∫ ∘ _______ x+√ ___ 1+x2 ∘ ___________ 1+∘ _______ x+√ ___ 1+x2

3.9.65 $\int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1+x^2} \, dx$

Optimal. Leaf size=65 \[ \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+6 \text {$\#$1}^4-4 \text {$\#$1}^2+2\& ,\frac {\text {$\#$1} \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )}{\text {$\#$1}^2-1}\& \right ] \]

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Rubi [F] time = 0.82, 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 {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1+x^2} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

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

[Out]

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

qrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(I + x), x]

Rubi steps

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

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Mathematica [F] time = 0.14, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1+x^2} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A] time = 0.00, size = 65, normalized size = 1.00 \begin {gather*} \text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^2}\&\right ] \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

IntegrateAlgebraic[(Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(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 + #1^2) &

]

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fricas [B] time = 1.57, size = 2463, normalized size = 37.89

result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))*log((2*(1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))^3 - 4*(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) + 2)*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(-(2*(1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))^3 - 4*(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) + 2)*sqrt(1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2)) + 6*sqrt(sqrt(x + s

qrt(x^2 + 1)) + 1)) + sqrt(-1/2*I*sqrt(2) - 1/2*sqrt(-4*I*sqrt(2) - 2))*log((2*(1/2*I*sqrt(2) - 1/2*sqrt(4*I*s

qrt(2) - 2))^3 - (-1/2*I*sqrt(2) - 1/2*sqrt(-4*I*sqrt(2) - 2))^2*(-I*sqrt(2) + sqrt(4*I*sqrt(2) - 2) - 4) - ((

1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))^2 - 1)*(I*sqrt(2) + sqrt(-4*I*sqrt(2) - 2)) + 2*I*sqrt(2) - 2*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(-(2*(1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))^3 - (

-1/2*I*sqrt(2) - 1/2*sqrt(-4*I*sqrt(2) - 2))^2*(-I*sqrt(2) + sqrt(4*I*sqrt(2) - 2) - 4) - ((1/2*I*sqrt(2) - 1/

2*sqrt(4*I*sqrt(2) - 2))^2 - 1)*(I*sqrt(2) + sqrt(-4*I*sqrt(2) - 2)) + 2*I*sqrt(2) - 2*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)) + 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(1/4*(2*(-1/2*I*sqrt(2) - 1/2*sqrt(-4*I*sqrt(2) - 2))^2*(-I*sqrt(2) + sqrt(4*I*sqrt(2) - 2

) - 4) + 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)) - 8*(1/2*I

*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))^2 - sqrt(-3*(1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))^2 - 3*(-1/2*I*sq

rt(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))*(-I*sqrt(2) + sqrt(4*I*sqrt(2) - 2) - 4) + 4*I*sqrt(2) - 4*

sqrt(4*I*sqrt(2) - 2) + 4) + 2*I*sqrt(2) - 2*sqrt(4*I*sqrt(2) - 2) - 8)*sqrt(2*sqrt(-3*(1/2*I*sqrt(2) - 1/2*sq

rt(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)) - 1/2*sqrt(2*sqrt(-3*(1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))^2 - 3*(-1/2*I*s

qrt(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(-1/4*(2*(-1/2*I*sqrt(2) - 1/2*sqrt(-4*I*sq

rt(2) - 2))^2*(-I*sqrt(2) + sqrt(4*I*sqrt(2) - 2) - 4) + 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)) - 8*(1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 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))*(-I*sqrt(2)

+ sqrt(4*I*sqrt(2) - 2) - 4) + 4*I*sqrt(2) - 4*sqrt(4*I*sqrt(2) - 2) + 4) + 2*I*sqrt(2) - 2*sqrt(4*I*sqrt(2)

- 2) - 8)*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*sqr

t(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*s

qrt(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*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

))*log(1/4*(2*(-1/2*I*sqrt(2) - 1/2*sqrt(-4*I*sqrt(2) - 2))^2*(-I*sqrt(2) + sqrt(4*I*sqrt(2) - 2) - 4) + 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)) - 8*(1/2*I*sqrt(2) - 1/2

*sqrt(4*I*sqrt(2) - 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*sq

rt(-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))*(-I*sqrt(2) + sqrt(4*I*sqrt(2) - 2) - 4) + 4*I*sqrt(2) - 4*sqrt(4*I*sqrt(

2) - 2) + 4) + 2*I*sqrt(2) - 2*sqrt(4*I*sqrt(2) - 2) - 8)*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*s

qrt(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) + sqrt(4*I*sqrt(2) - 2)) - 8

) + sqrt(4*I*sqrt(2) - 2) + sqrt(-4*I*sqrt(2) - 2))*log(-1/4*(2*(-1/2*I*sqrt(2) - 1/2*sqrt(-4*I*sqrt(2) - 2))^

2*(-I*sqrt(2) + sqrt(4*I*sqrt(2) - 2) - 4) + 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)) - 8*(1/2*I*sqrt(2) - 1/2*sqrt(4*I*sqrt(2) - 2))^2 + sqrt(-3*(1/2*I*sqrt(2) - 1/2*sqr

t(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))*(-I*sqrt(2) + sqrt(4*I*

sqrt(2) - 2) - 4) + 4*I*sqrt(2) - 4*sqrt(4*I*sqrt(2) - 2) + 4) + 2*I*sqrt(2) - 2*sqrt(4*I*sqrt(2) - 2) - 8)*sq

rt(-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((x+(x^2+1)^(1/2))^(1/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(x^2+1),x, algorithm="giac")

[Out]

Timed out

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maple [F] time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x +\sqrt {x^{2}+1}}\, \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{x^{2}+1}\, dx\]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}\,\sqrt {x+\sqrt {x^2+1}}}{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 + (x^2 + 1)^(1/2))^(1/2))/(x^2 + 1),x)

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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