3.96 \(\int \frac{\tan ^{-1}(\sqrt{x})}{\sqrt{x} (1+x)} \, dx\)

Optimal. Leaf size=8 \[ \tan ^{-1}\left (\sqrt{x}\right )^2 \]

[Out]

ArcTan[Sqrt[x]]^2

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Rubi [A]  time = 0.0304373, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {63, 203, 6686} \[ \tan ^{-1}\left (\sqrt{x}\right )^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[Sqrt[x]]^2

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])

Rule 6686

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

Mathematica [A]  time = 0.0055862, size = 8, normalized size = 1. \[ \tan ^{-1}\left (\sqrt{x}\right )^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[Sqrt[x]]^2

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Maple [A]  time = 0.004, size = 7, normalized size = 0.9 \begin{align*} \left ( \arctan \left ( \sqrt{x} \right ) \right ) ^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(x^(1/2))^2

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Maxima [A]  time = 0.936507, size = 8, normalized size = 1. \begin{align*} \arctan \left (\sqrt{x}\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(sqrt(x))^2

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Fricas [A]  time = 0.519548, size = 26, normalized size = 3.25 \begin{align*} \arctan \left (\sqrt{x}\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(sqrt(x))^2

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Sympy [A]  time = 2.04667, size = 7, normalized size = 0.88 \begin{align*} \operatorname{atan}^{2}{\left (\sqrt{x} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan(sqrt(x))**2

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Giac [A]  time = 1.07531, size = 8, normalized size = 1. \begin{align*} \arctan \left (\sqrt{x}\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(sqrt(x))^2