3.6.85 \(\int \frac {3+x}{\sqrt {1-x^2}} \, dx\) [585]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, 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 = {655, 222} \begin {gather*} 3 \text {ArcSin}(x)-\sqrt {1-x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + x)/Sqrt[1 - x^2],x]

[Out]

-Sqrt[1 - x^2] + 3*ArcSin[x]

Rule 222

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]

Rule 655

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]

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*}

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Mathematica [A]
time = 0.09, size = 34, normalized size = 1.89 \begin {gather*} -\sqrt {1-x^2}-6 \tan ^{-1}\left (\frac {\sqrt {1-x^2}}{1+x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + x)/Sqrt[1 - x^2],x]

[Out]

-Sqrt[1 - x^2] - 6*ArcTan[Sqrt[1 - x^2]/(1 + x)]

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Maple [A]
time = 0.40, size = 17, normalized size = 0.94

method	result	size
default	\(3 \arcsin \left (x \right )-\sqrt {-x^{2}+1}\)	\(17\)
risch	\(\frac {x^{2}-1}{\sqrt {-x^{2}+1}}+3 \arcsin \left (x \right )\)	\(21\)
meijerg	\(3 \arcsin \left (x \right )-\frac {-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-x^{2}+1}}{2 \sqrt {\pi }}\)	\(31\)
trager	\(-\sqrt {-x^{2}+1}+3 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+1}+x \right )\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.53, size = 16, normalized size = 0.89 \begin {gather*} -\sqrt {-x^{2} + 1} + 3 \, \arcsin \left (x\right ) \end {gather*}

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 = 2.90, size = 30, normalized size = 1.67 \begin {gather*} -\sqrt {-x^{2} + 1} - 6 \, \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \end {gather*}

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 = 0.05, size = 12, normalized size = 0.67 \begin {gather*} - \sqrt {1 - x^{2}} + 3 \operatorname {asin}{\left (x \right )} \end {gather*}

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 = 1.66, size = 16, normalized size = 0.89 \begin {gather*} -\sqrt {-x^{2} + 1} + 3 \, \arcsin \left (x\right ) \end {gather*}

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)

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Mupad [B]
time = 0.33, size = 16, normalized size = 0.89 \begin {gather*} 3\,\mathrm {asin}\left (x\right )-\sqrt {1-x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 3)/(1 - x^2)^(1/2),x)

[Out]

3*asin(x) - (1 - x^2)^(1/2)

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