### 3.275 $$\int \frac{e^{-3 \coth ^{-1}(a x)}}{\sqrt{c-a c x}} \, dx$$

Optimal. Leaf size=85 $\frac{2 x \sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}+\frac{6 \sqrt{1-\frac{1}{a x}}}{a \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}$

[Out]

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

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Rubi [A]  time = 0.143442, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.2, Rules used = {6176, 6181, 78, 37} $\frac{2 x \sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}+\frac{6 \sqrt{1-\frac{1}{a x}}}{a \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 6176

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

Rule 6181

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

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{e^{-3 \coth ^{-1}(a x)}}{\sqrt{c-a c x}} \, dx &=\frac{\left (\sqrt{1-\frac{1}{a x}} \sqrt{x}\right ) \int \frac{e^{-3 \coth ^{-1}(a x)}}{\sqrt{1-\frac{1}{a x}} \sqrt{x}} \, dx}{\sqrt{c-a c x}}\\ &=-\frac{\sqrt{1-\frac{1}{a x}} \operatorname{Subst}\left (\int \frac{1-\frac{x}{a}}{x^{3/2} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{\sqrt{\frac{1}{x}} \sqrt{c-a c x}}\\ &=\frac{2 \sqrt{1-\frac{1}{a x}} x}{\sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}+\frac{\left (3 \sqrt{1-\frac{1}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{a \sqrt{\frac{1}{x}} \sqrt{c-a c x}}\\ &=\frac{6 \sqrt{1-\frac{1}{a x}}}{a \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}+\frac{2 \sqrt{1-\frac{1}{a x}} x}{\sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}\\ \end{align*}

Mathematica [A]  time = 0.0312703, size = 48, normalized size = 0.56 $\frac{2 \sqrt{1-\frac{1}{a x}} (a x+3)}{a \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.041, size = 47, normalized size = 0.6 \begin{align*} 2\,{\frac{ \left ( ax+3 \right ) \left ( ax+1 \right ) }{ \left ( ax-1 \right ) a\sqrt{-acx+c}} \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)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.08734, size = 65, normalized size = 0.76 \begin{align*} \frac{2 \,{\left (a^{2} x^{2} + 4 \, a x + 3\right )}{\left (a x - 1\right )}}{{\left (a^{2} \sqrt{-c} x - a \sqrt{-c}\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)^(1/2),x, algorithm="maxima")

[Out]

2*(a^2*x^2 + 4*a*x + 3)*(a*x - 1)/((a^2*sqrt(-c)*x - a*sqrt(-c))*(a*x + 1)^(3/2))

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Fricas [A]  time = 1.53825, size = 99, normalized size = 1.16 \begin{align*} -\frac{2 \, \sqrt{-a c x + c}{\left (a x + 3\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))^(3/2)/(-a*c*x+c)^(1/2),x, algorithm="fricas")

[Out]

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

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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)**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.18049, size = 49, normalized size = 0.58 \begin{align*} \frac{2 \,{\left (\sqrt{-a c x - c} - \frac{2 \, c}{\sqrt{-a c x - c}}\right )}{\left | c \right |}}{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)^(1/2),x, algorithm="giac")

[Out]

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