### 3.741 $$\int \frac{e^{n \coth ^{-1}(a x)}}{(c-a^2 c x^2)^2} \, dx$$

Optimal. Leaf size=72 $\frac{2 e^{n \coth ^{-1}(a x)}}{a c^2 n \left (4-n^2\right )}-\frac{(n-2 a x) e^{n \coth ^{-1}(a x)}}{a c^2 \left (4-n^2\right ) \left (1-a^2 x^2\right )}$

[Out]

(2*E^(n*ArcCoth[a*x]))/(a*c^2*n*(4 - n^2)) - (E^(n*ArcCoth[a*x])*(n - 2*a*x))/(a*c^2*(4 - n^2)*(1 - a^2*x^2))

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Rubi [A]  time = 0.0778262, antiderivative size = 72, 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} $\frac{2 e^{n \coth ^{-1}(a x)}}{a c^2 n \left (4-n^2\right )}-\frac{(n-2 a x) e^{n \coth ^{-1}(a x)}}{a c^2 \left (4-n^2\right ) \left (1-a^2 x^2\right )}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*E^(n*ArcCoth[a*x]))/(a*c^2*n*(4 - n^2)) - (E^(n*ArcCoth[a*x])*(n - 2*a*x))/(a*c^2*(4 - n^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^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx &=-\frac{e^{n \coth ^{-1}(a x)} (n-2 a x)}{a c^2 \left (4-n^2\right ) \left (1-a^2 x^2\right )}+\frac{2 \int \frac{e^{n \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx}{c \left (4-n^2\right )}\\ &=\frac{2 e^{n \coth ^{-1}(a x)}}{a c^2 n \left (4-n^2\right )}-\frac{e^{n \coth ^{-1}(a x)} (n-2 a x)}{a c^2 \left (4-n^2\right ) \left (1-a^2 x^2\right )}\\ \end{align*}

Mathematica [A]  time = 0.161797, size = 55, normalized size = 0.76 $-\frac{\left (2 a^2 x^2-2 a n x+n^2-2\right ) e^{n \coth ^{-1}(a x)}}{a c^2 n \left (n^2-4\right ) \left (a^2 x^2-1\right )}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-((E^(n*ArcCoth[a*x])*(-2 + n^2 - 2*a*n*x + 2*a^2*x^2))/(a*c^2*n*(-4 + n^2)*(-1 + a^2*x^2)))

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Maple [A]  time = 0.043, size = 55, normalized size = 0.8 \begin{align*} -{\frac{{{\rm e}^{n{\rm arccoth} \left (ax\right )}} \left ( 2\,{a}^{2}{x}^{2}-2\,nax+{n}^{2}-2 \right ) }{ \left ({a}^{2}{x}^{2}-1 \right ){c}^{2}an \left ({n}^{2}-4 \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*arccoth(a*x))/(-a^2*c*x^2+c)^2,x)

[Out]

-exp(n*arccoth(a*x))*(2*a^2*x^2-2*a*n*x+n^2-2)/(a^2*x^2-1)/c^2/a/n/(n^2-4)

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

[Out]

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

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Fricas [A]  time = 1.73817, size = 165, normalized size = 2.29 \begin{align*} -\frac{{\left (2 \, a^{2} x^{2} + 2 \, a n x + n^{2} - 2\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{a c^{2} n^{3} - 4 \, a c^{2} n -{\left (a^{3} c^{2} n^{3} - 4 \, a^{3} c^{2} n\right )} x^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*a^2*x^2 + 2*a*n*x + n^2 - 2)*((a*x - 1)/(a*x + 1))^(1/2*n)/(a*c^2*n^3 - 4*a*c^2*n - (a^3*c^2*n^3 - 4*a^3*c
^2*n)*x^2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*acoth(a*x))/(-a**2*c*x**2+c)**2,x)

[Out]

Timed out

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

[Out]

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