### 3.585 $$\int \frac{3+x}{\sqrt{1-x^2}} \, dx$$

Optimal. Leaf size=18 $3 \sin ^{-1}(x)-\sqrt{1-x^2}$

[Out]

-Sqrt[1 - x^2] + 3*ArcSin[x]

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Rubi [A]  time = 0.0049978, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.133, Rules used = {641, 216} $3 \sin ^{-1}(x)-\sqrt{1-x^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(3 + x)/Sqrt[1 - x^2],x]

[Out]

-Sqrt[1 - x^2] + 3*ArcSin[x]

Rule 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rule 216

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

Rubi steps

\begin{align*} \int \frac{3+x}{\sqrt{1-x^2}} \, dx &=-\sqrt{1-x^2}+3 \int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=-\sqrt{1-x^2}+3 \sin ^{-1}(x)\\ \end{align*}

Mathematica [A]  time = 0.0107118, size = 18, normalized size = 1. $3 \sin ^{-1}(x)-\sqrt{1-x^2}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(3 + x)/Sqrt[1 - x^2],x]

[Out]

-Sqrt[1 - x^2] + 3*ArcSin[x]

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Maple [A]  time = 0.043, size = 17, normalized size = 0.9 \begin{align*} 3\,\arcsin \left ( x \right ) -\sqrt{-{x}^{2}+1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((3+x)/(-x^2+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.74777, size = 22, normalized size = 1.22 \begin{align*} -\sqrt{-x^{2} + 1} + 3 \, \arcsin \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+x)/(-x^2+1)^(1/2),x, algorithm="maxima")

[Out]

-sqrt(-x^2 + 1) + 3*arcsin(x)

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Fricas [A]  time = 1.72906, size = 70, normalized size = 3.89 \begin{align*} -\sqrt{-x^{2} + 1} - 6 \, \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+x)/(-x^2+1)^(1/2),x, algorithm="fricas")

[Out]

-sqrt(-x^2 + 1) - 6*arctan((sqrt(-x^2 + 1) - 1)/x)

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Sympy [A]  time = 0.136026, size = 12, normalized size = 0.67 \begin{align*} - \sqrt{1 - x^{2}} + 3 \operatorname{asin}{\left (x \right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+x)/(-x**2+1)**(1/2),x)

[Out]

-sqrt(1 - x**2) + 3*asin(x)

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Giac [A]  time = 1.38096, size = 22, normalized size = 1.22 \begin{align*} -\sqrt{-x^{2} + 1} + 3 \, \arcsin \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+x)/(-x^2+1)^(1/2),x, algorithm="giac")

[Out]

-sqrt(-x^2 + 1) + 3*arcsin(x)