\(\int \frac {e^{-\coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx\) [594]

Optimal result

Integrand size = 22, antiderivative size = 16 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=-\frac {e^{-\coth ^{-1}(a x)}}{a c} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 16, 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} \[ \int \frac {e^{-\coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=-\frac {e^{-\coth ^{-1}(a x)}}{a c} \]

[In]

Int[1/(E^ArcCoth[a*x]*(c - a^2*c*x^2)),x]

[Out]

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

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 = 0.17 (sec) , antiderivative size = 16, 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[1/(E^ArcCoth[a*x]*(c - a^2*c*x^2)),x]

[Out]

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

Maple [A] (verified)

Time = 0.50 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.50

method	result	size
gosper	\(-\frac {\sqrt {\frac {a x -1}{a x +1}}}{a c}\)	\(24\)
default	\(-\frac {\sqrt {\frac {a x -1}{a x +1}}}{a c}\)	\(24\)
trager	\(-\frac {\sqrt {-\frac {-a x +1}{a x +1}}}{a c}\)	\(26\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.44 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=-\frac {\sqrt {\frac {a x - 1}{a x + 1}}}{a c} \]

[In]

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

[Out]

-sqrt((a*x - 1)/(a*x + 1))/(a*c)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.44 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=-\frac {\sqrt {\frac {a x - 1}{a x + 1}}}{a c} \]

[In]

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

[Out]

-sqrt((a*x - 1)/(a*x + 1))/(a*c)

Giac [F]

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

[In]

integrate(((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 = 0.03 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.44 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx=-\frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{a\,c} \]

[In]

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

[Out]

-((a*x - 1)/(a*x + 1))^(1/2)/(a*c)