3.1.57 \(\int \frac {(1+x)^2}{\sqrt {1-x^2}} \, dx\)

Optimal. Leaf size=40 \[ -\frac {1}{2} \sqrt {1-x^2} (x+1)-\frac {3 \sqrt {1-x^2}}{2}+\frac {3}{2} \sin ^{-1}(x) \]

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Rubi [A]  time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {671, 641, 216} \begin {gather*} -\frac {1}{2} \sqrt {1-x^2} (x+1)-\frac {3 \sqrt {1-x^2}}{2}+\frac {3}{2} \sin ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x)^2/Sqrt[1 - x^2],x]

[Out]

(-3*Sqrt[1 - x^2])/2 - ((1 + x)*Sqrt[1 - x^2])/2 + (3*ArcSin[x])/2

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]

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 671

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)*(a + c*x^2)^(p
+ 1))/(c*(m + 2*p + 1)), x] + Dist[(2*c*d*(m + p))/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p, x]
, x] /; FreeQ[{a, c, d, e, p}, x] && EqQ[c*d^2 + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p
]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 25, normalized size = 0.62 \begin {gather*} \frac {1}{2} \left (3 \sin ^{-1}(x)-(x+4) \sqrt {1-x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x)^2/Sqrt[1 - x^2],x]

[Out]

(-((4 + x)*Sqrt[1 - x^2]) + 3*ArcSin[x])/2

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IntegrateAlgebraic [A]  time = 0.19, size = 41, normalized size = 1.02 \begin {gather*} \frac {1}{2} (-x-4) \sqrt {1-x^2}-3 \tan ^{-1}\left (\frac {\sqrt {1-x^2}}{x+1}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 + x)^2/Sqrt[1 - x^2],x]

[Out]

((-4 - x)*Sqrt[1 - x^2])/2 - 3*ArcTan[Sqrt[1 - x^2]/(1 + x)]

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fricas [A]  time = 0.40, size = 33, normalized size = 0.82 \begin {gather*} -\frac {1}{2} \, \sqrt {-x^{2} + 1} {\left (x + 4\right )} - 3 \, \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(-x^2 + 1)*(x + 4) - 3*arctan((sqrt(-x^2 + 1) - 1)/x)

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giac [A]  time = 0.19, size = 19, normalized size = 0.48 \begin {gather*} -\frac {1}{2} \, \sqrt {-x^{2} + 1} {\left (x + 4\right )} + \frac {3}{2} \, \arcsin \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(-x^2 + 1)*(x + 4) + 3/2*arcsin(x)

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maple [A]  time = 0.00, size = 29, normalized size = 0.72 \begin {gather*} -\frac {\sqrt {-x^{2}+1}\, x}{2}+\frac {3 \arcsin \relax (x )}{2}-2 \sqrt {-x^{2}+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.96, size = 28, normalized size = 0.70 \begin {gather*} -\frac {1}{2} \, \sqrt {-x^{2} + 1} x - 2 \, \sqrt {-x^{2} + 1} + \frac {3}{2} \, \arcsin \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.03, size = 21, normalized size = 0.52 \begin {gather*} \frac {3\,\mathrm {asin}\relax (x)}{2}-\left (\frac {x}{2}+2\right )\,\sqrt {1-x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)^2/(1 - x^2)^(1/2),x)

[Out]

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

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sympy [A]  time = 0.24, size = 27, normalized size = 0.68 \begin {gather*} - \frac {x \sqrt {1 - x^{2}}}{2} - 2 \sqrt {1 - x^{2}} + \frac {3 \operatorname {asin}{\relax (x )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*sqrt(1 - x**2)/2 - 2*sqrt(1 - x**2) + 3*asin(x)/2

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