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

Optimal. Leaf size=6 \[ \tan ^{-1}\left (\sqrt{x}\right ) \]

[Out]

ArcTan[Sqrt[x]]

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Rubi [A]  time = 0.0024988, antiderivative size = 6, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {12, 63, 203} \[ \tan ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(2*Sqrt[x]*(1 + x)),x]

[Out]

ArcTan[Sqrt[x]]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0030513, size = 6, normalized size = 1. \[ \tan ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(2*Sqrt[x]*(1 + x)),x]

[Out]

ArcTan[Sqrt[x]]

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Maple [A]  time = 0.003, size = 5, normalized size = 0.8 \begin{align*} \arctan \left ( \sqrt{x} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(x^(1/2))

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Maxima [A]  time = 1.52695, size = 5, normalized size = 0.83 \begin{align*} \arctan \left (\sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(sqrt(x))

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Fricas [A]  time = 1.50217, size = 23, normalized size = 3.83 \begin{align*} \arctan \left (\sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(sqrt(x))

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Sympy [A]  time = 0.205102, size = 5, normalized size = 0.83 \begin{align*} \operatorname{atan}{\left (\sqrt{x} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan(sqrt(x))

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Giac [A]  time = 1.11072, size = 5, normalized size = 0.83 \begin{align*} \arctan \left (\sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(sqrt(x))