3.9.55 \(\int \frac {\sqrt {1+x}}{\sqrt {1-x^2}} \, dx\) [855]

Optimal. Leaf size=11 \[ -2 \sqrt {1-x} \]

[Out]

-2*(1-x)^(1/2)

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Rubi [A]
time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {26, 32} \begin {gather*} -2 \sqrt {1-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + x]/Sqrt[1 - x^2],x]

[Out]

-2*Sqrt[1 - x]

Rule 26

Int[(u_.)*((a_) + (b_.)*(x_)^(n_.))^(m_.)*((c_) + (d_.)*(x_)^(j_))^(p_.), x_Symbol] :> Dist[(-b^2/d)^m, Int[u/
(a - b*x^n)^m, x], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[j, 2*n] && EqQ[p, -m] && EqQ[b^2*c + a^2*d, 0]
&& GtQ[a, 0] && LtQ[d, 0]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

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Mathematica [A]
time = 0.03, size = 20, normalized size = 1.82 \begin {gather*} -\frac {2 \sqrt {1-x^2}}{\sqrt {1+x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + x]/Sqrt[1 - x^2],x]

[Out]

(-2*Sqrt[1 - x^2])/Sqrt[1 + x]

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Maple [A]
time = 0.56, size = 17, normalized size = 1.55

method	result	size
default	\(-\frac {2 \sqrt {-x^{2}+1}}{\sqrt {1+x}}\)	\(17\)
gosper	\(\frac {2 \left (-1+x \right ) \sqrt {1+x}}{\sqrt {-x^{2}+1}}\)	\(20\)
risch	\(\frac {2 \sqrt {\frac {-x^{2}+1}{1+x}}\, \sqrt {1+x}\, \left (-1+x \right )}{\sqrt {-x^{2}+1}\, \sqrt {1-x}}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C] Result contains higher order function than in optimal. Order 3 vs. order 2.
time = 0.30, size = 12, normalized size = 1.09 \begin {gather*} \frac {2 \, {\left (x - 1\right )}}{\sqrt {-x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(x - 1)/sqrt(-x + 1)

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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 2.
time = 0.33, size = 16, normalized size = 1.45 \begin {gather*} -\frac {2 \, \sqrt {-x^{2} + 1}}{\sqrt {x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(-x^2 + 1)/sqrt(x + 1)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x + 1}}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(x + 1)/sqrt(-(x - 1)*(x + 1)), x)

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Giac [A]
time = 2.60, size = 15, normalized size = 1.36 \begin {gather*} 2 \, \sqrt {2} - 2 \, \sqrt {-x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(2) - 2*sqrt(-x + 1)

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Mupad [B]
time = 3.56, size = 16, normalized size = 1.45 \begin {gather*} -\frac {2\,\sqrt {1-x^2}}{\sqrt {x+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)^(1/2)/(1 - x^2)^(1/2),x)

[Out]

-(2*(1 - x^2)^(1/2))/(x + 1)^(1/2)

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