3.31.10 \(\int \frac {(1+x^4) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1-x^4) \sqrt {x+\sqrt {1+x^2}}} \, dx\) [3010]
Optimal. Leaf size=406 \[ \frac {\left (-8-3 x-32 x^2\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+(2-16 x) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\sqrt {1+x^2} \left ((-3-32 x) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-16 \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )}{24 \left (x+\sqrt {1+x^2}\right )^{3/2}}+\frac {1}{8} \tanh ^{-1}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )-\frac {1}{2} \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\& ,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^2}\& \right ]+\text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\& ,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}}{-1+3 \text {$\#$1}^2-3 \text {$\#$1}^4+\text {$\#$1}^6}\& \right ]+\frac {1}{2} \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\& ,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^2}\& \right ] \]
[Out]
Unintegrable
________________________________________________________________________________________
Rubi [F]
time = 1.81, 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}}}}{\left (1-x^4\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[((1 + x^4)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/((1 - x^4)*Sqrt[x + Sqrt[1 + x^2]]),x]
[Out]
-Defer[Int][Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/Sqrt[x + Sqrt[1 + x^2]], x] + (I/2)*Defer[Int][Sqrt[1 + Sqrt[x +
Sqrt[1 + x^2]]]/((I - x)*Sqrt[x + Sqrt[1 + x^2]]), x] + Defer[Int][Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/((1 - x)
*Sqrt[x + Sqrt[1 + x^2]]), x]/2 + (I/2)*Defer[Int][Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/((I + x)*Sqrt[x + Sqrt[1
+ x^2]]), x] + Defer[Int][Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/((1 + x)*Sqrt[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}}}}{\left (1-x^4\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx &=\int \left (-\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {x+\sqrt {1+x^2}}}+\frac {2 \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^4\right ) \sqrt {x+\sqrt {1+x^2}}}\right ) \, dx\\ &=2 \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^4\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx-\int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {x+\sqrt {1+x^2}}} \, dx\\ &=2 \int \left (\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 \left (1-x^2\right ) \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 \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {x+\sqrt {1+x^2}}} \, dx\\ &=-\int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^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+\int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx\\ &=-\int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {x+\sqrt {1+x^2}}} \, dx+\int \left (\frac {i \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 (i-x) \sqrt {x+\sqrt {1+x^2}}}+\frac {i \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 (i+x) \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} i \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(i-x) \sqrt {x+\sqrt {1+x^2}}} \, dx+\frac {1}{2} i \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(i+x) \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) \sqrt {x+\sqrt {1+x^2}}} \, dx-\int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {x+\sqrt {1+x^2}}} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 367, normalized size = 0.90 \begin {gather*} \frac {1}{24} \left (-\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \left (8+32 x^2+3 \sqrt {1+x^2}-2 \sqrt {x+\sqrt {1+x^2}}+16 \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}+x \left (3+32 \sqrt {1+x^2}+16 \sqrt {x+\sqrt {1+x^2}}\right )\right )}{\left (x+\sqrt {1+x^2}\right )^{3/2}}+3 \tanh ^{-1}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )-12 \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^2}\&\right ]+24 \text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}}{-1+3 \text {$\#$1}^2-3 \text {$\#$1}^4+\text {$\#$1}^6}\&\right ]+12 \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^2}\&\right ]\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[((1 + x^4)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/((1 - x^4)*Sqrt[x + Sqrt[1 + x^2]]),x]
[Out]
(-((Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]*(8 + 32*x^2 + 3*Sqrt[1 + x^2] - 2*Sqrt[x + Sqrt[1 + x^2]] + 16*Sqrt[1 +
x^2]*Sqrt[x + Sqrt[1 + x^2]] + x*(3 + 32*Sqrt[1 + x^2] + 16*Sqrt[x + Sqrt[1 + x^2]])))/(x + Sqrt[1 + x^2])^(3/
2)) + 3*ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]] - 12*RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 +
Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(-1 + #1^2) & ] + 24*RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (Log[S
qrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(-1 + 3*#1^2 - 3*#1^4 + #1^6) & ] + 12*RootSum[2 - 8*#1^2 + 8*#1^4
- 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(-1 + #1^2) & ])/24
________________________________________________________________________________________
Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{4}+1\right ) \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{\left (-x^{4}+1\right ) \sqrt {x +\sqrt {x^{2}+1}}}\, dx\]
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^4+1)/(x+(x^2+1)^(1/2))^(1/2),x)
[Out]
int((x^4+1)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^4+1)/(x+(x^2+1)^(1/2))^(1/2),x)
________________________________________________________________________________________
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)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^4+1)/(x+(x^2+1)^(1/2))^(1/2),x, algorithm="maxima")
[Out]
-integrate((x^4 + 1)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)/((x^4 - 1)*sqrt(x + sqrt(x^2 + 1))), x)
________________________________________________________________________________________
Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order
1.
time = 1.58, size = 7234, normalized size = 17.82 \begin {gather*} \text {Too large to display} \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^4+1)/(x+(x^2+1)^(1/2))^(1/2),x, algorithm="fricas")
[Out]
1/24*((16*x^2 - sqrt(x^2 + 1)*(16*x + 3) + 3*x - 8)*sqrt(x + sqrt(x^2 + 1)) - 2*x + 2*sqrt(x^2 + 1) - 16)*sqrt
(sqrt(x + sqrt(x^2 + 1)) + 1) - 1/2*sqrt(sqrt(2) + sqrt(sqrt(2) + 1) + 1)*log(((sqrt(2) + sqrt(sqrt(2) + 1) +
1)^2*(sqrt(2) - sqrt(sqrt(2) + 1) + 3) + (sqrt(2) - sqrt(sqrt(2) + 1) + 1)^3 + ((sqrt(2) - sqrt(sqrt(2) + 1) +
1)^2 - 4*sqrt(2) + 4*sqrt(sqrt(2) + 1) - 11)*(sqrt(2) + sqrt(sqrt(2) + 1) + 1) - 4*(sqrt(2) - sqrt(sqrt(2) +
1) + 1)^2 - 1)*sqrt(sqrt(2) + sqrt(sqrt(2) + 1) + 1) + 5*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + 1/2*sqrt(sqrt(2)
+ sqrt(sqrt(2) + 1) + 1)*log(-((sqrt(2) + sqrt(sqrt(2) + 1) + 1)^2*(sqrt(2) - sqrt(sqrt(2) + 1) + 3) + (sqrt(
2) - sqrt(sqrt(2) + 1) + 1)^3 + ((sqrt(2) - sqrt(sqrt(2) + 1) + 1)^2 - 4*sqrt(2) + 4*sqrt(sqrt(2) + 1) - 11)*(
sqrt(2) + sqrt(sqrt(2) + 1) + 1) - 4*(sqrt(2) - sqrt(sqrt(2) + 1) + 1)^2 - 1)*sqrt(sqrt(2) + sqrt(sqrt(2) + 1)
+ 1) + 5*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + 1/2*sqrt(sqrt(2) - sqrt(sqrt(2) + 1) + 1)*log(((sqrt(2) - sqrt(
sqrt(2) + 1) + 1)^3 - 6*(sqrt(2) - sqrt(sqrt(2) + 1) + 1)^2 + 7*sqrt(2) - 7*sqrt(sqrt(2) + 1) + 8)*sqrt(sqrt(2
) - sqrt(sqrt(2) + 1) + 1) + 5*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 1/2*sqrt(sqrt(2) - sqrt(sqrt(2) + 1) + 1)*
log(-((sqrt(2) - sqrt(sqrt(2) + 1) + 1)^3 - 6*(sqrt(2) - sqrt(sqrt(2) + 1) + 1)^2 + 7*sqrt(2) - 7*sqrt(sqrt(2)
+ 1) + 8)*sqrt(sqrt(2) - sqrt(sqrt(2) + 1) + 1) + 5*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 1/2*sqrt(sqrt(2) + s
qrt(sqrt(2) - 1) - 1)*log(((3*sqrt(2) - 3*sqrt(sqrt(2) - 1) - 5)*(sqrt(2) + sqrt(sqrt(2) - 1) - 1)^2 + 3*(sqrt
(2) - sqrt(sqrt(2) - 1) - 1)^3 + (3*(sqrt(2) - sqrt(sqrt(2) - 1) - 1)^2 + 12*sqrt(2) - 12*sqrt(sqrt(2) - 1) -
17)*(sqrt(2) + sqrt(sqrt(2) - 1) - 1) + 12*(sqrt(2) - sqrt(sqrt(2) - 1) - 1)^2 + 12*sqrt(2) - 12*sqrt(sqrt(2)
- 1) - 19)*sqrt(sqrt(2) + sqrt(sqrt(2) - 1) - 1) + 13*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + 1/2*sqrt(sqrt(2) +
sqrt(sqrt(2) - 1) - 1)*log(-((3*sqrt(2) - 3*sqrt(sqrt(2) - 1) - 5)*(sqrt(2) + sqrt(sqrt(2) - 1) - 1)^2 + 3*(sq
rt(2) - sqrt(sqrt(2) - 1) - 1)^3 + (3*(sqrt(2) - sqrt(sqrt(2) - 1) - 1)^2 + 12*sqrt(2) - 12*sqrt(sqrt(2) - 1)
- 17)*(sqrt(2) + sqrt(sqrt(2) - 1) - 1) + 12*(sqrt(2) - sqrt(sqrt(2) - 1) - 1)^2 + 12*sqrt(2) - 12*sqrt(sqrt(2
) - 1) - 19)*sqrt(sqrt(2) + sqrt(sqrt(2) - 1) - 1) + 13*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + 1/2*sqrt(sqrt(2)
- sqrt(sqrt(2) - 1) - 1)*log((3*(sqrt(2) - sqrt(sqrt(2) - 1) - 1)^3 + 14*(sqrt(2) - sqrt(sqrt(2) - 1) - 1)^2 +
17*sqrt(2) - 17*sqrt(sqrt(2) - 1) - 34)*sqrt(sqrt(2) - sqrt(sqrt(2) - 1) - 1) + 13*sqrt(sqrt(x + sqrt(x^2 + 1
)) + 1)) - 1/2*sqrt(sqrt(2) - sqrt(sqrt(2) - 1) - 1)*log(-(3*(sqrt(2) - sqrt(sqrt(2) - 1) - 1)^3 + 14*(sqrt(2)
- sqrt(sqrt(2) - 1) - 1)^2 + 17*sqrt(2) - 17*sqrt(sqrt(2) - 1) - 34)*sqrt(sqrt(2) - sqrt(sqrt(2) - 1) - 1) +
13*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - sqrt(1/2*I*sqrt(2) - 1/2*sqrt(-4*I*sqrt(2) - 2))*log((4*(-1/2*I*sqrt(2
) - 1/2*sqrt(4*I*sqrt(2) - 2))^3 + 2*(1/2*I*sqrt(2) - 1/2*sqrt(-4*I*sqrt(2) - 2))^2*(-I*sqrt(2) - sqrt(4*I*sqr
t(2) - 2) - 1) - (2*(-1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))^2 + 1)*(-I*sqrt(2) + sqrt(-4*I*sqrt(2) - 2))
- 4*I*sqrt(2) - 4*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(-(4*(-1/2*I*sqrt(2) - 1/2*sqrt(4
*I*sqrt(2) - 2))^3 + 2*(1/2*I*sqrt(2) - 1/2*sqrt(-4*I*sqrt(2) - 2))^2*(-I*sqrt(2) - sqrt(4*I*sqrt(2) - 2) - 1)
- (2*(-1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))^2 + 1)*(-I*sqrt(2) + sqrt(-4*I*sqrt(2) - 2)) - 4*I*sqrt(2)
- 4*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((4*(-1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2)
)^3 + 2*(-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) - 10)*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*sqr
t(4*I*sqrt(2) - 2))*log(-(4*(-1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))^3 + 2*(-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) - 10)*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(sqrt(2) + 1) + 1)^2
- 3/16*(sqrt(2) - sqrt(sqrt(2) + 1) + 1)^2 - 1/8*(sqrt(2) + sqrt(sqrt(2) + 1) + 1)*(sqrt(2) - sqrt(sqrt(2) +
1) - 3) + 1/2*sqrt(2) - 1/2*sqrt(sqrt(2) + 1) + 3/2) + 1)*log(1/2*((sqrt(2) + sqrt(sqrt(2) + 1) + 1)^2*(sqrt(2
) - sqrt(sqrt(2) + 1) + 3) + ((sqrt(2) - sqrt(sqrt(2) + 1) + 1)^2 - 4*sqrt(2) + 4*sqrt(sqrt(2) + 1) - 11)*(sqr
t(2) + sqrt(sqrt(2) + 1) + 1) + 2*(sqrt(2) - sqrt(sqrt(2) + 1) + 1)^2 + 4*sqrt(-3/16*(sqrt(2) + sqrt(sqrt(2) +
1) + 1)^2 - 3/16*(sqrt(2) - sqrt(sqrt(2) + 1) + 1)^2 - 1/8*(sqrt(2) + sqrt(sqrt(2) + 1) + 1)*(sqrt(2) - sqrt(
sqrt(2) + 1) - 3) + 1/2*sqrt(2) - 1/2*sqrt(sqrt(2) + 1) + 3/2)*((sqrt(2) + sqrt(sqrt(2) + 1) + 1)*(sqrt(2) - s
qrt(sqrt(2) + 1) + 3) + 2*sqrt(2) - 2*sqrt(sqrt...
________________________________________________________________________________________
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 + \sqrt {x^{2} + 1}} - \sqrt {x + \sqrt {x^{2} + 1}}}\, dx - \int \frac {x^{4} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{x^{4} \sqrt {x + \sqrt {x^{2} + 1}} - \sqrt {x + \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**4+1)/(x+(x**2+1)**(1/2))**(1/2),x)
[Out]
-Integral(sqrt(sqrt(x + sqrt(x**2 + 1)) + 1)/(x**4*sqrt(x + sqrt(x**2 + 1)) - sqrt(x + sqrt(x**2 + 1))), x) -
Integral(x**4*sqrt(sqrt(x + sqrt(x**2 + 1)) + 1)/(x**4*sqrt(x + sqrt(x**2 + 1)) - sqrt(x + sqrt(x**2 + 1))), x
)
________________________________________________________________________________________
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)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^4+1)/(x+(x^2+1)^(1/2))^(1/2),x, algorithm="giac")
[Out]
Timed out
________________________________________________________________________________________
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 )}{\left (x^4-1\right )\,\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^4 + 1))/((x^4 - 1)*(x + (x^2 + 1)^(1/2))^(1/2)),x)
[Out]
int(-(((x + (x^2 + 1)^(1/2))^(1/2) + 1)^(1/2)*(x^4 + 1))/((x^4 - 1)*(x + (x^2 + 1)^(1/2))^(1/2)), x)
________________________________________________________________________________________