3.1206 \(\int \frac{e^{-\tanh ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.178884, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6153, 6150, 43} \[ -\frac{\sqrt{c-a^2 c x^2}}{x \sqrt{1-a^2 x^2}}-\frac{a \log (x) \sqrt{c-a^2 c x^2}}{\sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6153

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

Rule 6150

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0230615, size = 41, normalized size = 0.59 \[ \frac{\sqrt{c-a^2 c x^2} \left (-a \log (x)-\frac{1}{x}\right )}{\sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.088, size = 49, normalized size = 0.7 \begin{align*}{\frac{a\ln \left ( x \right ) x+1}{x \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) }\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^2,x)

[Out]

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

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Maxima [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}}{{\left (a x + 1\right )} x^{2}}\,{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^2,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.46878, size = 556, normalized size = 8.06 \begin{align*} \left [\frac{{\left (a^{3} x^{3} - a x\right )} \sqrt{c} \log \left (\frac{a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} + \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}{\left (x^{4} - 1\right )} \sqrt{c} - c}{a^{2} x^{4} - x^{2}}\right ) - 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}{\left (x - 1\right )}}{2 \,{\left (a^{2} x^{3} - x\right )}}, -\frac{{\left (a^{3} x^{3} - a x\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}{\left (x^{2} + 1\right )} \sqrt{-c}}{a^{2} c x^{4} -{\left (a^{2} + 1\right )} c x^{2} + c}\right ) + \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}{\left (x - 1\right )}}{a^{2} x^{3} - x}\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^2,x, algorithm="fricas")

[Out]

[1/2*((a^3*x^3 - a*x)*sqrt(c)*log((a^2*c*x^6 + a^2*c*x^2 - c*x^4 + sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*(x^
4 - 1)*sqrt(c) - c)/(a^2*x^4 - x^2)) - 2*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*(x - 1))/(a^2*x^3 - x), -((a^
3*x^3 - a*x)*sqrt(-c)*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*(x^2 + 1)*sqrt(-c)/(a^2*c*x^4 - (a^2 + 1)
*c*x^2 + c)) + sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*(x - 1))/(a^2*x^3 - x)]

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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 )}}{x^{2} \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**2*x**2+1)**(1/2)/x**2,x)

[Out]

Integral(sqrt(-(a*x - 1)*(a*x + 1))*sqrt(-c*(a*x - 1)*(a*x + 1))/(x**2*(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}}{{\left (a x + 1\right )} x^{2}}\,{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^2,x, algorithm="giac")

[Out]

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