### 3.266 $$\int \frac{e^{-2 \coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx$$

Optimal. Leaf size=37 $\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )}{a c^{3/2}}$

[Out]

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

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Rubi [A]  time = 0.0927469, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.25, Rules used = {6167, 6130, 21, 63, 206} $\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )}{a c^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 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\\ &=-\frac{\int \frac{1}{(1+a x) \sqrt{c-a c x}} \, dx}{c}\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{2-\frac{x^2}{c}} \, dx,x,\sqrt{c-a c x}\right )}{a c^2}\\ &=\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )}{a c^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0184084, size = 37, normalized size = 1. $\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )}{a c^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.052, size = 29, normalized size = 0.8 \begin{align*}{\frac{\sqrt{2}}{a}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{-acx+c}{\frac{1}{\sqrt{c}}}} \right ){c}^{-{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.89117, size = 231, normalized size = 6.24 \begin{align*} \left [\frac{\sqrt{2} \log \left (\frac{a x - \frac{2 \, \sqrt{2} \sqrt{-a c x + c}}{\sqrt{c}} - 3}{a x + 1}\right )}{2 \, a c^{\frac{3}{2}}}, \frac{\sqrt{2} \sqrt{-\frac{1}{c}} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{-\frac{1}{c}}}{a x - 1}\right )}{a c}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*sqrt(2)*log((a*x - 2*sqrt(2)*sqrt(-a*c*x + c)/sqrt(c) - 3)/(a*x + 1))/(a*c^(3/2)), sqrt(2)*sqrt(-1/c)*arc
tan(sqrt(2)*sqrt(-a*c*x + c)*sqrt(-1/c)/(a*x - 1))/(a*c)]

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Sympy [A]  time = 19.8034, size = 41, normalized size = 1.11 \begin{align*} - \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{- a c x + c}}{2 \sqrt{- c}} \right )}}{a c \sqrt{- c}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15827, size = 49, normalized size = 1.32 \begin{align*} -\frac{\sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c}}{2 \, \sqrt{-c}}\right )}{a \sqrt{-c} c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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