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

Optimal. Leaf size=236 \[ \frac {\sqrt {\sqrt {x^2+1}+x} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1} \left (3072 x^3+4096 x^2+1814 x+1712\right )+\sqrt {x^2+1} \left (\sqrt {\sqrt {x^2+1}+x} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1} \left (3072 x^2+4096 x+278\right )+\left (30720 x^3-4096 x^2-36930 x-632\right ) \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right )+\left (30720 x^4-4096 x^3-21570 x^2-2680 x-24993\right ) \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{26880 \left (\sqrt {x^2+1}+x\right )^{5/2}}-\frac {263}{256} \tanh ^{-1}\left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right ) \]

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Rubi [F]  time = 0.47, 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+x^2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\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]]])/Sqrt[x + Sqrt[1 + x^2]],x]

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 4.51, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1+x^2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {x+\sqrt {1+x^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.35, size = 236, normalized size = 1.00 \begin {gather*} \frac {\left (-24993-2680 x-21570 x^2-4096 x^3+30720 x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (1712+1814 x+4096 x^2+3072 x^3\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\sqrt {1+x^2} \left (\left (-632-36930 x-4096 x^2+30720 x^3\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (278+4096 x+3072 x^2\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )}{26880 \left (x+\sqrt {1+x^2}\right )^{5/2}}-\frac {263}{256} \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]]])/Sqrt[x + Sqrt[1 + x^2]],x]

[Out]

((-24993 - 2680*x - 21570*x^2 - 4096*x^3 + 30720*x^4)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (1712 + 1814*x + 409
6*x^2 + 3072*x^3)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[1 + x^2]*((-632 - 36930*x -
4096*x^2 + 30720*x^3)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (278 + 4096*x + 3072*x^2)*Sqrt[x + Sqrt[1 + x^2]]*S
qrt[1 + Sqrt[x + Sqrt[1 + x^2]]]))/(26880*(x + Sqrt[1 + x^2])^(5/2)) - (263*ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 +
x^2]]]])/256

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fricas [A]  time = 0.51, size = 129, normalized size = 0.55 \begin {gather*} -\frac {1}{26880} \, {\left (672 \, x^{2} - 2 \, \sqrt {x^{2} + 1} {\left (336 \, x + 139\right )} - {\left (10752 \, x^{3} + 784 \, x^{2} - {\left (10752 \, x^{2} + 784 \, x + 24993\right )} \sqrt {x^{2} + 1} + 38049 \, x - 632\right )} \sqrt {x + \sqrt {x^{2} + 1}} - 1258 \, x - 1712\right )} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} - \frac {263}{512} \, \log \left (\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} + 1\right ) + \frac {263}{512} \, \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+(x^2+1)^(1/2))^(1/2),x, algorithm="fricas")

[Out]

-1/26880*(672*x^2 - 2*sqrt(x^2 + 1)*(336*x + 139) - (10752*x^3 + 784*x^2 - (10752*x^2 + 784*x + 24993)*sqrt(x^
2 + 1) + 38049*x - 632)*sqrt(x + sqrt(x^2 + 1)) - 1258*x - 1712)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) - 263/512*l
og(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 1) + 263/512*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+(x^2+1)^(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 \[\int \frac {\sqrt {x^{2}+1}\, \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{\sqrt {x +\sqrt {x^{2}+1}}}\, dx\]

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

[Out]

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

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

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

[Out]

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

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

[Out]

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

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