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

Optimal. Leaf size=47 \[ \frac{\sqrt{x^2+2} \text{EllipticF}\left (\tan ^{-1}(x),\frac{1}{2}\right )}{\sqrt{2} \sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}}} \]

[Out]

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

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Rubi [A]  time = 0.00898, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {418} \[ \frac{\sqrt{x^2+2} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{2} \sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 418

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

Rubi steps

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

Mathematica [C]  time = 0.0182237, size = 19, normalized size = 0.4 \[ -\frac{i \text{EllipticF}\left (i \sinh ^{-1}(x),\frac{1}{2}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*EllipticF[I*ArcSinh[x], 1/2])/Sqrt[2]

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Maple [C]  time = 0.022, size = 15, normalized size = 0.3 \begin{align*} -i{\it EllipticF} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ) \end{align*}

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]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x^{2} + 2} \sqrt{x^{2} + 1}}\,{d x} \end{align*}

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., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{x^{2} + 2} \sqrt{x^{2} + 1}}{x^{4} + 3 \, x^{2} + 2}, x\right ) \end{align*}

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]

integral(sqrt(x^2 + 2)*sqrt(x^2 + 1)/(x^4 + 3*x^2 + 2), x)

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

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**2 + 1)*sqrt(x**2 + 2)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x^{2} + 2} \sqrt{x^{2} + 1}}\,{d x} \end{align*}

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)