### 3.595 $$\int \frac{e^{-\coth ^{-1}(a x)}}{(c-a^2 c x^2)^2} \, dx$$

Optimal. Leaf size=55 $\frac{(2 a x+1) e^{-\coth ^{-1}(a x)}}{3 a c^2 \left (1-a^2 x^2\right )}-\frac{2 e^{-\coth ^{-1}(a x)}}{3 a c^2}$

[Out]

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

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Rubi [A]  time = 0.0622484, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.091, Rules used = {6185, 6183} $\frac{(2 a x+1) e^{-\coth ^{-1}(a x)}}{3 a c^2 \left (1-a^2 x^2\right )}-\frac{2 e^{-\coth ^{-1}(a x)}}{3 a c^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-2/(3*a*c^2*E^ArcCoth[a*x]) + (1 + 2*a*x)/(3*a*c^2*E^ArcCoth[a*x]*(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 = 0.140154, size = 48, normalized size = 0.87 $-\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a^2 x^2+2 a x-1\right )}{3 (a x-1) (a c x+c)^2}$

Warning: Unable to verify antiderivative.

[In]

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

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Maple [A]  time = 0.046, size = 49, normalized size = 0.9 \begin{align*} -{\frac{2\,{a}^{2}{x}^{2}+2\,ax-1}{ \left ( 3\,{a}^{2}{x}^{2}-3 \right ) a{c}^{2}}\sqrt{{\frac{ax-1}{ax+1}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.994994, size = 90, normalized size = 1.64 \begin{align*} \frac{1}{12} \, a{\left (\frac{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - 6 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{2}} - \frac{3}{a^{2} c^{2} \sqrt{\frac{a x - 1}{a x + 1}}}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.64564, size = 105, normalized size = 1.91 \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} - a c^{2}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((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 - a*c^2)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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