3.3.42 \(\int \frac {1}{\sqrt {-1+x^2} \sqrt {2+x^2}} \, dx\) [242]

Optimal. Leaf size=32 \[ \frac {\sqrt {1-x^2} F\left (\sin ^{-1}(x)|-\frac {1}{2}\right )}{\sqrt {2} \sqrt {-1+x^2}} \]

[Out]

1/2*EllipticF(x,1/2*I*2^(1/2))*(-x^2+1)^(1/2)*2^(1/2)/(x^2-1)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 32, 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 = {432, 430} \begin {gather*} \frac {\sqrt {1-x^2} F\left (\text {ArcSin}(x)\left |-\frac {1}{2}\right .\right )}{\sqrt {2} \sqrt {x^2-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]
))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && Gt
Q[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Rule 432

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*
x^2], Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + (d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] &&  !GtQ[c, 0]

Rubi steps

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

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Mathematica [A]
time = 0.19, size = 32, normalized size = 1.00 \begin {gather*} \frac {\sqrt {1-x^2} F\left (\sin ^{-1}(x)|-\frac {1}{2}\right )}{\sqrt {2} \sqrt {-1+x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.09, size = 34, normalized size = 1.06

method	result	size
default	\(-\frac {i \EllipticF \left (\frac {i x \sqrt {2}}{2}, i \sqrt {2}\right ) \sqrt {-x^{2}+1}}{\sqrt {x^{2}-1}}\)	\(34\)
elliptic	\(-\frac {i \sqrt {\left (x^{2}-1\right ) \left (x^{2}+2\right )}\, \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {-x^{2}+1}\, \EllipticF \left (\frac {i x \sqrt {2}}{2}, i \sqrt {2}\right )}{2 \sqrt {x^{2}-1}\, \sqrt {x^{2}+2}\, \sqrt {x^{4}+x^{2}-2}}\)	\(76\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*EllipticF(1/2*I*x*2^(1/2),I*2^(1/2))*(-x^2+1)^(1/2)/(x^2-1)^(1/2)

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

[Out]

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

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

[Out]

0

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

integrate(1/(sqrt(x^2 + 2)*sqrt(x^2 - 1)), 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 {x^2+2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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