\(\int \frac {e^{-\coth ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx\) [647]

Optimal result

Integrand size = 24, antiderivative size = 37 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \log (1+a x)}{\sqrt {c-a^2 c x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6327, 6328, 31} \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=\frac {x \sqrt {1-\frac {1}{a^2 x^2}} \log (a x+1)}{\sqrt {c-a^2 c x^2}} \]

[In]

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

[Out]

(Sqrt[1 - 1/(a^2*x^2)]*x*Log[1 + a*x])/Sqrt[c - a^2*c*x^2]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 6327

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

Rule 6328

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_.), x_Symbol] :> Dist[c^p/a^(2*p), Int[(u/x^(
2*p))*(-1 + a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] &&  !
IntegerQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) && IntegersQ[2*p, p + n/2]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \log (1+a x)}{\sqrt {c-a^2 c x^2}} \]

[In]

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

[Out]

(Sqrt[1 - 1/(a^2*x^2)]*x*Log[1 + a*x])/Sqrt[c - a^2*c*x^2]

Maple [A] (verified)

Time = 0.52 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.38

method	result	size
default	\(-\frac {\ln \left (a x +1\right ) \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {\frac {a x -1}{a x +1}}}{c \left (a x -1\right ) a}\)	\(51\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.59 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx=-\frac {\sqrt {-a^{2} c} \log \left (a x + 1\right )}{a^{2} c} \]

[In]

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

[Out]

-sqrt(-a^2*c)*log(a*x + 1)/(a^2*c)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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