3.207 \(\int \frac{\sqrt{a-b x^2}}{\sqrt{a^2-b^2 x^4}} \, dx\)

Optimal. Leaf size=64 \[ \frac{\sqrt{a-b x^2} \sqrt{a+b x^2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{\sqrt{b} \sqrt{a^2-b^2 x^4}} \]

[Out]

(Sqrt[a - b*x^2]*Sqrt[a + b*x^2]*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*Sqrt[a^2 - b^2*x^4])

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Rubi [A]  time = 0.0254866, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {1152, 217, 206} \[ \frac{\sqrt{a-b x^2} \sqrt{a+b x^2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{\sqrt{b} \sqrt{a^2-b^2 x^4}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a - b*x^2]/Sqrt[a^2 - b^2*x^4],x]

[Out]

(Sqrt[a - b*x^2]*Sqrt[a + b*x^2]*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*Sqrt[a^2 - b^2*x^4])

Rule 1152

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[(a + c*x^4)^FracPart[p]/((d + e*x
^2)^FracPart[p]*(a/d + (c*x^2)/e)^FracPart[p]), Int[(d + e*x^2)^(p + q)*(a/d + (c*x^2)/e)^p, x], x] /; FreeQ[{
a, c, d, e, p, q}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p]

Rule 217

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

Rule 206

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

Rubi steps

\begin{align*} \int \frac{\sqrt{a-b x^2}}{\sqrt{a^2-b^2 x^4}} \, dx &=\frac{\left (\sqrt{a-b x^2} \sqrt{a+b x^2}\right ) \int \frac{1}{\sqrt{a+b x^2}} \, dx}{\sqrt{a^2-b^2 x^4}}\\ &=\frac{\left (\sqrt{a-b x^2} \sqrt{a+b x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{\sqrt{a^2-b^2 x^4}}\\ &=\frac{\sqrt{a-b x^2} \sqrt{a+b x^2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{\sqrt{b} \sqrt{a^2-b^2 x^4}}\\ \end{align*}

Mathematica [A]  time = 0.0411674, size = 67, normalized size = 1.05 \[ \frac{\log \left (\sqrt{b} \sqrt{a-b x^2} \sqrt{a^2-b^2 x^4}+a b x-b^2 x^3\right )-\log \left (b x^2-a\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a - b*x^2]/Sqrt[a^2 - b^2*x^4],x]

[Out]

(-Log[-a + b*x^2] + Log[a*b*x - b^2*x^3 + Sqrt[b]*Sqrt[a - b*x^2]*Sqrt[a^2 - b^2*x^4]])/Sqrt[b]

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Maple [A]  time = 0.014, size = 54, normalized size = 0.8 \begin{align*}{\sqrt{-{b}^{2}{x}^{4}+{a}^{2}}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){\frac{1}{\sqrt{-b{x}^{2}+a}}}{\frac{1}{\sqrt{b{x}^{2}+a}}}{\frac{1}{\sqrt{b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-b*x^2+a)^(1/2)/(-b^2*x^4+a^2)^(1/2),x)

[Out]

1/(-b*x^2+a)^(1/2)/(b*x^2+a)^(1/2)/b^(1/2)*(-b^2*x^4+a^2)^(1/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-b x^{2} + a}}{\sqrt{-b^{2} x^{4} + a^{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^2+a)^(1/2)/(-b^2*x^4+a^2)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(-b*x^2 + a)/sqrt(-b^2*x^4 + a^2), x)

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Fricas [A]  time = 1.94114, size = 269, normalized size = 4.2 \begin{align*} \left [\frac{\log \left (\frac{2 \, b^{2} x^{4} - a b x^{2} - 2 \, \sqrt{-b^{2} x^{4} + a^{2}} \sqrt{-b x^{2} + a} \sqrt{b} x - a^{2}}{b x^{2} - a}\right )}{2 \, \sqrt{b}}, \frac{\sqrt{-b} \arctan \left (\frac{\sqrt{-b^{2} x^{4} + a^{2}} \sqrt{-b x^{2} + a} \sqrt{-b}}{b^{2} x^{3} - a b x}\right )}{b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^2+a)^(1/2)/(-b^2*x^4+a^2)^(1/2),x, algorithm="fricas")

[Out]

[1/2*log((2*b^2*x^4 - a*b*x^2 - 2*sqrt(-b^2*x^4 + a^2)*sqrt(-b*x^2 + a)*sqrt(b)*x - a^2)/(b*x^2 - a))/sqrt(b),
sqrt(-b)*arctan(sqrt(-b^2*x^4 + a^2)*sqrt(-b*x^2 + a)*sqrt(-b)/(b^2*x^3 - a*b*x))/b]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x**2+a)**(1/2)/(-b**2*x**4+a**2)**(1/2),x)

[Out]

Integral(sqrt(a - b*x**2)/sqrt(-(-a + b*x**2)*(a + b*x**2)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-b x^{2} + a}}{\sqrt{-b^{2} x^{4} + a^{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^2+a)^(1/2)/(-b^2*x^4+a^2)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(-b*x^2 + a)/sqrt(-b^2*x^4 + a^2), x)