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

Optimal. Leaf size=48 \[ 4 \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-4 \tanh ^{-1}\left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right ) \]

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Rubi [F]  time = 0.14, 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+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {1+x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.05, size = 48, normalized size = 1.00 \begin {gather*} 4 \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\tanh ^{-1}\left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

4*(Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]])

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IntegrateAlgebraic [A]  time = 0.10, size = 48, normalized size = 1.00 \begin {gather*} 4 \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-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]]]/Sqrt[1 + x^2],x]

[Out]

4*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - 4*ArcTanh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]]

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fricas [A]  time = 0.44, size = 58, normalized size = 1.21 \begin {gather*} 4 \, \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} - 2 \, \log \left (\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} + 1\right ) + 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((1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(x^2+1)^(1/2),x, algorithm="fricas")

[Out]

4*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) - 2*log(sqrt(sqrt(x + sqrt(x^2 + 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((1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(x^2+1)^(1/2),x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{\sqrt {x^{2}+1}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\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]

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

[Out]

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

[Out]

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

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