### 3.561 $$\int \frac{e^{\coth ^{-1}(a x)}}{(c-a^2 c x^2)^2} \, dx$$

Optimal. Leaf size=51 $\frac{2 e^{\coth ^{-1}(a x)}}{3 a c^2}-\frac{(1-2 a x) e^{\coth ^{-1}(a x)}}{3 a c^2 \left (1-a^2 x^2\right )}$

[Out]

(2*E^ArcCoth[a*x])/(3*a*c^2) - (E^ArcCoth[a*x]*(1 - 2*a*x))/(3*a*c^2*(1 - a^2*x^2))

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Rubi [A]  time = 0.0604559, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.1, Rules used = {6185, 6183} $\frac{2 e^{\coth ^{-1}(a x)}}{3 a c^2}-\frac{(1-2 a x) e^{\coth ^{-1}(a x)}}{3 a c^2 \left (1-a^2 x^2\right )}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*E^ArcCoth[a*x])/(3*a*c^2) - (E^ArcCoth[a*x]*(1 - 2*a*x))/(3*a*c^2*(1 - a^2*x^2))

Rule 6185

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((n + 2*a*(p + 1)*x)*(c + d*x^
2)^(p + 1)*E^(n*ArcCoth[a*x]))/(a*c*(n^2 - 4*(p + 1)^2)), x] - Dist[(2*(p + 1)*(2*p + 3))/(c*(n^2 - 4*(p + 1)^
2)), Int[(c + d*x^2)^(p + 1)*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] &&  !In
tegerQ[n/2] && LtQ[p, -1] && NeQ[p, -3/2] && NeQ[n^2 - 4*(p + 1)^2, 0] && (IntegerQ[p] ||  !IntegerQ[n])

Rule 6183

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[E^(n*ArcCoth[a*x])/(a*c*n), x] /; F
reeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] &&  !IntegerQ[n/2]

Rubi steps

\begin{align*} \int \frac{e^{\coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx &=-\frac{e^{\coth ^{-1}(a x)} (1-2 a x)}{3 a c^2 \left (1-a^2 x^2\right )}+\frac{2 \int \frac{e^{\coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx}{3 c}\\ &=\frac{2 e^{\coth ^{-1}(a x)}}{3 a c^2}-\frac{e^{\coth ^{-1}(a x)} (1-2 a x)}{3 a c^2 \left (1-a^2 x^2\right )}\\ \end{align*}

Mathematica [A]  time = 0.143481, size = 50, normalized size = 0.98 $\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a^2 x^2-2 a x-1\right )}{3 c^2 (a x-1)^2 (a x+1)}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[1 - 1/(a^2*x^2)]*x*(-1 - 2*a*x + 2*a^2*x^2))/(3*c^2*(-1 + a*x)^2*(1 + a*x))

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Maple [A]  time = 0.045, size = 49, normalized size = 1. \begin{align*}{\frac{2\,{a}^{2}{x}^{2}-2\,ax-1}{ \left ( 3\,{a}^{2}{x}^{2}-3 \right ){c}^{2}a}{\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.06119, size = 88, normalized size = 1.73 \begin{align*} \frac{1}{12} \, a{\left (\frac{3 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{2}} + \frac{\frac{6 \,{\left (a x - 1\right )}}{a x + 1} - 1}{a^{2} c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*a*(3*sqrt((a*x - 1)/(a*x + 1))/(a^2*c^2) + (6*(a*x - 1)/(a*x + 1) - 1)/(a^2*c^2*((a*x - 1)/(a*x + 1))^(3/
2)))

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Fricas [A]  time = 1.57265, size = 123, normalized size = 2.41 \begin{align*} \frac{{\left (2 \, a^{2} x^{2} - 2 \, a x - 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{3 \,{\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{a^{4} x^{4} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}} - 2 a^{2} x^{2} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}} + \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}\, dx}{c^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(a**4*x**4*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)) - 2*a**2*x**2*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)) + sqrt
(a*x/(a*x + 1) - 1/(a*x + 1))), x)/c**2

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Giac [A]  time = 1.10668, size = 104, normalized size = 2.04 \begin{align*} \frac{1}{12} \, a{\left (\frac{{\left (a x + 1\right )}{\left (\frac{6 \,{\left (a x - 1\right )}}{a x + 1} - 1\right )}}{{\left (a x - 1\right )} a^{2} c^{2} \sqrt{\frac{a x - 1}{a x + 1}}} + \frac{3 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{2}}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*a*((a*x + 1)*(6*(a*x - 1)/(a*x + 1) - 1)/((a*x - 1)*a^2*c^2*sqrt((a*x - 1)/(a*x + 1))) + 3*sqrt((a*x - 1)
/(a*x + 1))/(a^2*c^2))