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

Optimal. Leaf size=37 \[ \frac {1}{3} \left (\sqrt {x^2+1}+x\right )^{3/2}-\frac {1}{\sqrt {\sqrt {x^2+1}+x}} \]

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Rubi [A]  time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2117, 14} \begin {gather*} \frac {1}{3} \left (\sqrt {x^2+1}+x\right )^{3/2}-\frac {1}{\sqrt {\sqrt {x^2+1}+x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[x + Sqrt[1 + x^2]],x]

[Out]

-(1/Sqrt[x + Sqrt[1 + x^2]]) + (x + Sqrt[1 + x^2])^(3/2)/3

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2117

Int[((g_.) + (h_.)*((d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_.)*(x_)^2])^(n_))^(p_.), x_Symbol] :> Dist[1/(2*
e), Subst[Int[((g + h*x^n)^p*(d^2 + a*f^2 - 2*d*x + x^2))/(d - x)^2, x], x, d + e*x + f*Sqrt[a + c*x^2]], x] /
; FreeQ[{a, c, d, e, f, g, h, n}, x] && EqQ[e^2 - c*f^2, 0] && IntegerQ[p]

Rubi steps

\begin {align*} \int \sqrt {x+\sqrt {1+x^2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1+x^2}{x^{3/2}} \, dx,x,x+\sqrt {1+x^2}\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{x^{3/2}}+\sqrt {x}\right ) \, dx,x,x+\sqrt {1+x^2}\right )\\ &=-\frac {1}{\sqrt {x+\sqrt {1+x^2}}}+\frac {1}{3} \left (x+\sqrt {1+x^2}\right )^{3/2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 35, normalized size = 0.95 \begin {gather*} \frac {2 \left (x^2+\sqrt {x^2+1} x-1\right )}{3 \sqrt {\sqrt {x^2+1}+x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x + Sqrt[1 + x^2]],x]

[Out]

(2*(-1 + x^2 + x*Sqrt[1 + x^2]))/(3*Sqrt[x + Sqrt[1 + x^2]])

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IntegrateAlgebraic [A]  time = 0.04, size = 37, normalized size = 1.00 \begin {gather*} -\frac {1}{\sqrt {x+\sqrt {1+x^2}}}+\frac {1}{3} \left (x+\sqrt {1+x^2}\right )^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(1/Sqrt[x + Sqrt[1 + x^2]]) + (x + Sqrt[1 + x^2])^(3/2)/3

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fricas [A]  time = 0.46, size = 26, normalized size = 0.70 \begin {gather*} \frac {2}{3} \, {\left (2 \, x - \sqrt {x^{2} + 1}\right )} \sqrt {x + \sqrt {x^{2} + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(2*x - sqrt(x^2 + 1))*sqrt(x + sqrt(x^2 + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(x + sqrt(x^2 + 1)), x)

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maple [C]  time = 0.02, size = 57, normalized size = 1.54

method	result	size
meijerg	\(\frac {\frac {16 \sqrt {\pi }\, \sqrt {2}\, x^{\frac {3}{2}} \left (1-\frac {1}{x^{2}}\right ) \cosh \left (\frac {\arcsinh \left (\frac {1}{x}\right )}{2}\right )}{3}+\frac {16 \sqrt {\pi }\, \sqrt {2}\, \sqrt {x}\, \sqrt {1+\frac {1}{x^{2}}}\, \sinh \left (\frac {\arcsinh \left (\frac {1}{x}\right )}{2}\right )}{3}}{8 \sqrt {\pi }}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8/Pi^(1/2)*(16/3*Pi^(1/2)*2^(1/2)*x^(3/2)*(1-1/x^2)*cosh(1/2*arcsinh(1/x))+16/3*Pi^(1/2)*2^(1/2)*x^(1/2)*(1+
1/x^2)^(1/2)*sinh(1/2*arcsinh(1/x)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(x + sqrt(x^2 + 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \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),x)

[Out]

int((x + (x^2 + 1)^(1/2))^(1/2), x)

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sympy [A]  time = 0.29, size = 42, normalized size = 1.14 \begin {gather*} \frac {4 x \sqrt {x + \sqrt {x^{2} + 1}}}{3} - \frac {2 \sqrt {x + \sqrt {x^{2} + 1}} \sqrt {x^{2} + 1}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*x*sqrt(x + sqrt(x**2 + 1))/3 - 2*sqrt(x + sqrt(x**2 + 1))*sqrt(x**2 + 1)/3

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