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

Optimal. Leaf size=63 \[ -\frac {2 \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{\sqrt {\sqrt {x^2+1}+x}}-2 \tanh ^{-1}\left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right ) \]

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Rubi [F]  time = 0.45, 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} \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]]]/(Sqrt[1 + x^2]*Sqrt[x + Sqrt[1 + x^2]]),x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.11, size = 117, normalized size = 1.86 \begin {gather*} -\frac {2 \left (\sqrt {x^2+1}+\sqrt {\sqrt {x^2+1}+x}+\left (\sqrt {x^2+1}+x\right ) \sqrt {\sqrt {\sqrt {x^2+1}+x}+1} \tanh ^{-1}\left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right )+x\right )}{\left (\sqrt {x^2+1}+x\right ) \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(x + Sqrt[1 + x^2] + Sqrt[x + Sqrt[1 + x^2]] + (x + Sqrt[1 + x^2])*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]*ArcTa
nh[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]]))/((x + Sqrt[1 + x^2])*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])

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IntegrateAlgebraic [A]  time = 0.12, size = 63, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {x+\sqrt {1+x^2}}}-2 \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]*Sqrt[x + Sqrt[1 + x^2]]),x]

[Out]

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

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fricas [A]  time = 0.48, size = 78, normalized size = 1.24 \begin {gather*} 2 \, \sqrt {x + \sqrt {x^{2} + 1}} {\left (x - \sqrt {x^{2} + 1}\right )} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} - \log \left (\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} + 1\right ) + \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+(x^2+1)^(1/2))^(1/2),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(x^2+1)^(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 {\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]

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

[Out]

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

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mupad [B]  time = 1.49, size = 121, normalized size = 1.92 \begin {gather*} -\frac {2\,\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{\sqrt {x+\sqrt {x^2+1}}}-\frac {\ln \left (\sqrt {\frac {1}{x+\sqrt {x^2+1}}+\frac {1}{\sqrt {x+\sqrt {x^2+1}}}}+\frac {1}{\sqrt {x+\sqrt {x^2+1}}}+\frac {1}{2}\right )\,\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{\sqrt {x+\sqrt {x^2+1}}\,\sqrt {\frac {1}{x+\sqrt {x^2+1}}+\frac {1}{\sqrt {x+\sqrt {x^2+1}}}}} \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]

- (2*((x + (x^2 + 1)^(1/2))^(1/2) + 1)^(1/2))/(x + (x^2 + 1)^(1/2))^(1/2) - (log((1/(x + (x^2 + 1)^(1/2)) + 1/
(x + (x^2 + 1)^(1/2))^(1/2))^(1/2) + 1/(x + (x^2 + 1)^(1/2))^(1/2) + 1/2)*((x + (x^2 + 1)^(1/2))^(1/2) + 1)^(1
/2))/((x + (x^2 + 1)^(1/2))^(1/2)*(1/(x + (x^2 + 1)^(1/2)) + 1/(x + (x^2 + 1)^(1/2))^(1/2))^(1/2))

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

[Out]

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

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