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

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

[Out]

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

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Rubi [A]  time = 0.117439, 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 = {6192, 6193} \[ \frac{x \sqrt{c-a^2 c x^2}}{2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\sqrt{c-a^2 c x^2}}{a \sqrt{1-\frac{1}{a^2 x^2}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6192

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

Rule 6193

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

Rubi steps

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

Mathematica [A]  time = 0.0195669, size = 41, normalized size = 0.59 \[ \frac{(a x-2) \sqrt{c-a^2 c x^2}}{2 a \sqrt{1-\frac{1}{a^2 x^2}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.041, size = 44, normalized size = 0.6 \begin{align*}{\frac{x \left ( ax-2 \right ) }{2\,ax-2}\sqrt{-{a}^{2}c{x}^{2}+c}\sqrt{{\frac{ax-1}{ax+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*x+1))^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{-a^{2} c x^{2} + c} \sqrt{\frac{a x - 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*x+1))^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.79792, size = 46, normalized size = 0.67 \begin{align*} \frac{\sqrt{-a^{2} c}{\left (a x^{2} - 2 \, x\right )}}{2 \, a} \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*x+1))^(1/2),x, algorithm="fricas")

[Out]

1/2*sqrt(-a^2*c)*(a*x^2 - 2*x)/a

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{- c \left (a x - 1\right ) \left (a x + 1\right )}\, 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*x+1))**(1/2),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{-a^{2} c x^{2} + c} \sqrt{\frac{a x - 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*x+1))^(1/2),x, algorithm="giac")

[Out]

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