(-2*(sqrt(2) - 1)*sqrt(-2*sqrt(2) + 4) - 4*sqrt(2) + 8)*(-2*sqrt(2) + 4)^(3/4) - 1/4*((2
*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2) + 2)*sqrt(-2*(sqrt(2) - 1)*sqrt(-2*sqrt(2) + 4) - 4*sqrt(2) + 8
)*(-2*sqrt(2) + 4)^(3/4)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) - (sqrt(2) + 1)*sqrt(-2*sqrt(2) + 4) - sqrt(2) - 1)
+ 4*(x^2 + sqrt(2)*(x^2 + 1) + 1)*sqrt(-2*(sqrt(2) - 1)*sqrt(-2*sqrt(2) + 4) - 4*sqrt(2) + 8)*(-2*sqrt(2) + 4
)^(3/4)*arctan(1/24*sqrt(-9*((sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2) + 2)*sqrt(-2*(sqrt(2) - 1)*sqrt(-2
*sqrt(2) + 4) - 4*sqrt(2) + 8)*(-2*sqrt(2) + 4)^(1/4)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 18*(sqrt(2) + 2)*sqr
t(-2*sqrt(2) + 4) + 36*sqrt(x + sqrt(x^2 + 1)) + 36)*((2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2) + 2)*sq
rt(-2*(sqrt(2) - 1)*sqrt(-2*sqrt(2) + 4) - 4*sqrt(2) + 8)*(-2*sqrt(2) + 4)^(3/4) - 1/4*((2*sqrt(2) + 3)*sqrt(-
2*sqrt(2) + 4) + 2*sqrt(2) + 2)*sqrt(-2*(sqrt(2) - 1)*sqrt(-2*sqrt(2) + 4) - 4*sqrt(2) + 8)*(-2*sqrt(2) + 4)^(
3/4)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + (sqrt(2) + 1)*sqrt(-2*sqrt(2) + 4) + sqrt(2) + 1) + 32*sqrt(2)*(x^2 +
1)*sqrt(sqrt(sqrt(2) + 1) - 1)*arctan(1/2*(sqrt(2)*sqrt(sqrt(2) + 1) + sqrt(2))*sqrt(sqrt(x + sqrt(x^2 + 1))
+ sqrt(sqrt(2) + 1))*sqrt(sqrt(sqrt(2) + 1) - 1) - 1/2*(sqrt(2)*sqrt(sqrt(2) + 1) + sqrt(2))*sqrt(sqrt(x + sqr
t(x^2 + 1)) + 1)*sqrt(sqrt(sqrt(2) + 1) - 1)) - 8*sqrt(2)*(x^2 + 1)*sqrt(sqrt(sqrt(2) + 1) + 1)*log(sqrt(sqrt(
x + sqrt(x^2 + 1)) + 1) + sqrt(sqrt(sqrt(2) + 1) + 1)) + 8*sqrt(2)*(x^2 + 1)*sqrt(sqrt(sqrt(2) + 1) + 1)*log(s
qrt(sqrt(x + sqrt(x^2 + 1)) + 1) - sqrt(sqrt(sqrt(2) + 1) + 1)) + 8*sqrt(2)*(x^2 + 1)*sqrt(sqrt(sqrt(2) - 1) +
1)*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + sqrt(sqrt(sqrt(2) - 1) + 1)) - 8*sqrt(2)*(x^2 + 1)*sqrt(sqrt(sqrt(
2) - 1) + 1)*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) - sqrt(sqrt(sqrt(2) - 1) + 1)) + 8*sqrt(2)*(x^2 + 1)*sqrt(-
sqrt(sqrt(2) - 1) + 1)*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + sqrt(-sqrt(sqrt(2) - 1) + 1)) - 8*sqrt(2)*(x^2
+ 1)*sqrt(-sqrt(sqrt(2) - 1) + 1)*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) - sqrt(-sqrt(sqrt(2) - 1) + 1)) - (2*s
qrt(2)*(x^2 + 1) - (x^2 + 1)*sqrt(2*sqrt(2) + 4))*sqrt(2*sqrt(2*sqrt(2) + 4)*(sqrt(2) + 1) + 4*sqrt(2) + 8)*(2
*sqrt(2) + 4)^(1/4)*log(9*sqrt(2*sqrt(2*sqrt(2) + 4)*(sqrt(2) + 1) + 4*sqrt(2) + 8)*(sqrt(2*sqrt(2) + 4)*(sqrt
(2) - 2) + 2*sqrt(2) - 2)*(2*sqrt(2) + 4)^(1/4)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) - 18*sqrt(2*sqrt(2) + 4)*(sq
rt(2) - 2) + 36*sqrt(x + sqrt(x^2 + 1)) + 36) + (2*sqrt(2)*(x^2 + 1) - (x^2 + 1)*sqrt(2*sqrt(2) + 4))*sqrt(2*s
qrt(2*sqrt(2) + 4)*(sqrt(2) + 1) + 4*sqrt(2) + 8)*(2*sqrt(2) + 4)^(1/4)*log(-9*sqrt(2*sqrt(2*sqrt(2) + 4)*(sqr
t(2) + 1) + 4*sqrt(2) + 8)*(sqrt(2*sqrt(2) + 4)*(sqrt(2) - 2) + 2*sqrt(2) - 2)*(2*sqrt(2) + 4)^(1/4)*sqrt(sqrt
(x + sqrt(x^2 + 1)) + 1) - 18*sqrt(2*sqrt(2) + 4)*(sqrt(2) - 2) + 36*sqrt(x + sqrt(x^2 + 1)) + 36) + 4*(sqrt(2
)*(x^2 + 1) - (x^2 - sqrt(2)*(x^2 + 1) + 1)*sqrt(sqrt(2) + 2))*sqrt(-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2*sqr
t(2))*(sqrt(2) + 2)^(1/4)*log(1/2*sqrt(-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2*sqrt(2))*(sqrt(sqrt(2) + 2)*(sqr
t(2) - 2) - 2)*(sqrt(2) + 2)^(1/4)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + sqrt(x + sqrt(x^2 + 1)) + sqrt(sqrt(2)
+ 2) + 1) - 4*(sqrt(2)*(x^2 + 1) - (x^2 - sqrt(2)*(x^2 + 1) + 1)*sqrt(sqrt(2) + 2))*sqrt(-2*sqrt(sqrt(2) + 2)*
(sqrt(2) - 1) + 2*sqrt(2))*(sqrt(2) + 2)^(1/4)*log(-1/2*sqrt(-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2*sqrt(2))*(
sqrt(sqrt(2) + 2)*(sqrt(2) - 2) - 2)*(sqrt(2) + 2)^(1/4)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + sqrt(x + sqrt(x^2
+ 1)) + sqrt(sqrt(2) + 2) + 1) - (2*sqrt(2)*(x^2 + 1) + (x^2 + 1)*sqrt(-2*sqrt(2) + 4))*sqrt(-2*(sqrt(2) - 1)
*sqrt(-2*sqrt(2) + 4) - 4*sqrt(2) + 8)*(-2*sqrt(2) + 4)^(1/4)*log(9*((sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 2*sq
rt(2) + 2)*sqrt(-2*(sqrt(2) - 1)*sqrt(-2*sqrt(2) + 4) - 4*sqrt(2) + 8)*(-2*sqrt(2) + 4)^(1/4)*sqrt(sqrt(x + sq
rt(x^2 + 1)) + 1) + 18*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 36*sqrt(x + sqrt(x^2 + 1)) + 36) + (2*sqrt(2)*(x^2
+ 1) + (x^2 + 1)*sqrt(-2*sqrt(2) + 4))*sqrt(-2*(sqrt(2) - 1)*sqrt(-2*sqrt(2) + 4) - 4*sqrt(2) + 8)*(-2*sqrt(2
) + 4)^(1/4)*log(-9*((sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2) + 2)*sqrt(-2*(sqrt(2) - 1)*sqrt(-2*sqrt(2)
+ 4) - 4*sqrt(2) + 8)*(-2*sqrt(2) + 4)^(1/4)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 18*(sqrt(2) + 2)*sqrt(-2*sqr
t(2) + 4) + 36*sqrt(x + sqrt(x^2 + 1)) + 36) + 4*2^(1/8)*(sqrt(2)*(x^2 + 1) + 2^(1/4)*(x^2 + 1))*sqrt(-2*2^(1/
4)*(sqrt(2) + 1) + 2*sqrt(2) + 4)*log(1/2*2^(1/8)*sqrt(-2*2^(1/4)*(sqrt(2) + 1) + 2*sqrt(2) + 4)*(2^(3/4) + 2)
*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + sqrt(x + sqrt(x^2 + 1)) + 2^(1/4) + 1) - 4*2^(1/8)*(sqrt(2)*(x^2 + 1) + 2
^(1/4)*(x^2 + 1))*sqrt(-2*2^(1/4)*(sqrt(2) + 1) + 2*sqrt(2) + 4)*log(-1/2*2^(1/8)*sqrt(-2*2^(1/4)*(sqrt(2) + 1
) + 2*sqrt(2) + 4)*(2^(3/4) + 2)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + sqrt(x + sqrt(x^2 + 1)) + 2^(1/4) + 1) -
64*(x^2 + 1)*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 1) + 64*(x^2 + 1)*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) -
1) + 32*(5*x^2 - sqrt(x^2 + 1)*x + 5)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1))/(x^2 + 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^4+1)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(x^2+1)^(1/2)/(-x^4+1),x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{4}+1\right ) \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{\sqrt {x^{2}+1}\, \left (-x^{4}+1\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {{\left (x^{4} + 1\right )} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{{\left (x^{4} - 1\right )} \sqrt {x^{2} + 1}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{x^{4} \sqrt {x^{2} + 1} - \sqrt {x^{2} + 1}}\, dx - \int \frac {x^{4} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{x^{4} \sqrt {x^{2} + 1} - \sqrt {x^{2} + 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(sqrt(sqrt(x + sqrt(x**2 + 1)) + 1)/(x**4*sqrt(x**2 + 1) - sqrt(x**2 + 1)), x) - Integral(x**4*sqrt(s
qrt(x + sqrt(x**2 + 1)) + 1)/(x**4*sqrt(x**2 + 1) - sqrt(x**2 + 1)), x)

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