\(\int \frac {e^{\coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx\) [232]

Optimal result

Integrand size = 18, antiderivative size = 129 \[ \int \frac {e^{\coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx=-\frac {a \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x}{\left (a-\frac {1}{x}\right ) (c-a c x)^{3/2}}-\frac {\left (1-\frac {1}{a x}\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}\right )}{\sqrt {2} a \left (\frac {1}{a x}\right )^{3/2} (c-a c x)^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.90 \[ \int \frac {e^{\coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx=\frac {\sqrt {1-\frac {1}{a x}} x \left (2 \sqrt {a} \sqrt {1+\frac {1}{a x}}+\sqrt {2} \sqrt {\frac {1}{x}} (-1+a x) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )\right )}{2 \sqrt {a} c (-1+a x) \sqrt {c-a c x}} \] Input:

Integrate[E^ArcCoth[a*x]/(c - a*c*x)^(3/2),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.88, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6727, 27, 105, 104, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {e^{\coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx\)

\(\Big \downarrow \) 6727

\(\displaystyle -\frac {\left (1-\frac {1}{a x}\right )^{3/2} \int \frac {a^2 \sqrt {1+\frac {1}{a x}}}{\left (a-\frac {1}{x}\right )^2 \sqrt {\frac {1}{x}}}d\frac {1}{x}}{\left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {a^2 \left (1-\frac {1}{a x}\right )^{3/2} \int \frac {\sqrt {1+\frac {1}{a x}}}{\left (a-\frac {1}{x}\right )^2 \sqrt {\frac {1}{x}}}d\frac {1}{x}}{\left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}}\)

\(\Big \downarrow \) 105

\(\displaystyle -\frac {a^2 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {\int \frac {1}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \sqrt {\frac {1}{x}}}d\frac {1}{x}}{2 a}+\frac {\sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}}{a \left (a-\frac {1}{x}\right )}\right )}{\left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}}\)

\(\Big \downarrow \) 104

\(\displaystyle -\frac {a^2 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {\int \frac {1}{a-\frac {2}{x^2}}d\frac {\sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{a x}}}}{a}+\frac {\sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}}{a \left (a-\frac {1}{x}\right )}\right )}{\left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {a^2 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )}{\sqrt {2} a^{3/2}}+\frac {\sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}}{a \left (a-\frac {1}{x}\right )}\right )}{\left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}}\)

Input:

Int[E^ArcCoth[a*x]/(c - a*c*x)^(3/2),x]
 

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[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] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 
1)*(b*e - a*f))), x] - Simp[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 219

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] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 6727

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

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.91

method	result	size
default	\(-\frac {\sqrt {-c \left (a x -1\right )}\, \left (\sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) a c x -\sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c +2 \sqrt {-c \left (a x +1\right )}\, \sqrt {c}\right )}{2 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x -1\right ) \sqrt {-c \left (a x +1\right )}\, c^{\frac {5}{2}} a}\)	\(118\)

Input:

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

Output:

-1/2/((a*x-1)/(a*x+1))^(1/2)/(a*x-1)*(-c*(a*x-1))^(1/2)*(2^(1/2)*arctan(1/ 
2*(-c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2))*a*c*x-2^(1/2)*arctan(1/2*(-c*(a*x+1) 
)^(1/2)*2^(1/2)/c^(1/2))*c+2*(-c*(a*x+1))^(1/2)*c^(1/2))/(-c*(a*x+1))^(1/2 
)/c^(5/2)/a
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.22 \[ \int \frac {e^{\coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx=\left [-\frac {\sqrt {2} {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + 4 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{4 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}}, -\frac {\sqrt {2} {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, {\left (a c x - c\right )}}\right ) + 2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}}\right ] \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.47 \[ \int \frac {e^{\coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx=\frac {\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x - c}}{2 \, \sqrt {c}}\right )}{\sqrt {c}} + \frac {2 \, \sqrt {-a c x - c}}{a c x - c}}{2 \, a {\left | c \right |}} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.54 \[ \int \frac {e^{\coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx=\frac {\sqrt {c}\, i \left (2 \sqrt {a x +1}-\sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )\right ) a x +\sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )\right )\right )}{2 a \,c^{2} \left (a x -1\right )} \] Input:

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

Output:

(sqrt(c)*i*(2*sqrt(a*x + 1) - sqrt(2)*log(tan(asin(sqrt( - a*x + 1)/sqrt(2 
))/2))*a*x + sqrt(2)*log(tan(asin(sqrt( - a*x + 1)/sqrt(2))/2))))/(2*a*c** 
2*(a*x - 1))