\(\int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx\) [537]

Optimal result

Integrand size = 27, antiderivative size = 134 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=-\frac {8 \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {2 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}}}+\frac {2 \sqrt {c-\frac {c}{a x}} \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )}{\sqrt {1-\frac {1}{a x}}} \]

[Out]

2*arctanh((1+1/a/x)^(1/2))*(c-c/a/x)^(1/2)/(1-1/a/x)^(1/2)-8*(c-c/a/x)^(1/2)/(1-1/a/x)^(1/2)/(1+1/a/x)^(1/2)-2
*(1+1/a/x)^(1/2)*(c-c/a/x)^(1/2)/(1-1/a/x)^(1/2)

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6317, 6315, 89, 65, 214} \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=\frac {2 \text {arctanh}\left (\sqrt {\frac {1}{a x}+1}\right ) \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}}}-\frac {2 \sqrt {\frac {1}{a x}+1} \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}}}-\frac {8 \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}} \]

[In]

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

[Out]

(-8*Sqrt[c - c/(a*x)])/(Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) - (2*Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a*x)])/Sqrt[1
- 1/(a*x)] + (2*Sqrt[c - c/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]])/Sqrt[1 - 1/(a*x)]

Rule 65

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 89

Int[(((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_))/((a_.) + (b_.)*(x_)), x_Symbol] :> Int[ExpandIntegr
and[(e + f*x)^FractionalPart[p], (c + d*x)^n*((e + f*x)^IntegerPart[p]/(a + b*x)), x], x] /; FreeQ[{a, b, c, d
, e, f}, x] && IGtQ[n, 0] && LtQ[p, -1] && FractionQ[p]

Rule 214

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

Rule 6315

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

Rule 6317

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

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c-\frac {c}{a x}} \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a x}}}{x} \, dx}{\sqrt {1-\frac {1}{a x}}} \\ & = -\frac {\sqrt {c-\frac {c}{a x}} \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^2}{x \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}} \\ & = -\frac {\sqrt {c-\frac {c}{a x}} \text {Subst}\left (\int \left (-\frac {4}{a \left (1+\frac {x}{a}\right )^{3/2}}+\frac {1}{a \sqrt {1+\frac {x}{a}}}+\frac {1}{x \sqrt {1+\frac {x}{a}}}\right ) \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}} \\ & = -\frac {8 \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {2 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}}}-\frac {\sqrt {c-\frac {c}{a x}} \text {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}} \\ & = -\frac {8 \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {2 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}}}-\frac {\left (2 a \sqrt {c-\frac {c}{a x}}\right ) \text {Subst}\left (\int \frac {1}{-a+a x^2} \, dx,x,\sqrt {1+\frac {1}{a x}}\right )}{\sqrt {1-\frac {1}{a x}}} \\ & = -\frac {8 \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {2 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}}}{\sqrt {1-\frac {1}{a x}}}+\frac {2 \sqrt {c-\frac {c}{a x}} \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )}{\sqrt {1-\frac {1}{a x}}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.83 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.98 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=-\frac {2 a \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} x (1+5 a x)}{-1+a^2 x^2}-\sqrt {c} \log (1-a x)+\sqrt {c} \log \left (2 a^2 \sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} x^2+c \left (-1-a x+2 a^2 x^2\right )\right ) \]

[In]

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

[Out]

(-2*a*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x*(1 + 5*a*x))/(-1 + a^2*x^2) - Sqrt[c]*Log[1 - a*x] + Sqrt[c]*L
og[2*a^2*Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2 + c*(-1 - a*x + 2*a^2*x^2)]

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.13

method	result	size
default	\(-\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (10 a^{\frac {3}{2}} x \sqrt {\left (a x +1\right ) x}-\ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a^{2} x^{2}-\ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a x +2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}\right )}{\left (a x -1\right )^{2} \sqrt {a}\, \sqrt {\left (a x +1\right ) x}}\)	\(151\)
risch	\(-\frac {2 \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}{a x -1}+\frac {\left (\frac {a \ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right )}{\sqrt {a^{2} c}}-\frac {8 \sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-\left (x +\frac {1}{a}\right ) a c}}{a c \left (x +\frac {1}{a}\right )}\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\left (a x +1\right ) a c x}}{a x -1}\)	\(180\)

[In]

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

[Out]

-((a*x-1)/(a*x+1))^(3/2)*(a*x+1)/(a*x-1)^2*(c*(a*x-1)/a/x)^(1/2)*(10*a^(3/2)*x*((a*x+1)*x)^(1/2)-ln(1/2*(2*((a
*x+1)*x)^(1/2)*a^(1/2)+2*a*x+1)/a^(1/2))*a^2*x^2-ln(1/2*(2*((a*x+1)*x)^(1/2)*a^(1/2)+2*a*x+1)/a^(1/2))*a*x+2*(
(a*x+1)*x)^(1/2)*a^(1/2))/a^(1/2)/((a*x+1)*x)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 277, normalized size of antiderivative = 2.07 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=\left [\frac {{\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) - 4 \, {\left (5 \, a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a x - 1\right )}}, -\frac {{\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (5 \, a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{a x - 1}\right ] \]

[In]

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

[Out]

[1/2*((a*x - 1)*sqrt(c)*log(-(8*a^3*c*x^3 - 7*a*c*x + 4*(2*a^3*x^3 + 3*a^2*x^2 + a*x)*sqrt(c)*sqrt((a*x - 1)/(
a*x + 1))*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) - 4*(5*a*x + 1)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(
a*x)))/(a*x - 1), -((a*x - 1)*sqrt(-c)*arctan(2*(a^2*x^2 + a*x)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x
- c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) + 2*(5*a*x + 1)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a*
x - 1)]

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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