\(\int \frac {e^{n \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx\) [740]

Optimal result

Integrand size = 22, antiderivative size = 18 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=\frac {e^{n \coth ^{-1}(a x)}}{a c n} \]

[Out]

exp(n*arccoth(a*x))/a/c/n

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {6318} \[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=\frac {e^{n \coth ^{-1}(a x)}}{a c n} \]

[In]

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

[Out]

E^(n*ArcCoth[a*x])/(a*c*n)

Rule 6318

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*} \text {integral}& = \frac {e^{n \coth ^{-1}(a x)}}{a c n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=\frac {e^{n \coth ^{-1}(a x)}}{a c n} \]

[In]

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

[Out]

E^(n*ArcCoth[a*x])/(a*c*n)

Maple [A] (verified)

Time = 1.08 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

method	result	size
gosper	\(\frac {{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}}{a c n}\)	\(18\)
parallelrisch	\(\frac {{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}}{a c n}\)	\(18\)
risch	\(\frac {\left (a x -1\right )^{-\frac {n}{2}} \left (a x +1\right )^{\frac {n}{2}}}{c a n}\)	\(29\)

[In]

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

[Out]

exp(n*arccoth(a*x))/a/c/n

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.50 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=\frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a c n} \]

[In]

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

[Out]

((a*x + 1)/(a*x - 1))^(1/2*n)/(a*c*n)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.51 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.72 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=\begin {cases} \frac {x}{c} & \text {for}\: a = 0 \wedge n = 0 \\\frac {x e^{\frac {i \pi n}{2}}}{c} & \text {for}\: a = 0 \\- \frac {\log {\left (x - \frac {1}{a} \right )}}{2 a c} + \frac {\log {\left (x + \frac {1}{a} \right )}}{2 a c} & \text {for}\: n = 0 \\\frac {e^{n \operatorname {acoth}{\left (a x \right )}}}{a c n} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((x/c, Eq(a, 0) & Eq(n, 0)), (x*exp(I*pi*n/2)/c, Eq(a, 0)), (-log(x - 1/a)/(2*a*c) + log(x + 1/a)/(2*
a*c), Eq(n, 0)), (exp(n*acoth(a*x))/(a*c*n), True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.67 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=\frac {e^{\left (\frac {1}{2} \, n \log \left (a x + 1\right ) - \frac {1}{2} \, n \log \left (a x - 1\right )\right )}}{a c n} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 4.06 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.17 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=\frac {{\left (\frac {1}{a\,x}+1\right )}^{n/2}}{a\,c\,n\,{\left (1-\frac {1}{a\,x}\right )}^{n/2}} \]

[In]

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

[Out]

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