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

Optimal. Leaf size=242 \[ \frac {\sqrt {\sqrt {x^2+1}+x} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1} \left (2560 x^3+2048 x^2+345 x+184\right )+\sqrt {x^2+1} \left (\sqrt {\sqrt {x^2+1}+x} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1} \left (2560 x^2+2048 x-935\right )+\left (35840 x^3-3072 x^2+175104 x+282\right ) \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right )+\left (35840 x^4-3072 x^3+193024 x^2-1254 x+78032\right ) \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{80640 \sqrt {x^2+1} x+40320 \left (2 x^2+1\right )}-\frac {251}{128} \tanh ^{-1}\left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right ) \]

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Rubi [F]  time = 0.15, 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+x^2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.65, size = 373, normalized size = 1.54 \begin {gather*} \frac {\left (x^2+\sqrt {x^2+1} x+1\right ) \left (79065 \left (2 x^2+2 \sqrt {x^2+1} x+1\right ) \log \left (1-\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right )-79065 \left (2 x^2+2 \sqrt {x^2+1} x+1\right ) \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}+1\right )+2 \sqrt {\sqrt {\sqrt {x^2+1}+x}+1} \left (35840 x^4+512 \left (5 \sqrt {\sqrt {x^2+1}+x} \sqrt {x^2+1}-6 \sqrt {x^2+1}+4 \sqrt {\sqrt {x^2+1}+x}+377\right ) x^2+\left (2048 \sqrt {\sqrt {x^2+1}+x} \sqrt {x^2+1}+175104 \sqrt {x^2+1}+345 \sqrt {\sqrt {x^2+1}+x}-1254\right ) x+282 \sqrt {x^2+1}-935 \sqrt {x^2+1} \sqrt {\sqrt {x^2+1}+x}+184 \sqrt {\sqrt {x^2+1}+x}+512 \left (70 \sqrt {x^2+1}+5 \sqrt {\sqrt {x^2+1}+x}-6\right ) x^3+78032\right )\right )}{80640 \sqrt {x^2+1} \left (\sqrt {x^2+1}+x\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((1 + x^2 + x*Sqrt[1 + x^2])*(2*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]*(78032 + 35840*x^4 + 282*Sqrt[1 + x^2] + 184
*Sqrt[x + Sqrt[1 + x^2]] - 935*Sqrt[1 + x^2]*Sqrt[x + Sqrt[1 + x^2]] + 512*x^3*(-6 + 70*Sqrt[1 + x^2] + 5*Sqrt
[x + Sqrt[1 + x^2]]) + 512*x^2*(377 - 6*Sqrt[1 + x^2] + 4*Sqrt[x + Sqrt[1 + x^2]] + 5*Sqrt[1 + x^2]*Sqrt[x + S
qrt[1 + x^2]]) + x*(-1254 + 175104*Sqrt[1 + x^2] + 345*Sqrt[x + Sqrt[1 + x^2]] + 2048*Sqrt[1 + x^2]*Sqrt[x + S
qrt[1 + x^2]])) + 79065*(1 + 2*x^2 + 2*x*Sqrt[1 + x^2])*Log[1 - Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]] - 79065*(1
+ 2*x^2 + 2*x*Sqrt[1 + x^2])*Log[1 + Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]]))/(80640*Sqrt[1 + x^2]*(x + Sqrt[1 + x
^2])^3)

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IntegrateAlgebraic [A]  time = 0.31, size = 242, normalized size = 1.00 \begin {gather*} \frac {\left (78032-1254 x+193024 x^2-3072 x^3+35840 x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (184+345 x+2048 x^2+2560 x^3\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\sqrt {1+x^2} \left (\left (282+175104 x-3072 x^2+35840 x^3\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (-935+2048 x+2560 x^2\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )}{80640 x \sqrt {1+x^2}+40320 \left (1+2 x^2\right )}-\frac {251}{128} \tanh ^{-1}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((78032 - 1254*x + 193024*x^2 - 3072*x^3 + 35840*x^4)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (184 + 345*x + 2048*
x^2 + 2560*x^3)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[1 + x^2]*((282 + 175104*x - 3
072*x^2 + 35840*x^3)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (-935 + 2048*x + 2560*x^2)*Sqrt[x + Sqrt[1 + x^2]]*Sq
rt[1 + Sqrt[x + Sqrt[1 + x^2]]]))/(80640*x*Sqrt[1 + x^2] + 40320*(1 + 2*x^2)) - (251*ArcTanh[Sqrt[1 + Sqrt[x +
Sqrt[1 + x^2]]]])/128

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fricas [A]  time = 0.52, size = 118, normalized size = 0.49 \begin {gather*} -\frac {1}{40320} \, {\left (1120 \, x^{2} - 2 \, \sqrt {x^{2} + 1} {\left (9520 \, x + 141\right )} + {\left (1680 \, x^{2} - 5 \, \sqrt {x^{2} + 1} {\left (336 \, x - 187\right )} - 2215 \, x - 184\right )} \sqrt {x + \sqrt {x^{2} + 1}} + 1818 \, x - 78032\right )} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} - \frac {251}{256} \, \log \left (\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} + 1\right ) + \frac {251}{256} \, \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^2+1)^(1/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2),x, algorithm="fricas")

[Out]

-1/40320*(1120*x^2 - 2*sqrt(x^2 + 1)*(9520*x + 141) + (1680*x^2 - 5*sqrt(x^2 + 1)*(336*x - 187) - 2215*x - 184
)*sqrt(x + sqrt(x^2 + 1)) + 1818*x - 78032)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) - 251/256*log(sqrt(sqrt(x + sqrt
(x^2 + 1)) + 1) + 1) + 251/256*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^2+1)^(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 = 0.03, size = 0, normalized size = 0.00 \[\int \sqrt {x^{2}+1}\, \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(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 = -0.00 \begin {gather*} \int \sqrt {\sqrt {x+\sqrt {x^2+1}}+1}\,\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)^(1/2),x)

[Out]

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

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

[Out]

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

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