### 3.369 $$\int \frac{e^{n \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx$$

Optimal. Leaf size=48 $-\frac{\left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-1} \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}}}{a c^2 (n+2)}$

[Out]

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

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Rubi [A]  time = 0.109206, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.167, Rules used = {6175, 6180, 37} $-\frac{\left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-1} \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}}}{a c^2 (n+2)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 6175

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> Dist[d^p, Int[u*x^p*(1 + c/(d*
x))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c^2 - d^2, 0] &&  !IntegerQ[n/2] && Inte
gerQ[p]

Rule 6180

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

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.167723, size = 33, normalized size = 0.69 $-\frac{(a x+1) e^{n \coth ^{-1}(a x)}}{a c^2 (n+2) (a x-1)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.041, size = 33, normalized size = 0.7 \begin{align*} -{\frac{{{\rm e}^{n{\rm arccoth} \left (ax\right )}} \left ( ax+1 \right ) }{ \left ( ax-1 \right ){c}^{2} \left ( 2+n \right ) a}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 c x - 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*c*x+c)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.67622, size = 122, normalized size = 2.54 \begin{align*} -\frac{{\left (a x + 1\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}}{a c^{2} n - 2 \, a c^{2} -{\left (a^{2} c^{2} n - 2 \, a^{2} c^{2}\right )} x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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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 c x - 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*c*x+c)^2,x, algorithm="giac")

[Out]

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