3.3.44 \(\int \frac {1}{\sqrt {2-2 x^2} \sqrt {-1+x^2}} \, dx\) [244]

Optimal. Leaf size=29 \[ -\frac {\sqrt {-1+x^2} \tanh ^{-1}(x)}{\sqrt {2} \sqrt {1-x^2}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {23, 213} \begin {gather*} -\frac {\sqrt {x^2-1} \tanh ^{-1}(x)}{\sqrt {2} \sqrt {1-x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 23

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((c_) + (d_.)*(v_))^(n_), x_Symbol] :> Dist[(a + b*v)^m/(c + d*v)^m, Int[u*
(c + d*v)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] &&  !(IntegerQ[m] || IntegerQ[n
] || GtQ[b/d, 0])

Rule 213

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

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 40, normalized size = 1.38 \begin {gather*} \frac {\left (-1+x^2\right ) (\log (1-x)-\log (1+x))}{2 \sqrt {2} \sqrt {-\left (-1+x^2\right )^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.11, size = 24, normalized size = 0.83

method	result	size
default	\(\frac {\sqrt {-x^{2}+1}\, \sqrt {2}\, \arctanh \left (x \right )}{2 \sqrt {x^{2}-1}}\)	\(24\)
meijerg	\(\frac {\sqrt {2}\, \sqrt {-\mathrm {signum}\left (x^{2}-1\right )}\, \arctanh \left (x \right )}{2 \sqrt {\mathrm {signum}\left (x^{2}-1\right )}}\)	\(26\)
risch	\(\frac {\sqrt {\frac {-2 x^{2}+2}{x^{2}-1}}\, \sqrt {x^{2}-1}\, \left (\frac {\sqrt {-2}\, \ln \left (x +1\right )}{4}-\frac {\sqrt {-2}\, \ln \left (x -1\right )}{4}\right )}{\sqrt {-2 x^{2}+2}}\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.99, size = 34, normalized size = 1.17 \begin {gather*} \frac {1}{4} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {x^{2} - 1} \sqrt {-2 \, x^{2} + 2} x}{x^{4} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\sqrt {2} \int \frac {1}{\sqrt {1 - x^{2}} \sqrt {x^{2} - 1}}\, dx}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2)*Integral(1/(sqrt(1 - x**2)*sqrt(x**2 - 1)), x)/2

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {1}{\sqrt {x^2-1}\,\sqrt {2-2\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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