### 3.231 $$\int \frac{e^{\coth ^{-1}(a x)}}{\sqrt{c-a c x}} \, dx$$

Optimal. Leaf size=118 $\frac{2 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}{\sqrt{c-a c x}}-\frac{2 \sqrt{2} \sqrt{1-\frac{1}{a x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{\sqrt{a} \sqrt{\frac{1}{x}} \sqrt{c-a c x}}$

[Out]

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

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Rubi [A]  time = 0.166492, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.278, Rules used = {6176, 6181, 94, 93, 206} $\frac{2 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}{\sqrt{c-a c x}}-\frac{2 \sqrt{2} \sqrt{1-\frac{1}{a x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{\sqrt{a} \sqrt{\frac{1}{x}} \sqrt{c-a c x}}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^ArcCoth[a*x]/Sqrt[c - a*c*x],x]

[Out]

(2*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x)/Sqrt[c - a*c*x] - (2*Sqrt[2]*Sqrt[1 - 1/(a*x)]*ArcTanh[(Sqrt[2]*Sqrt
[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/(Sqrt[a]*Sqrt[x^(-1)]*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 94

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

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*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^{\coth ^{-1}(a x)}}{\sqrt{c-a c x}} \, dx &=\frac{\left (\sqrt{1-\frac{1}{a x}} \sqrt{x}\right ) \int \frac{e^{\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{\sqrt{1+\frac{x}{a}}}{x^{3/2} \left (1-\frac{x}{a}\right )} \, dx,x,\frac{1}{x}\right )}{\sqrt{\frac{1}{x}} \sqrt{c-a c x}}\\ &=\frac{2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x}{\sqrt{c-a c x}}-\frac{\left (2 \sqrt{1-\frac{1}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{a \sqrt{\frac{1}{x}} \sqrt{c-a c x}}\\ &=\frac{2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x}{\sqrt{c-a c x}}-\frac{\left (4 \sqrt{1-\frac{1}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{2 x^2}{a}} \, dx,x,\frac{\sqrt{\frac{1}{x}}}{\sqrt{1+\frac{1}{a x}}}\right )}{a \sqrt{\frac{1}{x}} \sqrt{c-a c x}}\\ &=\frac{2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x}{\sqrt{c-a c x}}-\frac{2 \sqrt{2} \sqrt{1-\frac{1}{a x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{1+\frac{1}{a x}}}\right )}{\sqrt{a} \sqrt{\frac{1}{x}} \sqrt{c-a c x}}\\ \end{align*}

Mathematica [A]  time = 0.0616523, size = 99, normalized size = 0.84 $\frac{2 x \sqrt{1-\frac{1}{a x}} \left (\sqrt{a} \sqrt{\frac{1}{a x}+1}-\sqrt{2} \sqrt{\frac{1}{x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )\right )}{\sqrt{a} \sqrt{c-a c x}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.154, size = 83, normalized size = 0.7 \begin{align*} 2\,{\frac{\sqrt{-c \left ( ax-1 \right ) }}{\sqrt{-c \left ( ax+1 \right ) }ca} \left ( \sqrt{c}\sqrt{2}\arctan \left ( 1/2\,{\frac{\sqrt{-c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ) -\sqrt{-c \left ( ax+1 \right ) } \right ){\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.57809, size = 568, normalized size = 4.81 \begin{align*} \left [\frac{\sqrt{2}{\left (a c x - c\right )} \sqrt{-\frac{1}{c}} \log \left (-\frac{a^{2} x^{2} - 2 \, \sqrt{2} \sqrt{-a c x + c}{\left (a x + 1\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{-\frac{1}{c}} + 2 \, a x - 3}{a^{2} x^{2} - 2 \, a x + 1}\right ) - 2 \, \sqrt{-a c x + c}{\left (a x + 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c x - a c}, -\frac{2 \,{\left (\sqrt{-a c x + c}{\left (a x + 1\right )} \sqrt{\frac{a x - 1}{a x + 1}} - \frac{\sqrt{2}{\left (a c x - c\right )} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x - 1\right )} \sqrt{c}}\right )}{\sqrt{c}}\right )}}{a^{2} c x - a c}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[(sqrt(2)*(a*c*x - c)*sqrt(-1/c)*log(-(a^2*x^2 - 2*sqrt(2)*sqrt(-a*c*x + c)*(a*x + 1)*sqrt((a*x - 1)/(a*x + 1)
)*sqrt(-1/c) + 2*a*x - 3)/(a^2*x^2 - 2*a*x + 1)) - 2*sqrt(-a*c*x + c)*(a*x + 1)*sqrt((a*x - 1)/(a*x + 1)))/(a^
2*c*x - a*c), -2*(sqrt(-a*c*x + c)*(a*x + 1)*sqrt((a*x - 1)/(a*x + 1)) - sqrt(2)*(a*c*x - c)*arctan(sqrt(2)*sq
rt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1))/((a*x - 1)*sqrt(c)))/sqrt(c))/(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(1/((a*x-1)/(a*x+1))**(1/2)/(-a*c*x+c)**(1/2),x)

[Out]

Timed out

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Giac [C]  time = 1.20736, size = 123, normalized size = 1.04 \begin{align*} \frac{2 \,{\left (\frac{\sqrt{2} \sqrt{c} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x - c}}{2 \, \sqrt{c}}\right ) - \sqrt{-a c x - c}}{\mathrm{sgn}\left (-a c x - c\right )} - \frac{-i \, \sqrt{2} \sqrt{-c} \arctan \left (-i\right ) + \sqrt{2} \sqrt{-c}}{\mathrm{sgn}\left (c\right )}\right )}}{a c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*((sqrt(2)*sqrt(c)*arctan(1/2*sqrt(2)*sqrt(-a*c*x - c)/sqrt(c)) - sqrt(-a*c*x - c))/sgn(-a*c*x - c) - (-I*sqr
t(2)*sqrt(-c)*arctan(-I) + sqrt(2)*sqrt(-c))/sgn(c))/(a*c)