### 3.257 $$\int \frac{e^{-\coth ^{-1}(a x)}}{\sqrt{c-a c x}} \, dx$$

Optimal. Leaf size=29 $\frac{2 (a x+1) e^{-\coth ^{-1}(a x)}}{a \sqrt{c-a c x}}$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0378088, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.05, Rules used = {6174} $\frac{2 (a x+1) e^{-\coth ^{-1}(a x)}}{a \sqrt{c-a c x}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 6174

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

Rubi steps

\begin{align*} \int \frac{e^{-\coth ^{-1}(a x)}}{\sqrt{c-a c x}} \, dx &=\frac{2 e^{-\coth ^{-1}(a x)} (1+a x)}{a \sqrt{c-a c x}}\\ \end{align*}

Mathematica [A]  time = 0.0226005, size = 28, normalized size = 0.97 $\frac{2 x \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-a c x}}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.041, size = 35, normalized size = 1.2 \begin{align*} 2\,{\frac{ax+1}{a\sqrt{-acx+c}}\sqrt{{\frac{ax-1}{ax+1}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.10366, size = 39, normalized size = 1.34 \begin{align*} -\frac{2 \,{\left (a \sqrt{-c} x + \sqrt{-c}\right )}}{\sqrt{a x + 1} a c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.88169, size = 99, normalized size = 3.41 \begin{align*} -\frac{2 \, \sqrt{-a c x + c}{\left (a x + 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c x - a c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{a x - 1}{a x + 1}}}{\sqrt{- c \left (a x - 1\right )}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.13755, size = 66, normalized size = 2.28 \begin{align*} -\frac{2 \,{\left (\frac{\sqrt{2} \sqrt{-c}}{a} + \frac{{\left (-a c x - c\right )}^{\frac{3}{2}}}{{\left (a c x + c\right )} a}\right )}{\left | c \right |} \mathrm{sgn}\left (a x + 1\right )}{c^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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