\(\int \frac {e^{\coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx\) [231]

Optimal result

Integrand size = 18, antiderivative size = 118 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=\frac {2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x}{\sqrt {c-a c x}}-\frac {2 \sqrt {2} \sqrt {1-\frac {1}{a x}} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )}{\sqrt {a} \sqrt {\frac {1}{x}} \sqrt {c-a c x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6311, 6316, 96, 95, 212} \[ \int \frac {e^{\coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=\frac {2 x \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}{\sqrt {c-a c x}}-\frac {2 \sqrt {2} \sqrt {1-\frac {1}{a x}} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )}{\sqrt {a} \sqrt {\frac {1}{x}} \sqrt {c-a c x}} \]

[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 95

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 96

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[m, 1] ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 6311

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 6316

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_), x_Symbol] :> Dist[(-c^p)*x^m*(1/x)^m, S
ubst[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]

Rubi steps \begin{align*} \text {integral}& = \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}} \text {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 ) \text {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 ) \text {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}} \text {arctanh}\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] (verified)

Time = 0.07 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.84 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=\frac {2 \sqrt {1-\frac {1}{a x}} x \left (\sqrt {a} \sqrt {1+\frac {1}{a x}}-\sqrt {2} \sqrt {\frac {1}{x}} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )\right )}{\sqrt {a} \sqrt {c-a c x}} \]

[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])

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.70

method	result	size
default	\(\frac {2 \sqrt {-c \left (a x -1\right )}\, \left (\sqrt {c}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )-\sqrt {-c \left (a x +1\right )}\right )}{\sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x +1\right )}\, c a}\)	\(83\)
risch	\(\frac {2 a x -2}{a \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x -1\right )}}+\frac {2 \sqrt {2}\, \arctan \left (\frac {\sqrt {-a c x -c}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {-c \left (a x +1\right )}\, \left (a x -1\right )}{a \sqrt {c}\, \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x -1\right )}}\)	\(115\)

[In]

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

[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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 239, normalized size of antiderivative = 2.03 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=\left [\frac {\sqrt {2} {\left (a c x - c\right )} \sqrt {-\frac {1}{c}} \log \left (-\frac {a^{2} x^{2} - 2 \, \sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {-\frac {1}{c}} + 2 \, a x - 3}{a^{2} x^{2} - 2 \, a x + 1}\right ) - 2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c x - a c}, -\frac {2 \, {\left (\sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}} - \frac {\sqrt {2} {\left (a c x - c\right )} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x - 1\right )} \sqrt {c}}\right )}{\sqrt {c}}\right )}}{a^{2} c x - a c}\right ] \]

[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)]

Sympy [F]

\[ \int \frac {e^{\coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=\int \frac {1}{\sqrt {\frac {a x - 1}{a x + 1}} \sqrt {- c \left (a x - 1\right )}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {e^{\coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=\int { \frac {1}{\sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.80 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=\frac {2 \, c {\left (\frac {\sqrt {2} {\left (\sqrt {c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {c}}\right ) - \sqrt {-c}\right )}}{c} - \frac {\sqrt {2} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x - c}}{2 \, \sqrt {c}}\right ) - \sqrt {-a c x - c}}{c}\right )}}{a {\left | c \right |} \mathrm {sgn}\left (a x + 1\right )} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{\coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=\int \frac {1}{\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]

[In]

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

[Out]

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