\(\int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx\) [248]

Optimal result

Integrand size = 20, antiderivative size = 177 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=-\frac {6 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{\left (a-\frac {1}{x}\right ) \sqrt {c-a c x}}+\frac {2 a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x}{\left (a-\frac {1}{x}\right ) \sqrt {c-a c x}}-\frac {3 \sqrt {2} \sqrt {1-\frac {1}{a x}} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )}{\sqrt {a} \sqrt {\frac {1}{x}} \sqrt {c-a c x}} \]

[Out]

2*a*(1+1/a/x)^(3/2)*x*(1-1/a/x)^(1/2)/(a-1/x)/(-a*c*x+c)^(1/2)-6*(1-1/a/x)^(1/2)*(1+1/a/x)^(1/2)/(a-1/x)/(-a*c
*x+c)^(1/2)-3*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 = 0.13 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6311, 6316, 96, 95, 212} \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=-\frac {3 \sqrt {2} \sqrt {1-\frac {1}{a x}} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )}{\sqrt {a} \sqrt {\frac {1}{x}} \sqrt {c-a c x}}+\frac {2 a x \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}{\left (a-\frac {1}{x}\right ) \sqrt {c-a c x}}-\frac {6 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}{\left (a-\frac {1}{x}\right ) \sqrt {c-a c x}} \]

[In]

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

[Out]

(-6*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)])/((a - x^(-1))*Sqrt[c - a*c*x]) + (2*a*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))
^(3/2)*x)/((a - x^(-1))*Sqrt[c - a*c*x]) - (3*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^{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}} \text {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{3/2}}{x^{3/2} \left (1-\frac {x}{a}\right )^2} \, dx,x,\frac {1}{x}\right )}{\sqrt {\frac {1}{x}} \sqrt {c-a c x}} \\ & = \frac {2 a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x}{\left (a-\frac {1}{x}\right ) \sqrt {c-a c x}}-\frac {\left (6 \sqrt {1-\frac {1}{a x}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{\sqrt {x} \left (1-\frac {x}{a}\right )^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}} \sqrt {1+\frac {1}{a x}}}{\left (a-\frac {1}{x}\right ) \sqrt {c-a c x}}+\frac {2 a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x}{\left (a-\frac {1}{x}\right ) \sqrt {c-a c x}}-\frac {\left (3 \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 {6 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{\left (a-\frac {1}{x}\right ) \sqrt {c-a c x}}+\frac {2 a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x}{\left (a-\frac {1}{x}\right ) \sqrt {c-a c x}}-\frac {\left (6 \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 {6 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{\left (a-\frac {1}{x}\right ) \sqrt {c-a c x}}+\frac {2 a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x}{\left (a-\frac {1}{x}\right ) \sqrt {c-a c x}}-\frac {3 \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 = 0.16 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.66 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=\frac {\sqrt {1-\frac {1}{a x}} x \left (2 \sqrt {a} \sqrt {1+\frac {1}{a x}} (-2+a x)-3 \sqrt {2} \sqrt {\frac {1}{x}} (-1+a x) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )\right )}{\sqrt {a} (-1+a x) \sqrt {c-a c x}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.45 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.77

method	result	size
default	\(-\frac {\sqrt {-c \left (a x -1\right )}\, \left (-3 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) a c x +2 a x \sqrt {c}\, \sqrt {-c \left (a x +1\right )}+3 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c -4 \sqrt {-c \left (a x +1\right )}\, \sqrt {c}\right )}{\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) c^{\frac {3}{2}} \sqrt {-c \left (a x +1\right )}\, a}\)	\(136\)
risch	\(\frac {2 a x -2}{a \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x -1\right )}}+\frac {\left (-\frac {2 \sqrt {-a c x -c}}{a \left (-a c x +c \right )}+\frac {3 \sqrt {2}\, \arctan \left (\frac {\sqrt {-a c x -c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{a \sqrt {c}}\right ) \sqrt {-c \left (a x +1\right )}\, \left (a x -1\right )}{\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x -1\right )}}\)	\(142\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.63 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=\left [\frac {3 \, \sqrt {2} {\left (a^{2} c x^{2} - 2 \, a c x + c\right )} \sqrt {-\frac {1}{c}} \log \left (-\frac {a^{2} x^{2} - 2 \, \sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {-\frac {1}{c}} + 2 \, a x - 3}{a^{2} x^{2} - 2 \, a x + 1}\right ) - 4 \, {\left (a^{2} x^{2} - a x - 2\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, {\left (a^{3} c x^{2} - 2 \, a^{2} c x + a c\right )}}, -\frac {2 \, {\left (a^{2} x^{2} - a x - 2\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}} - \frac {3 \, \sqrt {2} {\left (a^{2} c x^{2} - 2 \, a c x + c\right )} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x - 1\right )} \sqrt {c}}\right )}{\sqrt {c}}}{a^{3} c x^{2} - 2 \, a^{2} c x + a c}\right ] \]

[In]

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

[Out]

[1/2*(3*sqrt(2)*(a^2*c*x^2 - 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)) - 4*(a^2*x^2 - a*x - 2)*sqrt(-a*c*x + c)*
sqrt((a*x - 1)/(a*x + 1)))/(a^3*c*x^2 - 2*a^2*c*x + a*c), -(2*(a^2*x^2 - a*x - 2)*sqrt(-a*c*x + c)*sqrt((a*x -
1)/(a*x + 1)) - 3*sqrt(2)*(a^2*c*x^2 - 2*a*c*x + c)*arctan(sqrt(2)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1))
/((a*x - 1)*sqrt(c)))/sqrt(c))/(a^3*c*x^2 - 2*a^2*c*x + a*c)]

Sympy [F]

\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=\int \frac {1}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} \sqrt {- c \left (a x - 1\right )}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=\int { \frac {1}{\sqrt {-a c x + c} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.42 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=-\frac {3 \, \sqrt {2} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x - c}}{2 \, \sqrt {c}}\right ) - 2 \, \sqrt {-a c x - c} + \frac {2 \, \sqrt {-a c x - c} c}{a c x - c}}{a {\left | c \right |}} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=\int \frac {1}{\sqrt {c-a\,c\,x}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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