3.11.23 \(\int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1+x^2) \sqrt {x+\sqrt {1+x^2}}} \, dx\)
Optimal. Leaf size=77 \[ \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+6 \text {$\#$1}^4-4 \text {$\#$1}^2+2\& ,\frac {\text {$\#$1} \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )}{\text {$\#$1}^6-3 \text {$\#$1}^4+3 \text {$\#$1}^2-1}\& \right ] \]
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Rubi [F] time = 1.46, 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 {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/((1 + x^2)*Sqrt[x + Sqrt[1 + x^2]]),x]
[Out]
(I/2)*Defer[Int][Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/((I - x)*Sqrt[x + Sqrt[1 + x^2]]), x] + (I/2)*Defer[Int][Sq
rt[1 + Sqrt[x + Sqrt[1 + x^2]]]/((I + x)*Sqrt[x + Sqrt[1 + x^2]]), x]
Rubi steps
\begin {align*} \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right ) \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\\ &=\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\\ \end {align*}
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Mathematica [F] time = 3.44, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Integrate[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/((1 + x^2)*Sqrt[x + Sqrt[1 + x^2]]),x]
[Out]
Integrate[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/((1 + x^2)*Sqrt[x + Sqrt[1 + x^2]]), x]
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IntegrateAlgebraic [A] time = 0.19, size = 77, normalized size = 1.00 \begin {gather*} \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 ] \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/((1 + x^2)*Sqrt[x + Sqrt[1 + x^2]]),x]
[Out]
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 + 3*#1^2
- 3*#1^4 + #1^6) & ]
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fricas [B] time = 1.42, size = 2495, normalized size = 32.40
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(x^2+1)/(x+(x^2+1)^(1/2))^(1/2),x, algorithm="fricas")
[Out]
-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*sqr
t(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*sq
rt(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*sq
rt(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*sqrt(4*I*sqrt(2) - 2))*log(-(4*(-1/2*I*sqr
t(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(2*sqrt(-3*(1/2*I*sqrt(2) - 1/2*sqrt(-4*I*sqrt(2) - 2))^2 - 3*(-1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2
) - 2))^2 - 1/2*(I*sqrt(2) + sqrt(4*I*sqrt(2) - 2))*(-I*sqrt(2) + sqrt(-4*I*sqrt(2) - 2)) - 8) + sqrt(4*I*sqrt
(2) - 2) + sqrt(-4*I*sqrt(2) - 2))*log(1/2*(2*(1/2*I*sqrt(2) - 1/2*sqrt(-4*I*sqrt(2) - 2))^2*(-I*sqrt(2) - sqr
t(4*I*sqrt(2) - 2) - 1) - 2*(-1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))^2 - (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)) - sqrt(-3*(1/2*I*sqrt(2) - 1/2*sqrt(-4*I*sqrt(2) -
2))^2 - 3*(-1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))^2 - 1/2*(I*sqrt(2) + sqrt(4*I*sqrt(2) - 2))*(-I*sqrt(2
) + sqrt(-4*I*sqrt(2) - 2)) - 8)*((-I*sqrt(2) - sqrt(4*I*sqrt(2) - 2) - 1)*(-I*sqrt(2) + sqrt(-4*I*sqrt(2) - 2
)) - I*sqrt(2) - sqrt(4*I*sqrt(2) - 2) + 2) - I*sqrt(2) - sqrt(4*I*sqrt(2) - 2) + 4)*sqrt(2*sqrt(-3*(1/2*I*sqr
t(2) - 1/2*sqrt(-4*I*sqrt(2) - 2))^2 - 3*(-1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))^2 - 1/2*(I*sqrt(2) + sqr
t(4*I*sqrt(2) - 2))*(-I*sqrt(2) + sqrt(-4*I*sqrt(2) - 2)) - 8) + sqrt(4*I*sqrt(2) - 2) + sqrt(-4*I*sqrt(2) - 2
)) + 12*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 1/2*sqrt(2*sqrt(-3*(1/2*I*sqrt(2) - 1/2*sqrt(-4*I*sqrt(2) - 2))^2
- 3*(-1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))^2 - 1/2*(I*sqrt(2) + sqrt(4*I*sqrt(2) - 2))*(-I*sqrt(2) + sq
rt(-4*I*sqrt(2) - 2)) - 8) + sqrt(4*I*sqrt(2) - 2) + sqrt(-4*I*sqrt(2) - 2))*log(-1/2*(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 - (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)) - sqrt(
-3*(1/2*I*sqrt(2) - 1/2*sqrt(-4*I*sqrt(2) - 2))^2 - 3*(-1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))^2 - 1/2*(I*
sqrt(2) + sqrt(4*I*sqrt(2) - 2))*(-I*sqrt(2) + sqrt(-4*I*sqrt(2) - 2)) - 8)*((-I*sqrt(2) - sqrt(4*I*sqrt(2) -
2) - 1)*(-I*sqrt(2) + sqrt(-4*I*sqrt(2) - 2)) - I*sqrt(2) - sqrt(4*I*sqrt(2) - 2) + 2) - I*sqrt(2) - sqrt(4*I*
sqrt(2) - 2) + 4)*sqrt(2*sqrt(-3*(1/2*I*sqrt(2) - 1/2*sqrt(-4*I*sqrt(2) - 2))^2 - 3*(-1/2*I*sqrt(2) - 1/2*sqrt
(4*I*sqrt(2) - 2))^2 - 1/2*(I*sqrt(2) + sqrt(4*I*sqrt(2) - 2))*(-I*sqrt(2) + sqrt(-4*I*sqrt(2) - 2)) - 8) + sq
rt(4*I*sqrt(2) - 2) + sqrt(-4*I*sqrt(2) - 2)) + 12*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + 1/2*sqrt(-2*sqrt(-3*(1
/2*I*sqrt(2) - 1/2*sqrt(-4*I*sqrt(2) - 2))^2 - 3*(-1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))^2 - 1/2*(I*sqrt(
2) + sqrt(4*I*sqrt(2) - 2))*(-I*sqrt(2) + sqrt(-4*I*sqrt(2) - 2)) - 8) + sqrt(4*I*sqrt(2) - 2) + sqrt(-4*I*sqr
t(2) - 2))*log(1/2*(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 - (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)) + sqrt(-3*(1/2*I*sqrt(2) - 1/2*sqrt(-4*I*sqrt(2) - 2))^2 - 3*(-1/2*I*sqrt(
2) - 1/2*sqrt(4*I*sqrt(2) - 2))^2 - 1/2*(I*sqrt(2) + sqrt(4*I*sqrt(2) - 2))*(-I*sqrt(2) + sqrt(-4*I*sqrt(2) -
2)) - 8)*((-I*sqrt(2) - sqrt(4*I*sqrt(2) - 2) - 1)*(-I*sqrt(2) + sqrt(-4*I*sqrt(2) - 2)) - I*sqrt(2) - sqrt(4*
I*sqrt(2) - 2) + 2) - I*sqrt(2) - sqrt(4*I*sqrt(2) - 2) + 4)*sqrt(-2*sqrt(-3*(1/2*I*sqrt(2) - 1/2*sqrt(-4*I*sq
rt(2) - 2))^2 - 3*(-1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))^2 - 1/2*(I*sqrt(2) + sqrt(4*I*sqrt(2) - 2))*(-I
*sqrt(2) + sqrt(-4*I*sqrt(2) - 2)) - 8) + sqrt(4*I*sqrt(2) - 2) + sqrt(-4*I*sqrt(2) - 2)) + 12*sqrt(sqrt(x + s
qrt(x^2 + 1)) + 1)) - 1/2*sqrt(-2*sqrt(-3*(1/2*I*sqrt(2) - 1/2*sqrt(-4*I*sqrt(2) - 2))^2 - 3*(-1/2*I*sqrt(2) -
1/2*sqrt(4*I*sqrt(2) - 2))^2 - 1/2*(I*sqrt(2) + sqrt(4*I*sqrt(2) - 2))*(-I*sqrt(2) + sqrt(-4*I*sqrt(2) - 2))
- 8) + sqrt(4*I*sqrt(2) - 2) + sqrt(-4*I*sqrt(2) - 2))*log(-1/2*(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 - (2*(-1/2*I*s
qrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))^2 + 1)*(-I*sqrt(2) + sqrt(-4*I*sqrt(2) - 2)) + sqrt(-3*(1/2*I*sqrt(2) - 1/
2*sqrt(-4*I*sqrt(2) - 2))^2 - 3*(-1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))^2 - 1/2*(I*sqrt(2) + sqrt(4*I*sqr
t(2) - 2))*(-I*sqrt(2) + sqrt(-4*I*sqrt(2) - 2)) - 8)*((-I*sqrt(2) - sqrt(4*I*sqrt(2) - 2) - 1)*(-I*sqrt(2) +
sqrt(-4*I*sqrt(2) - 2)) - I*sqrt(2) - sqrt(4*I*sqrt(2) - 2) + 2) - I*sqrt(2) - sqrt(4*I*sqrt(2) - 2) + 4)*sqrt
(-2*sqrt(-3*(1/2*I*sqrt(2) - 1/2*sqrt(-4*I*sqrt(2) - 2))^2 - 3*(-1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))^2
- 1/2*(I*sqrt(2) + sqrt(4*I*sqrt(2) - 2))*(-I*sqrt(2) + sqrt(-4*I*sqrt(2) - 2)) - 8) + sqrt(4*I*sqrt(2) - 2) +
sqrt(-4*I*sqrt(2) - 2)) + 12*sqrt(sqrt(x + sqrt(x^2 + 1)) + 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((1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(x^2+1)/(x+(x^2+1)^(1/2))^(1/2),x, algorithm="giac")
[Out]
Timed out
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{\left (x^{2}+1\right ) \sqrt {x +\sqrt {x^{2}+1}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(x^2+1)/(x+(x^2+1)^(1/2))^(1/2),x)
[Out]
int((1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(x^2+1)/(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*} \int \frac {\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{{\left (x^{2} + 1\right )} \sqrt {x + \sqrt {x^{2} + 1}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(x^2+1)/(x+(x^2+1)^(1/2))^(1/2),x, algorithm="maxima")
[Out]
integrate(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)/((x^2 + 1)*sqrt(x + sqrt(x^2 + 1))), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{\left (x^2+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^2 + 1)*(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 + (x^2 + 1)^(1/2))^(1/2)), 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}}{\sqrt {x + \sqrt {x^{2} + 1}} \left (x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((1+(x+(x**2+1)**(1/2))**(1/2))**(1/2)/(x**2+1)/(x+(x**2+1)**(1/2))**(1/2),x)
[Out]
Integral(sqrt(sqrt(x + sqrt(x**2 + 1)) + 1)/(sqrt(x + sqrt(x**2 + 1))*(x**2 + 1)), x)
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