3.19.29 \(\int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1+x^2)^{3/2}} \, dx\)

Optimal. Leaf size=124 \[ \frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+6 \text {$\#$1}^4-4 \text {$\#$1}^2+2\& ,\frac {\log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )}{\text {$\#$1}^7-3 \text {$\#$1}^5+3 \text {$\#$1}^3-\text {$\#$1}}\& \right ]-\frac {\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{x^2+\sqrt {x^2+1} x+1} \]

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Rubi [F]  time = 0.17, 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 )^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \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}}}}{\left (1+x^2\right )^{3/2}} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.19, size = 124, normalized size = 1.00 \begin {gather*} \frac {1}{4} \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 ]-\frac {\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{x^2+\sqrt {x^2+1} x+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/(1 + x^2)^(3/2),x]

[Out]

-(Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/(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) & ]/4

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IntegrateAlgebraic [A]  time = 0.19, size = 124, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1+x^2+x \sqrt {1+x^2}}+\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 ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/(1 + x^2)^(3/2),x]

[Out]

-(Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/(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) & ]/4

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fricas [B]  time = 0.57, size = 1964, normalized size = 15.84

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)^(3/2),x, algorithm="fricas")

[Out]

1/32*(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*s
qrt(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 + s
qrt(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*sq
rt(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*sqrt(2) + 8)*(-2*sqrt(2) + 4)^(3/4)*arctan(1/8*sqrt(((sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2) + 2)*s
qrt(-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)*sq
rt(-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*sqrt(2) + 4) + 36*sqrt(x + sqrt(x^2 + 1)) + 36)*((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)*s
qrt(-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) + (2*sqrt(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)*(sqrt(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*sqrt(2*sq
rt(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 + sqr
t(x^2 + 1)) + 1) - 18*sqrt(2*sqrt(2) + 4)*(sqrt(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*(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*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*(sq
rt(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*sqrt(2) + 4) + 36*sqrt(x + sqrt(x^2 + 1)) + 36) - 32*(
x^2 - sqrt(x^2 + 1)*x + 1)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1))/(x^2 + 1)

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giac [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 )}^{\frac {3}{2}}}\,{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)^(3/2),x, algorithm="giac")

[Out]

integrate(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)/(x^2 + 1)^(3/2), x)

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{\left (x^{2}+1\right )^{\frac {3}{2}}}\, dx\]

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)^(3/2),x)

[Out]

int((1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(x^2+1)^(3/2),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 )}^{\frac {3}{2}}}\,{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)^(3/2),x, algorithm="maxima")

[Out]

integrate(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)/(x^2 + 1)^(3/2), 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 )}^{3/2}} \,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)^(3/2),x)

[Out]

int(((x + (x^2 + 1)^(1/2))^(1/2) + 1)^(1/2)/(x^2 + 1)^(3/2), 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}}{\left (x^{2} + 1\right )^{\frac {3}{2}}}\, 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)**(3/2),x)

[Out]

Integral(sqrt(sqrt(x + sqrt(x**2 + 1)) + 1)/(x**2 + 1)**(3/2), x)

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