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

Optimal. Leaf size=36 \[ -\frac{2 \sqrt{c-a c x}}{a c}-\frac{4}{a \sqrt{c-a c x}} \]

[Out]

-4/(a*Sqrt[c - a*c*x]) - (2*Sqrt[c - a*c*x])/(a*c)

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Rubi [A]  time = 0.0839535, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6167, 6130, 21, 43} \[ -\frac{2 \sqrt{c-a c x}}{a c}-\frac{4}{a \sqrt{c-a c x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-4/(a*Sqrt[c - a*c*x]) - (2*Sqrt[c - a*c*x])/(a*c)

Rule 6167

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

Rule 6130

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

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{e^{2 \coth ^{-1}(a x)}}{\sqrt{c-a c x}} \, dx &=-\int \frac{e^{2 \tanh ^{-1}(a x)}}{\sqrt{c-a c x}} \, dx\\ &=-\int \frac{1+a x}{(1-a x) \sqrt{c-a c x}} \, dx\\ &=-\left (c \int \frac{1+a x}{(c-a c x)^{3/2}} \, dx\right )\\ &=-\left (c \int \left (\frac{2}{(c-a c x)^{3/2}}-\frac{1}{c \sqrt{c-a c x}}\right ) \, dx\right )\\ &=-\frac{4}{a \sqrt{c-a c x}}-\frac{2 \sqrt{c-a c x}}{a c}\\ \end{align*}

Mathematica [A]  time = 0.0251678, size = 21, normalized size = 0.58 \[ \frac{2 a x-6}{a \sqrt{c-a c x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6 + 2*a*x)/(a*Sqrt[c - a*c*x])

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Maple [A]  time = 0.041, size = 20, normalized size = 0.6 \begin{align*} 2\,{\frac{ax-3}{a\sqrt{-acx+c}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.00103, size = 41, normalized size = 1.14 \begin{align*} -\frac{2 \,{\left (\sqrt{-a c x + c} + \frac{2 \, c}{\sqrt{-a c x + c}}\right )}}{a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.5983, size = 63, normalized size = 1.75 \begin{align*} -\frac{2 \, \sqrt{-a c x + c}{\left (a x - 3\right )}}{a^{2} c x - a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 14.7232, size = 51, normalized size = 1.42 \begin{align*} \begin{cases} - \frac{\frac{2}{\sqrt{- a c x + c}} - \frac{2 \left (- \frac{c}{\sqrt{- a c x + c}} - \sqrt{- a c x + c}\right )}{c}}{a} & \text{for}\: a \neq 0 \\- \frac{x}{\sqrt{c}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-(2/sqrt(-a*c*x + c) - 2*(-c/sqrt(-a*c*x + c) - sqrt(-a*c*x + c))/c)/a, Ne(a, 0)), (-x/sqrt(c), Tru
e))

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Giac [A]  time = 1.16585, size = 43, normalized size = 1.19 \begin{align*} -\frac{4}{\sqrt{-a c x + c} a} - \frac{2 \, \sqrt{-a c x + c}}{a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/(sqrt(-a*c*x + c)*a) - 2*sqrt(-a*c*x + c)/(a*c)