3.1.20 \(\int \frac {x}{\sqrt {1+x^2} \sqrt {1+\sqrt {1+x^2}}} \, dx\) [20]

Optimal. Leaf size=17 \[ 2 \sqrt {1+\sqrt {1+x^2}} \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {6818} \begin {gather*} 2 \sqrt {\sqrt {x^2+1}+1} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*Sqrt[1 + Sqrt[1 + x^2]]

Rule 6818

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*(y^(m + 1)/(m + 1)), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 17, normalized size = 1.00 \begin {gather*} 2 \sqrt {1+\sqrt {1+x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*Sqrt[1 + Sqrt[1 + x^2]]

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Maple [A]
time = 0.06, size = 14, normalized size = 0.82

method	result	size
derivativedivides	\(2 \sqrt {1+\sqrt {x^{2}+1}}\)	\(14\)
default	\(2 \sqrt {1+\sqrt {x^{2}+1}}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^2+1)^(1/2)/(1+(x^2+1)^(1/2))^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 3.78, size = 13, normalized size = 0.76 \begin {gather*} 2 \, \sqrt {\sqrt {x^{2} + 1} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(sqrt(x^2 + 1) + 1)

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Fricas [A]
time = 0.49, size = 13, normalized size = 0.76 \begin {gather*} 2 \, \sqrt {\sqrt {x^{2} + 1} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(sqrt(x^2 + 1) + 1)

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Sympy [A]
time = 0.11, size = 14, normalized size = 0.82 \begin {gather*} 2 \sqrt {\sqrt {x^{2} + 1} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(sqrt(x**2 + 1) + 1)

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Giac [A]
time = 0.59, size = 13, normalized size = 0.76 \begin {gather*} 2 \, \sqrt {\sqrt {x^{2} + 1} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(sqrt(x^2 + 1) + 1)

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Mupad [B]
time = 0.21, size = 13, normalized size = 0.76 \begin {gather*} 2\,\sqrt {\sqrt {x^2+1}+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*((x^2 + 1)^(1/2) + 1)^(1/2)

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