3.1204 \(\int e^{-\tanh ^{-1}(a x)} \sqrt{c-a^2 c x^2} \, dx\)

Optimal. Leaf size=69 \[ \frac{x \sqrt{c-a^2 c x^2}}{\sqrt{1-a^2 x^2}}-\frac{a x^2 \sqrt{c-a^2 c x^2}}{2 \sqrt{1-a^2 x^2}} \]

[Out]

(x*Sqrt[c - a^2*c*x^2])/Sqrt[1 - a^2*x^2] - (a*x^2*Sqrt[c - a^2*c*x^2])/(2*Sqrt[1 - a^2*x^2])

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Rubi [A]  time = 0.0713259, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {6143, 6140} \[ \frac{x \sqrt{c-a^2 c x^2}}{\sqrt{1-a^2 x^2}}-\frac{a x^2 \sqrt{c-a^2 c x^2}}{2 \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c - a^2*c*x^2]/E^ArcTanh[a*x],x]

[Out]

(x*Sqrt[c - a^2*c*x^2])/Sqrt[1 - a^2*x^2] - (a*x^2*Sqrt[c - a^2*c*x^2])/(2*Sqrt[1 - a^2*x^2])

Rule 6143

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(c^IntPart[p]*(c + d*x^2)^Frac
Part[p])/(1 - a^2*x^2)^FracPart[p], Int[(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x
] && EqQ[a^2*c + d, 0] &&  !(IntegerQ[p] || GtQ[c, 0])

Rule 6140

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[(1 - a*x)^(p - n/2)*
(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])

Rubi steps

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

Mathematica [A]  time = 0.0155949, size = 40, normalized size = 0.58 \[ \frac{\left (x-\frac{a x^2}{2}\right ) \sqrt{c-a^2 c x^2}}{\sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c - a^2*c*x^2]/E^ArcTanh[a*x],x]

[Out]

((x - (a*x^2)/2)*Sqrt[c - a^2*c*x^2])/Sqrt[1 - a^2*x^2]

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Maple [A]  time = 0.027, size = 48, normalized size = 0.7 \begin{align*}{\frac{x \left ( ax-2 \right ) }{ \left ( 2\,ax-2 \right ) \left ( ax+1 \right ) }\sqrt{-{a}^{2}c{x}^{2}+c}\sqrt{-{a}^{2}{x}^{2}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.00756, size = 65, normalized size = 0.94 \begin{align*} -\frac{{\left (a^{2} \sqrt{c} x^{2} - 2 \, a \sqrt{c} x + 2 \, \sqrt{c}\right )}{\left (a x + 1\right )}{\left (a x - 1\right )}}{2 \,{\left (a^{3} x^{2} - a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(a^2*sqrt(c)*x^2 - 2*a*sqrt(c)*x + 2*sqrt(c))*(a*x + 1)*(a*x - 1)/(a^3*x^2 - a)

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Fricas [A]  time = 2.42799, size = 99, normalized size = 1.43 \begin{align*} \frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}{\left (a x^{2} - 2 \, x\right )}}{2 \,{\left (a^{2} x^{2} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*(a*x^2 - 2*x)/(a^2*x^2 - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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