\(\int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx\) [369]

Optimal result

Integrand size = 18, antiderivative size = 48 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx=-\frac {\left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}}}{a c^2 (2+n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6310, 6315, 37} \[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx=-\frac {\left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}}{a c^2 (n+2)} \]

[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 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]

Rule 6310

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 6315

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]

Rubi steps \begin{align*} \text {integral}& = \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 {\text {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] (verified)

Time = 0.29 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.69 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx=-\frac {e^{n \coth ^{-1}(a x)} (1+a x)}{a c^2 (2+n) (-1+a x)} \]

[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)))

Maple [A] (verified)

Time = 1.76 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.69

method	result	size
gosper	\(-\frac {{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} \left (a x +1\right )}{\left (a x -1\right ) c^{2} \left (2+n \right ) a}\)	\(33\)
parallelrisch	\(\frac {-x \,{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} a -{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}}{c^{2} \left (a x -1\right ) \left (2+n \right ) a}\)	\(41\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.21 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx=\frac {{\left (a x + 1\right )} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a c^{2} n + 2 \, a c^{2} - {\left (a^{2} c^{2} n + 2 \, a^{2} c^{2}\right )} x} \]

[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)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 10.17 (sec) , antiderivative size = 187, normalized size of antiderivative = 3.90 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx=\begin {cases} - \frac {x}{c^{2}} & \text {for}\: a = 0 \wedge n = -2 \\\frac {x e^{\frac {i \pi n}{2}}}{c^{2}} & \text {for}\: a = 0 \\- \frac {a x \operatorname {acoth}{\left (a x \right )}}{a^{2} c^{2} x e^{2 \operatorname {acoth}{\left (a x \right )}} - a c^{2} e^{2 \operatorname {acoth}{\left (a x \right )}}} - \frac {\operatorname {acoth}{\left (a x \right )}}{a^{2} c^{2} x e^{2 \operatorname {acoth}{\left (a x \right )}} - a c^{2} e^{2 \operatorname {acoth}{\left (a x \right )}}} & \text {for}\: n = -2 \\- \frac {a x e^{n \operatorname {acoth}{\left (a x \right )}}}{a^{2} c^{2} n x + 2 a^{2} c^{2} x - a c^{2} n - 2 a c^{2}} - \frac {e^{n \operatorname {acoth}{\left (a x \right )}}}{a^{2} c^{2} n x + 2 a^{2} c^{2} x - a c^{2} n - 2 a c^{2}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-x/c**2, Eq(a, 0) & Eq(n, -2)), (x*exp(I*pi*n/2)/c**2, Eq(a, 0)), (-a*x*acoth(a*x)/(a**2*c**2*x*exp
(2*acoth(a*x)) - a*c**2*exp(2*acoth(a*x))) - acoth(a*x)/(a**2*c**2*x*exp(2*acoth(a*x)) - a*c**2*exp(2*acoth(a*
x))), Eq(n, -2)), (-a*x*exp(n*acoth(a*x))/(a**2*c**2*n*x + 2*a**2*c**2*x - a*c**2*n - 2*a*c**2) - exp(n*acoth(
a*x))/(a**2*c**2*n*x + 2*a**2*c**2*x - a*c**2*n - 2*a*c**2), True))

Maxima [F]

\[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (a c x - c\right )}^{2}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (a c x - c\right )}^{2}} \,d x } \]

[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)

Mupad [B] (verification not implemented)

Time = 4.92 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.67 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx=-\frac {{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}\,\left (a\,x+1\right )}{a\,c^2\,\left (a\,x-1\right )\,\left (n+2\right )} \]

[In]

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

[Out]

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