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

Optimal. Leaf size=195 \[ \frac {\sqrt {\sqrt {x^2+1}+x} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1} (6 x+16)+\sqrt {x^2+1} \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1} (60 x-8)+6 \sqrt {\sqrt {x^2+1}+x} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right )+\left (60 x^2-8 x-75\right ) \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{105 \sqrt {\sqrt {x^2+1}+x}}-\tanh ^{-1}\left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right ) \]

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Rubi [F]  time = 0.16, 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 {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {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]]],x]

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 1.90, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {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]]],x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.21, size = 195, normalized size = 1.00 \begin {gather*} \frac {\left (-75-8 x+60 x^2\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+(16+6 x) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\sqrt {1+x^2} \left ((-8+60 x) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+6 \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )}{105 \sqrt {x+\sqrt {1+x^2}}}-\tanh ^{-1}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-75 - 8*x + 60*x^2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (16 + 6*x)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x +
Sqrt[1 + x^2]]] + Sqrt[1 + x^2]*((-8 + 60*x)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + 6*Sqrt[x + Sqrt[1 + x^2]]*Sq
rt[1 + Sqrt[x + Sqrt[1 + x^2]]]))/(105*Sqrt[x + Sqrt[1 + x^2]]) - ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]]

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fricas [A]  time = 0.66, size = 98, normalized size = 0.50 \begin {gather*} \frac {1}{105} \, {\left ({\left (135 \, x - 75 \, \sqrt {x^{2} + 1} - 8\right )} \sqrt {x + \sqrt {x^{2} + 1}} + 6 \, x + 6 \, \sqrt {x^{2} + 1} + 16\right )} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} - \frac {1}{2} \, \log \left (\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} + 1\right ) + \frac {1}{2} \, \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((x+(x^2+1)^(1/2))^(1/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2),x, algorithm="fricas")

[Out]

1/105*((135*x - 75*sqrt(x^2 + 1) - 8)*sqrt(x + sqrt(x^2 + 1)) + 6*x + 6*sqrt(x^2 + 1) + 16)*sqrt(sqrt(x + sqrt
(x^2 + 1)) + 1) - 1/2*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 1) + 1/2*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 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, algorithm="giac")

[Out]

Timed out

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

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)

[Out]

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

[Out]

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

[Out]

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

[Out]

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

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