\(\int e^{-\text {arctanh}(a x)} (c-\frac {c}{a x})^{3/2} \, dx\) [555]

Optimal result

Integrand size = 24, antiderivative size = 126 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=-\frac {2 \left (c-\frac {c}{a x}\right )^{3/2} x \sqrt {1+a x}}{(1-a x)^{3/2}}+\frac {a \left (c-\frac {c}{a x}\right )^{3/2} x^2 \sqrt {1+a x}}{(1-a x)^{3/2}}-\frac {5 \sqrt {a} \left (c-\frac {c}{a x}\right )^{3/2} x^{3/2} \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{(1-a x)^{3/2}} \] Output:

-2*(c-c/a/x)^(3/2)*x*(a*x+1)^(1/2)/(-a*x+1)^(3/2)+a*(c-c/a/x)^(3/2)*x^2*(a 
*x+1)^(1/2)/(-a*x+1)^(3/2)-5*a^(1/2)*(c-c/a/x)^(3/2)*x^(3/2)*arcsinh(a^(1/ 
2)*x^(1/2))/(-a*x+1)^(3/2)
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.56 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=-\frac {c \sqrt {c-\frac {c}{a x}} \left ((-2+a x) \sqrt {1+a x}-5 \sqrt {a} \sqrt {x} \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )\right )}{a \sqrt {1-a x}} \] Input:

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

Output:

-((c*Sqrt[c - c/(a*x)]*((-2 + a*x)*Sqrt[1 + a*x] - 5*Sqrt[a]*Sqrt[x]*ArcSi 
nh[Sqrt[a]*Sqrt[x]]))/(a*Sqrt[1 - a*x]))
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.69, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {6684, 6679, 100, 27, 90, 63, 222}

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.

\(\displaystyle \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx\)

\(\Big \downarrow \) 6684

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

\(\Big \downarrow \) 6679

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

\(\Big \downarrow \) 100

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 90

\(\displaystyle \frac {x^{3/2} \left (c-\frac {c}{a x}\right )^{3/2} \left (-a \left (\frac {5}{2} \int \frac {1}{\sqrt {x} \sqrt {a x+1}}dx-\sqrt {x} \sqrt {a x+1}\right )-\frac {2 \sqrt {a x+1}}{\sqrt {x}}\right )}{(1-a x)^{3/2}}\)

\(\Big \downarrow \) 63

\(\displaystyle \frac {x^{3/2} \left (c-\frac {c}{a x}\right )^{3/2} \left (-a \left (5 \int \frac {1}{\sqrt {a x+1}}d\sqrt {x}-\sqrt {x} \sqrt {a x+1}\right )-\frac {2 \sqrt {a x+1}}{\sqrt {x}}\right )}{(1-a x)^{3/2}}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {x^{3/2} \left (-a \left (\frac {5 \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a}}-\sqrt {x} \sqrt {a x+1}\right )-\frac {2 \sqrt {a x+1}}{\sqrt {x}}\right ) \left (c-\frac {c}{a x}\right )^{3/2}}{(1-a x)^{3/2}}\)

Input:

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

Output:

((c - c/(a*x))^(3/2)*x^(3/2)*((-2*Sqrt[1 + a*x])/Sqrt[x] - a*(-(Sqrt[x]*Sq 
rt[1 + a*x]) + (5*ArcSinh[Sqrt[a]*Sqrt[x]])/Sqrt[a])))/(1 - a*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 63

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2/b   S 
ubst[Int[1/Sqrt[c + d*(x^2/b)], x], x, Sqrt[b*x]], x] /; FreeQ[{b, c, d}, x 
] && GtQ[c, 0]
 

rule 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 100

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d 
*e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1))   Int[(c + d*x)^ 
(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( 
p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x 
, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n 
 + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))
 

rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt 
[a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
 

rule 6679

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

rule 6684

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

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.87

method	result	size
default	\(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c \sqrt {-a^{2} x^{2}+1}\, \left (2 a^{\frac {3}{2}} x \sqrt {-x \left (a x +1\right )}+5 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}}\right ) a x -4 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}\right )}{2 a^{\frac {3}{2}} \left (a x -1\right ) \sqrt {-x \left (a x +1\right )}}\)	\(109\)
risch	\(-\frac {\left (a^{2} x^{2}-a x -2\right ) c \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{\sqrt {-\left (a x +1\right ) a c x}\, \sqrt {-a^{2} x^{2}+1}\, a}+\frac {5 \arctan \left (\frac {\sqrt {a^{2} c}\, \left (x +\frac {1}{2 a}\right )}{\sqrt {-a^{2} c \,x^{2}-a c x}}\right ) c \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{2 \sqrt {a^{2} c}\, \sqrt {-a^{2} x^{2}+1}}\)	\(176\)

Input:

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

Output:

1/2*(c*(a*x-1)/a/x)^(1/2)*c/a^(3/2)*(-a^2*x^2+1)^(1/2)*(2*a^(3/2)*x*(-x*(a 
*x+1))^(1/2)+5*arctan(1/2/a^(1/2)*(2*a*x+1)/(-x*(a*x+1))^(1/2))*a*x-4*a^(1 
/2)*(-x*(a*x+1))^(1/2))/(a*x-1)/(-x*(a*x+1))^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 268, normalized size of antiderivative = 2.13 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=\left [\frac {5 \, {\left (a c x - c\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, \sqrt {-a^{2} x^{2} + 1} {\left (a c x - 2 \, c\right )} \sqrt {\frac {a c x - c}{a x}}}{4 \, {\left (a^{2} x - a\right )}}, -\frac {5 \, {\left (a c x - c\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \, \sqrt {-a^{2} x^{2} + 1} {\left (a c x - 2 \, c\right )} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a^{2} x - a\right )}}\right ] \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.50 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx=\frac {\sqrt {c}\, c i \left (4 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a x -8 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}-20 \,\mathrm {log}\left (\sqrt {a x +1}\, i +\sqrt {x}\, \sqrt {a}\, i \right ) a x -9 a x \right )}{4 a^{2} x} \] Input:

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

Output:

(sqrt(c)*c*i*(4*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a*x - 8*sqrt(x)*sqrt(a)*sqrt 
(a*x + 1) - 20*log(sqrt(a*x + 1)*i + sqrt(x)*sqrt(a)*i)*a*x - 9*a*x))/(4*a 
**2*x)