\(\int \frac {e^{n \coth ^{-1}(a x)}}{(c-a^2 c x^2)^{3/2}} \, dx\) [747]

Optimal result

Integrand size = 24, antiderivative size = 46 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=-\frac {e^{n \coth ^{-1}(a x)} (n-a x)}{a c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {6319} \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=-\frac {(n-a x) e^{n \coth ^{-1}(a x)}}{a c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}} \]

[In]

Int[E^(n*ArcCoth[a*x])/(c - a^2*c*x^2)^(3/2),x]

[Out]

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

Rule 6319

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(n - a*x)*(E^(n*ArcCoth[a*x])/
(a*c*(n^2 - 1)*Sqrt[c + d*x^2])), x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] &&  !IntegerQ[n]

Rubi steps \begin{align*} \text {integral}& = -\frac {e^{n \coth ^{-1}(a x)} (n-a x)}{a c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.93 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {e^{n \coth ^{-1}(a x)} (n-a x)}{a c \left (-1+n^2\right ) \sqrt {c-a^2 c x^2}} \]

[In]

Integrate[E^(n*ArcCoth[a*x])/(c - a^2*c*x^2)^(3/2),x]

[Out]

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

Maple [A] (verified)

Time = 0.53 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.07

method	result	size
gosper	\(\frac {\left (a x -1\right ) \left (a x +1\right ) \left (a x -n \right ) {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}}{\left (n^{2}-1\right ) a \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}\)	\(49\)

[In]

int(exp(n*arccoth(a*x))/(-a^2*c*x^2+c)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.74 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=-\frac {\sqrt {-a^{2} c x^{2} + c} {\left (a x - n\right )} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a c^{2} n^{2} - a c^{2} - {\left (a^{3} c^{2} n^{2} - a^{3} c^{2}\right )} x^{2}} \]

[In]

integrate(exp(n*arccoth(a*x))/(-a^2*c*x^2+c)^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int \frac {e^{n \operatorname {acoth}{\left (a x \right )}}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(exp(n*acoth(a*x))/(-a**2*c*x**2+c)**(3/2),x)

[Out]

Integral(exp(n*acoth(a*x))/(-c*(a*x - 1)*(a*x + 1))**(3/2), x)

Maxima [F]

\[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(exp(n*arccoth(a*x))/(-a^2*c*x^2+c)^(3/2),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(exp(n*arccoth(a*x))/(-a^2*c*x^2+c)^(3/2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 4.08 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.70 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=-\frac {\left (\frac {x}{c\,\left (n^2-1\right )}-\frac {n}{a\,c\,\left (n^2-1\right )}\right )\,{\left (\frac {a\,x+1}{a\,x}\right )}^{n/2}}{\sqrt {c-a^2\,c\,x^2}\,{\left (\frac {a\,x-1}{a\,x}\right )}^{n/2}} \]

[In]

int(exp(n*acoth(a*x))/(c - a^2*c*x^2)^(3/2),x)

[Out]

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