\(\int \frac {e^{\coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx\) [560]

Optimal result

Integrand size = 20, antiderivative size = 13 \[ \int \frac {e^{\coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=\frac {e^{\coth ^{-1}(a x)}}{a c} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {6318} \[ \int \frac {e^{\coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=\frac {e^{\coth ^{-1}(a x)}}{a c} \]

[In]

Int[E^ArcCoth[a*x]/(c - a^2*c*x^2),x]

[Out]

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

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^{\coth ^{-1}(a x)}}{a c} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=\frac {e^{\coth ^{-1}(a x)}}{a c} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.50 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.77

method	result	size
gosper	\(\frac {1}{\sqrt {\frac {a x -1}{a x +1}}\, a c}\)	\(23\)
default	\(\frac {1}{\sqrt {\frac {a x -1}{a x +1}}\, a c}\)	\(23\)
trager	\(\frac {\left (a x +1\right ) \sqrt {-\frac {-a x +1}{a x +1}}}{a c \left (a x -1\right )}\)	\(37\)

[In]

int(1/((a*x-1)/(a*x+1))^(1/2)/(-a^2*c*x^2+c),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.62 \[ \int \frac {e^{\coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=\frac {{\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c x - a c} \]

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/2)/(-a^2*c*x^2+c),x, algorithm="fricas")

[Out]

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

Sympy [F]

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

[In]

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

[Out]

-Integral(1/(a**2*x**2*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)) - sqrt(a*x/(a*x + 1) - 1/(a*x + 1))), x)/c

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.69 \[ \int \frac {e^{\coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=\frac {1}{a c \sqrt {\frac {a x - 1}{a x + 1}}} \]

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/2)/(-a^2*c*x^2+c),x, algorithm="maxima")

[Out]

1/(a*c*sqrt((a*x - 1)/(a*x + 1)))

Giac [F]

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

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/2)/(-a^2*c*x^2+c),x, algorithm="giac")

[Out]

undef

Mupad [B] (verification not implemented)

Time = 3.83 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.69 \[ \int \frac {e^{\coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=\frac {1}{a\,c\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \]

[In]

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

[Out]

1/(a*c*((a*x - 1)/(a*x + 1))^(1/2))