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

Optimal. Leaf size=140 \[ \frac {1}{2} \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+8 \text {$\#$1}^4-8 \text {$\#$1}^2+2\& ,\frac {\text {$\#$1} \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )}{\text {$\#$1}^4-2 \text {$\#$1}^2+2}\& \right ]-\frac {1}{2} \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+4 \text {$\#$1}^4-2\& ,\frac {\log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )}{\text {$\#$1}^3-2 \text {$\#$1}}\& \right ] \]

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Rubi [F]  time = 1.20, 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 {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[1 + x^2]),x]

[Out]

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 {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right ) \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\\ \end {align*}

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Mathematica [A]  time = 0.32, size = 138, normalized size = 0.99 \begin {gather*} \frac {1}{2} \left (\text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+8 \text {$\#$1}^4-8 \text {$\#$1}^2+2\&,\frac {\text {$\#$1} \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )}{\text {$\#$1}^4-2 \text {$\#$1}^2+2}\&\right ]-\text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+4 \text {$\#$1}^4-2\&,\frac {\log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )}{\text {$\#$1}^3-2 \text {$\#$1}}\&\right ]\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]/(-2*#1 + #1^3) & ] + Roo
tSum[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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IntegrateAlgebraic [A]  time = 0.20, size = 140, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )}{-2 \text {$\#$1}+\text {$\#$1}^3}\&\right ]+\frac {1}{2} \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}}{2-2 \text {$\#$1}^2+\text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]/(-2*#1 + #1^3) & ] +
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 = 0.78, size = 1973, normalized size = 14.09

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^2+1)^(1/2),x, algorithm="fricas")

[Out]

1/2*sqrt(-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2*sqrt(2))*(sqrt(2) + 2)^(3/4)*sqrt(sqrt(2) + 1)*(sqrt(2) - 1)*a
rctan(1/2*sqrt(sqrt(-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2*sqrt(2))*(sqrt(sqrt(2) + 2)*(sqrt(2) - 2) - 2)*(sqr
t(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*s
qrt(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)*(sqrt(2) - 1) + 2*sqrt(2))*(sqrt(2) + 2)^(3/4)*sqr
t(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)) + 1/2*sqrt(-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2*sqrt(2))*(sqrt(2) + 2)^(3/4)*sqrt(sqrt(2) + 1)*(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(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(s
qrt(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) + sqrt(sqrt(2) +
 1)*(sqrt(2) - 1)) - 1/8*(sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + sqrt(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(sq
rt(2) + 2) + 1) + 1/8*(sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + sqrt(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) - 1/8*2^(1/8)*sqrt(-2*2^(1/4)*(sqrt(2) + 1) + 2*sqrt(2) + 4)*(sqrt(2) + 2^(1/4))*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 + sqr
t(x^2 + 1)) + 2^(1/4) + 1) + 1/8*2^(1/8)*sqrt(-2*2^(1/4)*(sqrt(2) + 1) + 2*sqrt(2) + 4)*(sqrt(2) + 2^(1/4))*lo
g(-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) + 1/2*2^(3/8)*sqrt(-2*2^(1/4)*(sqrt(2) + 1) + 2*sqrt(2) + 4)*sqrt(sqr
t(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(sq
rt(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)*(s
qrt(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
)) + 1/2*2^(3/8)*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)) - sqrt(2)*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(sq
rt(sqrt(2) + 1) - 1) - 1/2*(sqrt(2)*sqrt(sqrt(2) + 1) + sqrt(2))*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)*sqrt(sqrt(s
qrt(2) + 1) - 1)) + 1/4*sqrt(2)*sqrt(sqrt(sqrt(2) + 1) + 1)*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + sqrt(sqrt(
sqrt(2) + 1) + 1)) - 1/4*sqrt(2)*sqrt(sqrt(sqrt(2) + 1) + 1)*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) - sqrt(sqrt
(sqrt(2) + 1) + 1)) - 1/4*sqrt(2)*sqrt(sqrt(sqrt(2) - 1) + 1)*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + sqrt(sqr
t(sqrt(2) - 1) + 1)) + 1/4*sqrt(2)*sqrt(sqrt(sqrt(2) - 1) + 1)*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) - sqrt(sq
rt(sqrt(2) - 1) + 1)) - 1/4*sqrt(2)*sqrt(-sqrt(sqrt(2) - 1) + 1)*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + sqrt(
-sqrt(sqrt(2) - 1) + 1)) + 1/4*sqrt(2)*sqrt(-sqrt(sqrt(2) - 1) + 1)*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) - sq
rt(-sqrt(sqrt(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^2+1)^(1/2),x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{\left (-x^{2}+1\right ) \sqrt {x^{2}+1}}\, dx\]

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^2+1)^(1/2),x)

[Out]

int((1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^2+1)/(x^2+1)^(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}}{\sqrt {x^{2} + 1} {\left (x^{2} - 1\right )}}\,{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^2+1)^(1/2),x, algorithm="maxima")

[Out]

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

[Out]

-int(((x + (x^2 + 1)^(1/2))^(1/2) + 1)^(1/2)/((x^2 - 1)*(x^2 + 1)^(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}}{x^{2} \sqrt {x^{2} + 1} - \sqrt {x^{2} + 1}}\, 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**2+1)**(1/2),x)

[Out]

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

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