### 3.711 $$\int \frac{e^{-\coth ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x^2} \, dx$$

Optimal. Leaf size=72 $\frac{\sqrt{c-a^2 c x^2}}{a x^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\log (x) \sqrt{c-a^2 c x^2}}{x \sqrt{1-\frac{1}{a^2 x^2}}}$

[Out]

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

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Rubi [A]  time = 0.215674, antiderivative size = 72, 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 = {6192, 6193, 43} $\frac{\sqrt{c-a^2 c x^2}}{a x^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\log (x) \sqrt{c-a^2 c x^2}}{x \sqrt{1-\frac{1}{a^2 x^2}}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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]

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

Mathematica [A]  time = 0.0224395, size = 44, normalized size = 0.61 $\frac{\sqrt{c-a^2 c x^2} \left (a \log (x)+\frac{1}{x}\right )}{a x \sqrt{1-\frac{1}{a^2 x^2}}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.133, size = 48, normalized size = 0.7 \begin{align*}{\frac{a\ln \left ( x \right ) x+1}{ \left ( ax-1 \right ) x}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) }\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^2,x)

[Out]

(a*ln(x)*x+1)*(-c*(a^2*x^2-1))^(1/2)*((a*x-1)/(a*x+1))^(1/2)/(a*x-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{\frac{a x - 1}{a x + 1}}}{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*x+1))^(1/2)/x^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.62168, size = 50, normalized size = 0.69 \begin{align*} \frac{\sqrt{-a^{2} c}{\left (a x \log \left (x\right ) + 1\right )}}{a 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^2,x, algorithm="fricas")

[Out]

sqrt(-a^2*c)*(a*x*log(x) + 1)/(a*x)

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

[Out]

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

[Out]

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