3.240 \(\int \frac{e^{2 \coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx\)

Optimal. Leaf size=38 \[ \frac{2}{a c \sqrt{c-a c x}}-\frac{4}{3 a (c-a c x)^{3/2}} \]

[Out]

-4/(3*a*(c - a*c*x)^(3/2)) + 2/(a*c*Sqrt[c - a*c*x])

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Rubi [A]  time = 0.0859509, antiderivative size = 38, 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} \[ \frac{2}{a c \sqrt{c-a c x}}-\frac{4}{3 a (c-a c x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcCoth[a*x])/(c - a*c*x)^(3/2),x]

[Out]

-4/(3*a*(c - a*c*x)^(3/2)) + 2/(a*c*Sqrt[c - a*c*x])

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)}}{(c-a c x)^{3/2}} \, dx &=-\int \frac{e^{2 \tanh ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx\\ &=-\int \frac{1+a x}{(1-a x) (c-a c x)^{3/2}} \, dx\\ &=-\left (c \int \frac{1+a x}{(c-a c x)^{5/2}} \, dx\right )\\ &=-\left (c \int \left (\frac{2}{(c-a c x)^{5/2}}-\frac{1}{c (c-a c x)^{3/2}}\right ) \, dx\right )\\ &=-\frac{4}{3 a (c-a c x)^{3/2}}+\frac{2}{a c \sqrt{c-a c x}}\\ \end{align*}

Mathematica [A]  time = 0.0448822, size = 34, normalized size = 0.89 \[ -\frac{2 (3 a x-1) \sqrt{c-a c x}}{3 a c^2 (a x-1)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(2*ArcCoth[a*x])/(c - a*c*x)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.042, size = 21, normalized size = 0.6 \begin{align*} -{\frac{6\,ax-2}{3\,a} \left ( -acx+c \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02367, size = 35, normalized size = 0.92 \begin{align*} -\frac{2 \,{\left (3 \, a c x - c\right )}}{3 \,{\left (-a c x + c\right )}^{\frac{3}{2}} 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)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.56212, size = 96, normalized size = 2.53 \begin{align*} -\frac{2 \, \sqrt{-a c x + c}{\left (3 \, a x - 1\right )}}{3 \,{\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}} \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)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 24.2837, size = 29, normalized size = 0.76 \begin{align*} - \frac{4}{3 a \left (- a c x + c\right )^{\frac{3}{2}}} + \frac{2}{a c \sqrt{- a c x + 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)**(3/2),x)

[Out]

-4/(3*a*(-a*c*x + c)**(3/2)) + 2/(a*c*sqrt(-a*c*x + c))

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Giac [A]  time = 1.1544, size = 49, normalized size = 1.29 \begin{align*} \frac{2 \,{\left (3 \, a c x - c\right )}}{3 \,{\left (a c x - c\right )} \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)^(3/2),x, algorithm="giac")

[Out]

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