### 3.860 $$\int \frac{1+x}{\sqrt{1-x^2}} \, dx$$

Optimal. Leaf size=16 $\sin ^{-1}(x)-\sqrt{1-x^2}$

[Out]

-Sqrt[1 - x^2] + ArcSin[x]

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Rubi [A]  time = 0.0043927, antiderivative size = 16, 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} $\sin ^{-1}(x)-\sqrt{1-x^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(1 + x)/Sqrt[1 - x^2],x]

[Out]

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

Mathematica [A]  time = 0.0057832, size = 16, normalized size = 1. $\sin ^{-1}(x)-\sqrt{1-x^2}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(1 + x)/Sqrt[1 - x^2],x]

[Out]

-Sqrt[1 - x^2] + ArcSin[x]

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Maple [A]  time = 0.042, size = 15, normalized size = 0.9 \begin{align*} \arcsin \left ( x \right ) -\sqrt{-{x}^{2}+1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.64176, size = 19, normalized size = 1.19 \begin{align*} -\sqrt{-x^{2} + 1} + \arcsin \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 1) + arcsin(x)

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Fricas [B]  time = 2.03078, size = 70, normalized size = 4.38 \begin{align*} -\sqrt{-x^{2} + 1} - 2 \, \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.137741, size = 10, normalized size = 0.62 \begin{align*} - \sqrt{1 - x^{2}} + \operatorname{asin}{\left (x \right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(1 - x**2) + asin(x)

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Giac [A]  time = 1.27079, size = 19, normalized size = 1.19 \begin{align*} -\sqrt{-x^{2} + 1} + \arcsin \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 1) + arcsin(x)