### 3.332 $$\int \frac{\tan ^{-1}(\sqrt{x})}{\sqrt{x}} \, dx$$

Optimal. Leaf size=20 $2 \sqrt{x} \tan ^{-1}\left (\sqrt{x}\right )-\log (x+1)$

[Out]

2*Sqrt[x]*ArcTan[Sqrt[x]] - Log[1 + x]

________________________________________________________________________________________

Rubi [A]  time = 0.0075875, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.167, Rules used = {5033, 31} $2 \sqrt{x} \tan ^{-1}\left (\sqrt{x}\right )-\log (x+1)$

Antiderivative was successfully veriﬁed.

[In]

Int[ArcTan[Sqrt[x]]/Sqrt[x],x]

[Out]

2*Sqrt[x]*ArcTan[Sqrt[x]] - Log[1 + x]

Rule 5033

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcTan
[c*x^n]))/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[(x^(n - 1)*(d*x)^(m + 1))/(1 + c^2*x^(2*n)), x], x]
/; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0068592, size = 20, normalized size = 1. $2 \sqrt{x} \tan ^{-1}\left (\sqrt{x}\right )-\log (x+1)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[ArcTan[Sqrt[x]]/Sqrt[x],x]

[Out]

2*Sqrt[x]*ArcTan[Sqrt[x]] - Log[1 + x]

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 17, normalized size = 0.9 \begin{align*} -\ln \left ( 1+x \right ) +2\,\arctan \left ( \sqrt{x} \right ) \sqrt{x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-ln(1+x)+2*arctan(x^(1/2))*x^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 0.926823, size = 22, normalized size = 1.1 \begin{align*} 2 \, \sqrt{x} \arctan \left (\sqrt{x}\right ) - \log \left (x + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x)*arctan(sqrt(x)) - log(x + 1)

________________________________________________________________________________________

Fricas [A]  time = 2.0133, size = 54, normalized size = 2.7 \begin{align*} 2 \, \sqrt{x} \arctan \left (\sqrt{x}\right ) - \log \left (x + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x)*arctan(sqrt(x)) - log(x + 1)

________________________________________________________________________________________

Sympy [A]  time = 0.480636, size = 17, normalized size = 0.85 \begin{align*} 2 \sqrt{x} \operatorname{atan}{\left (\sqrt{x} \right )} - \log{\left (x + 1 \right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x)*atan(sqrt(x)) - log(x + 1)

________________________________________________________________________________________

Giac [A]  time = 1.04905, size = 22, normalized size = 1.1 \begin{align*} 2 \, \sqrt{x} \arctan \left (\sqrt{x}\right ) - \log \left (x + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x)*arctan(sqrt(x)) - log(x + 1)