\(\int \frac {e^{n \coth ^{-1}(a x)}}{(c-a^2 c x^2)^2} \, dx\) [741]

Optimal result

Integrand size = 22, antiderivative size = 72 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {2 e^{n \coth ^{-1}(a x)}}{a c^2 n \left (4-n^2\right )}-\frac {e^{n \coth ^{-1}(a x)} (n-2 a x)}{a c^2 \left (4-n^2\right ) \left (1-a^2 x^2\right )} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6320, 6318} \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {2 e^{n \coth ^{-1}(a x)}}{a c^2 n \left (4-n^2\right )}-\frac {(n-2 a x) e^{n \coth ^{-1}(a x)}}{a c^2 \left (4-n^2\right ) \left (1-a^2 x^2\right )} \]

[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 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]

Rule 6320

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])

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

Time = 0.32 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.76 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=-\frac {e^{n \coth ^{-1}(a x)} \left (-2+n^2-2 a n x+2 a^2 x^2\right )}{a c^2 n \left (-4+n^2\right ) \left (-1+a^2 x^2\right )} \]

[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)))

Maple [A] (verified)

Time = 4.00 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.76

method	result	size
gosper	\(-\frac {{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} \left (2 a^{2} x^{2}-2 a n x +n^{2}-2\right )}{\left (a^{2} x^{2}-1\right ) c^{2} a n \left (n^{2}-4\right )}\)	\(55\)
risch	\(-\frac {\left (2 a^{2} x^{2}-2 a n x +n^{2}-2\right ) \left (a x -1\right )^{-\frac {n}{2}} \left (a x +1\right )^{\frac {n}{2}}}{\left (a^{2} x^{2}-1\right ) c^{2} a n \left (n^{2}-4\right )}\)	\(66\)
parallelrisch	\(\frac {-2 x^{2} {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} a^{2}+2 x \,{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} a n -{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} n^{2}+2 \,{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}}{c^{2} \left (a^{2} x^{2}-1\right ) a n \left (n^{2}-4\right )}\)	\(78\)

[In]

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

[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)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.10 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {{\left (2 \, a^{2} x^{2} - 2 \, a n x + n^{2} - 2\right )} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a c^{2} n^{3} - 4 \, a c^{2} n - {\left (a^{3} c^{2} n^{3} - 4 \, a^{3} c^{2} n\right )} x^{2}} \]

[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)

Sympy [F]

\[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=\begin {cases} \frac {x e^{\frac {i \pi n}{2}}}{c^{2}} & \text {for}\: a = 0 \\- \frac {a^{2} x^{2} \operatorname {acoth}{\left (a x \right )}}{4 a^{3} c^{2} x^{2} e^{2 \operatorname {acoth}{\left (a x \right )}} - 4 a c^{2} e^{2 \operatorname {acoth}{\left (a x \right )}}} - \frac {2 a x \operatorname {acoth}{\left (a x \right )}}{4 a^{3} c^{2} x^{2} e^{2 \operatorname {acoth}{\left (a x \right )}} - 4 a c^{2} e^{2 \operatorname {acoth}{\left (a x \right )}}} + \frac {a x}{4 a^{3} c^{2} x^{2} e^{2 \operatorname {acoth}{\left (a x \right )}} - 4 a c^{2} e^{2 \operatorname {acoth}{\left (a x \right )}}} - \frac {\operatorname {acoth}{\left (a x \right )}}{4 a^{3} c^{2} x^{2} e^{2 \operatorname {acoth}{\left (a x \right )}} - 4 a c^{2} e^{2 \operatorname {acoth}{\left (a x \right )}}} + \frac {2}{4 a^{3} c^{2} x^{2} e^{2 \operatorname {acoth}{\left (a x \right )}} - 4 a c^{2} e^{2 \operatorname {acoth}{\left (a x \right )}}} & \text {for}\: n = -2 \\- \frac {a^{2} x^{2} \log {\left (x - \frac {1}{a} \right )}}{4 a^{3} c^{2} x^{2} - 4 a c^{2}} + \frac {a^{2} x^{2} \log {\left (x + \frac {1}{a} \right )}}{4 a^{3} c^{2} x^{2} - 4 a c^{2}} - \frac {2 a x}{4 a^{3} c^{2} x^{2} - 4 a c^{2}} + \frac {\log {\left (x - \frac {1}{a} \right )}}{4 a^{3} c^{2} x^{2} - 4 a c^{2}} - \frac {\log {\left (x + \frac {1}{a} \right )}}{4 a^{3} c^{2} x^{2} - 4 a c^{2}} & \text {for}\: n = 0 \\\frac {\int \frac {e^{2 \operatorname {acoth}{\left (a x \right )}}}{a^{4} x^{4} - 2 a^{2} x^{2} + 1}\, dx}{c^{2}} & \text {for}\: n = 2 \\- \frac {2 a^{2} x^{2} e^{n \operatorname {acoth}{\left (a x \right )}}}{a^{3} c^{2} n^{3} x^{2} - 4 a^{3} c^{2} n x^{2} - a c^{2} n^{3} + 4 a c^{2} n} + \frac {2 a n x e^{n \operatorname {acoth}{\left (a x \right )}}}{a^{3} c^{2} n^{3} x^{2} - 4 a^{3} c^{2} n x^{2} - a c^{2} n^{3} + 4 a c^{2} n} - \frac {n^{2} e^{n \operatorname {acoth}{\left (a x \right )}}}{a^{3} c^{2} n^{3} x^{2} - 4 a^{3} c^{2} n x^{2} - a c^{2} n^{3} + 4 a c^{2} n} + \frac {2 e^{n \operatorname {acoth}{\left (a x \right )}}}{a^{3} c^{2} n^{3} x^{2} - 4 a^{3} c^{2} n x^{2} - a c^{2} n^{3} + 4 a c^{2} n} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((x*exp(I*pi*n/2)/c**2, Eq(a, 0)), (-a**2*x**2*acoth(a*x)/(4*a**3*c**2*x**2*exp(2*acoth(a*x)) - 4*a*c
**2*exp(2*acoth(a*x))) - 2*a*x*acoth(a*x)/(4*a**3*c**2*x**2*exp(2*acoth(a*x)) - 4*a*c**2*exp(2*acoth(a*x))) +
a*x/(4*a**3*c**2*x**2*exp(2*acoth(a*x)) - 4*a*c**2*exp(2*acoth(a*x))) - acoth(a*x)/(4*a**3*c**2*x**2*exp(2*aco
th(a*x)) - 4*a*c**2*exp(2*acoth(a*x))) + 2/(4*a**3*c**2*x**2*exp(2*acoth(a*x)) - 4*a*c**2*exp(2*acoth(a*x))),
Eq(n, -2)), (-a**2*x**2*log(x - 1/a)/(4*a**3*c**2*x**2 - 4*a*c**2) + a**2*x**2*log(x + 1/a)/(4*a**3*c**2*x**2
- 4*a*c**2) - 2*a*x/(4*a**3*c**2*x**2 - 4*a*c**2) + log(x - 1/a)/(4*a**3*c**2*x**2 - 4*a*c**2) - log(x + 1/a)/
(4*a**3*c**2*x**2 - 4*a*c**2), Eq(n, 0)), (Integral(exp(2*acoth(a*x))/(a**4*x**4 - 2*a**2*x**2 + 1), x)/c**2,
Eq(n, 2)), (-2*a**2*x**2*exp(n*acoth(a*x))/(a**3*c**2*n**3*x**2 - 4*a**3*c**2*n*x**2 - a*c**2*n**3 + 4*a*c**2*
n) + 2*a*n*x*exp(n*acoth(a*x))/(a**3*c**2*n**3*x**2 - 4*a**3*c**2*n*x**2 - a*c**2*n**3 + 4*a*c**2*n) - n**2*ex
p(n*acoth(a*x))/(a**3*c**2*n**3*x**2 - 4*a**3*c**2*n*x**2 - a*c**2*n**3 + 4*a*c**2*n) + 2*exp(n*acoth(a*x))/(a
**3*c**2*n**3*x**2 - 4*a**3*c**2*n*x**2 - a*c**2*n**3 + 4*a*c**2*n), True))

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [B] (verification not implemented)

Time = 4.22 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.47 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {{\left (\frac {a\,x+1}{a\,x}\right )}^{n/2}\,\left (\frac {2\,x^2}{a\,c^2\,n\,\left (n^2-4\right )}-\frac {2\,x}{a^2\,c^2\,\left (n^2-4\right )}+\frac {n^2-2}{a^3\,c^2\,n\,\left (n^2-4\right )}\right )}{\left (\frac {1}{a^2}-x^2\right )\,{\left (\frac {a\,x-1}{a\,x}\right )}^{n/2}} \]

[In]

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

[Out]

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