### 3.276 $$\int \frac{e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx$$

Optimal. Leaf size=29 $-\frac{2 (a x+1) e^{-3 \coth ^{-1}(a x)}}{a (c-a c x)^{3/2}}$

[Out]

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

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Rubi [A]  time = 0.0380544, 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} $-\frac{2 (a x+1) e^{-3 \coth ^{-1}(a x)}}{a (c-a c x)^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/(E^(3*ArcCoth[a*x])*(c - a*c*x)^(3/2)),x]

[Out]

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

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^{-3 \coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx &=-\frac{2 e^{-3 \coth ^{-1}(a x)} (1+a x)}{a (c-a c x)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0286373, size = 41, normalized size = 1.41 $-\frac{2 x \left (1-\frac{1}{a x}\right )^{3/2}}{\sqrt{\frac{1}{a x}+1} (c-a c x)^{3/2}}$

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(E^(3*ArcCoth[a*x])*(c - a*c*x)^(3/2)),x]

[Out]

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

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Maple [A]  time = 0.046, size = 35, normalized size = 1.2 \begin{align*} -2\,{\frac{ax+1}{a \left ( -acx+c \right ) ^{3/2}} \left ({\frac{ax-1}{ax+1}} \right ) ^{3/2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.09125, size = 61, normalized size = 2.1 \begin{align*} -\frac{2 \,{\left (a \sqrt{-c} x + \sqrt{-c}\right )}{\left (a x - 1\right )}}{{\left (a^{2} c^{2} x - a c^{2}\right )}{\left (a x + 1\right )}^{\frac{3}{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(3/2)/(-a*c*x+c)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.54254, size = 90, normalized size = 3.1 \begin{align*} -\frac{2 \, \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{2} x - a c^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(3/2)/(-a*c*x+c)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.15873, size = 55, normalized size = 1.9 \begin{align*} \frac{{\left (\frac{\sqrt{2}}{a \sqrt{-c}} - \frac{2}{\sqrt{-a c x - c} 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))^(3/2)/(-a*c*x+c)^(3/2),x, algorithm="giac")

[Out]

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