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

Optimal. Leaf size=196 \[ \frac {\sqrt {\sqrt {x^2+1}+x} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1} (8 x-15)+\sqrt {x^2+1} \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1} (48 x-16)+8 \sqrt {\sqrt {x^2+1}+x} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right )+\left (48 x^2-16 x-6\right ) \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{60 \sqrt {x^2+1}+60 x}+\frac {1}{4} \tanh ^{-1}\left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right ) \]

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Rubi [F]  time = 0.03, 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 \sqrt {1+\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]]],x]

[Out]

Defer[Int][Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]], x]

Rubi steps

\begin {align*} \int \sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \, dx &=\int \sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.27, size = 179, normalized size = 0.91 \begin {gather*} \frac {1}{120} \left (48 \left (\sqrt {\sqrt {x^2+1}+x}+1\right )^{5/2}-80 \left (\sqrt {\sqrt {x^2+1}+x}+1\right )^{3/2}-\frac {30 \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{\sqrt {\sqrt {x^2+1}+x}}-\frac {60 \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{\sqrt {x^2+1}+x}-15 \log \left (1-\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right )+15 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}+1\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]],x]

[Out]

((-60*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(x + Sqrt[1 + x^2]) - (30*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/Sqrt[x +
Sqrt[1 + x^2]] - 80*(1 + Sqrt[x + Sqrt[1 + x^2]])^(3/2) + 48*(1 + Sqrt[x + Sqrt[1 + x^2]])^(5/2) - 15*Log[1 -
Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]] + 15*Log[1 + Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]])/120

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IntegrateAlgebraic [A]  time = 0.19, size = 196, normalized size = 1.00 \begin {gather*} \frac {\left (-6-16 x+48 x^2\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+(-15+8 x) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\sqrt {1+x^2} \left ((-16+48 x) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+8 \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )}{60 x+60 \sqrt {1+x^2}}+\frac {1}{4} \tanh ^{-1}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]],x]

[Out]

((-6 - 16*x + 48*x^2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (-15 + 8*x)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x
+ Sqrt[1 + x^2]]] + Sqrt[1 + x^2]*((-16 + 48*x)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + 8*Sqrt[x + Sqrt[1 + x^2]]*
Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]))/(60*x + 60*Sqrt[1 + x^2]) + ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]]/4

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fricas [A]  time = 0.48, size = 98, normalized size = 0.50 \begin {gather*} \frac {1}{60} \, {\left ({\left (15 \, x - 15 \, \sqrt {x^{2} + 1} + 8\right )} \sqrt {x + \sqrt {x^{2} + 1}} + 54 \, x - 6 \, \sqrt {x^{2} + 1} - 16\right )} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} + \frac {1}{8} \, \log \left (\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} + 1\right ) - \frac {1}{8} \, \log \left (\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} - 1\right ) \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, algorithm="fricas")

[Out]

1/60*((15*x - 15*sqrt(x^2 + 1) + 8)*sqrt(x + sqrt(x^2 + 1)) + 54*x - 6*sqrt(x^2 + 1) - 16)*sqrt(sqrt(x + sqrt(
x^2 + 1)) + 1) + 1/8*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 1) - 1/8*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) -
1)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 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, algorithm="giac")

[Out]

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

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \sqrt {\sqrt {x+\sqrt {x^2+1}}+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)

[Out]

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

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

[Out]

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

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