3.1007 \(\int \sqrt {\frac {x^2}{1+x^2}} \, dx\)

Optimal. Leaf size=20 \[ \frac {\sqrt {x^2} \sqrt {x^2+1}}{x} \]

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1958, 15, 261} \[ \frac {\sqrt {x^2} \sqrt {x^2+1}}{x} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x^2/(1 + x^2)],x]

[Out]

(Sqrt[x^2]*Sqrt[1 + x^2])/x

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 1958

Int[(u_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> Int[(u*(e*(a + b*x
^n))^p)/(c + d*x^n)^p, x] /; FreeQ[{a, b, c, d, e, n, p}, x] && GtQ[b*d*e, 0] && GtQ[c - (a*d)/b, 0]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 17, normalized size = 0.85 \[ \frac {x}{\sqrt {\frac {x^2}{x^2+1}}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x^2/(1 + x^2)],x]

[Out]

x/Sqrt[x^2/(1 + x^2)]

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fricas [A]  time = 0.66, size = 22, normalized size = 1.10 \[ \frac {{\left (x^{2} + 1\right )} \sqrt {\frac {x^{2}}{x^{2} + 1}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^2 + 1)*sqrt(x^2/(x^2 + 1))/x

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giac [A]  time = 0.44, size = 15, normalized size = 0.75 \[ \sqrt {x^{2} + 1} \mathrm {sgn}\relax (x) - \mathrm {sgn}\relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(x^2 + 1)*sgn(x) - sgn(x)

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maple [A]  time = 0.00, size = 23, normalized size = 1.15 \[ \frac {\left (x^{2}+1\right ) \sqrt {\frac {x^{2}}{x^{2}+1}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.08, size = 7, normalized size = 0.35 \[ \sqrt {x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(x^2 + 1)

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mupad [B]  time = 3.41, size = 13, normalized size = 0.65 \[ \frac {\sqrt {x^4+x^2}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2/(x^2 + 1))^(1/2),x)

[Out]

(x^2 + x^4)^(1/2)/x

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sympy [B]  time = 0.44, size = 36, normalized size = 1.80 \[ x \sqrt {x^{2}} \sqrt {\frac {1}{x^{2} + 1}} + \frac {\sqrt {x^{2}} \sqrt {\frac {1}{x^{2} + 1}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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