\(\int e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx\) [645]

Optimal result

Integrand size = 25, antiderivative size = 174 \[ \int e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx=\frac {8 \sqrt {c-\frac {c}{a x}} x^2}{\sqrt {1-a x} \sqrt {1+a x}}-\frac {47 \sqrt {c-\frac {c}{a x}} x \sqrt {1+a x}}{4 a \sqrt {1-a x}}+\frac {\sqrt {c-\frac {c}{a x}} x^2 \sqrt {1+a x}}{2 \sqrt {1-a x}}+\frac {47 \sqrt {c-\frac {c}{a x}} \sqrt {x} \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{4 a^{3/2} \sqrt {1-a x}} \] Output:

8*(c-c/a/x)^(1/2)*x^2/(-a*x+1)^(1/2)/(a*x+1)^(1/2)-47/4*(c-c/a/x)^(1/2)*x* 
(a*x+1)^(1/2)/a/(-a*x+1)^(1/2)+1/2*(c-c/a/x)^(1/2)*x^2*(a*x+1)^(1/2)/(-a*x 
+1)^(1/2)+47/4*(c-c/a/x)^(1/2)*x^(1/2)*arcsinh(a^(1/2)*x^(1/2))/a^(3/2)/(- 
a*x+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.53 \[ \int e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx=\frac {\sqrt {c-\frac {c}{a x}} \sqrt {x} \left (\sqrt {a} \sqrt {x} \left (-47-13 a x+2 a^2 x^2\right )+47 \sqrt {1+a x} \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )\right )}{4 a^{3/2} \sqrt {1-a^2 x^2}} \] Input:

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

Output:

(Sqrt[c - c/(a*x)]*Sqrt[x]*(Sqrt[a]*Sqrt[x]*(-47 - 13*a*x + 2*a^2*x^2) + 4 
7*Sqrt[1 + a*x]*ArcSinh[Sqrt[a]*Sqrt[x]]))/(4*a^(3/2)*Sqrt[1 - a^2*x^2])
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.62, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {6684, 6679, 100, 27, 90, 60, 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.

\(\displaystyle \int x e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} \, dx\)

\(\Big \downarrow \) 6684

\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \int e^{-3 \text {arctanh}(a x)} \sqrt {x} \sqrt {1-a x}dx}{\sqrt {1-a x}}\)

\(\Big \downarrow \) 6679

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

\(\Big \downarrow \) 100

\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {8 x^{3/2}}{\sqrt {a x+1}}-\frac {2 \int \frac {a^2 \sqrt {x} (11-a x)}{2 \sqrt {a x+1}}dx}{a^2}\right )}{\sqrt {1-a x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {8 x^{3/2}}{\sqrt {a x+1}}-\int \frac {\sqrt {x} (11-a x)}{\sqrt {a x+1}}dx\right )}{\sqrt {1-a x}}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (-\frac {47}{4} \int \frac {\sqrt {x}}{\sqrt {a x+1}}dx+\frac {1}{2} x^{3/2} \sqrt {a x+1}+\frac {8 x^{3/2}}{\sqrt {a x+1}}\right )}{\sqrt {1-a x}}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (-\frac {47}{4} \left (\frac {\sqrt {x} \sqrt {a x+1}}{a}-\frac {\int \frac {1}{\sqrt {x} \sqrt {a x+1}}dx}{2 a}\right )+\frac {1}{2} x^{3/2} \sqrt {a x+1}+\frac {8 x^{3/2}}{\sqrt {a x+1}}\right )}{\sqrt {1-a x}}\)

\(\Big \downarrow \) 63

\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (-\frac {47}{4} \left (\frac {\sqrt {x} \sqrt {a x+1}}{a}-\frac {\int \frac {1}{\sqrt {a x+1}}d\sqrt {x}}{a}\right )+\frac {1}{2} x^{3/2} \sqrt {a x+1}+\frac {8 x^{3/2}}{\sqrt {a x+1}}\right )}{\sqrt {1-a x}}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {\sqrt {x} \left (-\frac {47}{4} \left (\frac {\sqrt {x} \sqrt {a x+1}}{a}-\frac {\text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{a^{3/2}}\right )+\frac {1}{2} x^{3/2} \sqrt {a x+1}+\frac {8 x^{3/2}}{\sqrt {a x+1}}\right ) \sqrt {c-\frac {c}{a x}}}{\sqrt {1-a x}}\)

Input:

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

Output:

(Sqrt[c - c/(a*x)]*Sqrt[x]*((8*x^(3/2))/Sqrt[1 + a*x] + (x^(3/2)*Sqrt[1 + 
a*x])/2 - (47*((Sqrt[x]*Sqrt[1 + a*x])/a - ArcSinh[Sqrt[a]*Sqrt[x]]/a^(3/2 
)))/4))/Sqrt[1 - a*x]
 

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 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, 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 = 0.22 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.91

method	result	size
default	\(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (4 a^{\frac {5}{2}} x^{2} \sqrt {-x \left (a x +1\right )}-26 a^{\frac {3}{2}} x \sqrt {-x \left (a x +1\right )}-47 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}}\right ) a x -94 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}-47 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}}\right )\right ) \sqrt {-a^{2} x^{2}+1}}{8 a^{\frac {3}{2}} \left (a x +1\right ) \sqrt {-x \left (a x +1\right )}\, \left (a x -1\right )}\)	\(158\)
risch	\(\frac {\left (2 a x -15\right ) \left (a x +1\right ) x \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{4 a \sqrt {-\left (a x +1\right ) a c x}\, \sqrt {-a^{2} x^{2}+1}}+\frac {\left (\frac {47 \arctan \left (\frac {\sqrt {a^{2} c}\, \left (x +\frac {1}{2 a}\right )}{\sqrt {-a^{2} c \,x^{2}-a c x}}\right )}{8 a \sqrt {a^{2} c}}+\frac {8 \sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2} c +\left (x +\frac {1}{a}\right ) a c}}{a^{3} c \left (x +\frac {1}{a}\right )}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{\sqrt {-a^{2} x^{2}+1}}\)	\(217\)

Input:

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

Output:

-1/8*(c*(a*x-1)/a/x)^(1/2)*x*(4*a^(5/2)*x^2*(-x*(a*x+1))^(1/2)-26*a^(3/2)* 
x*(-x*(a*x+1))^(1/2)-47*arctan(1/2/a^(1/2)*(2*a*x+1)/(-x*(a*x+1))^(1/2))*a 
*x-94*a^(1/2)*(-x*(a*x+1))^(1/2)-47*arctan(1/2/a^(1/2)*(2*a*x+1)/(-x*(a*x+ 
1))^(1/2)))*(-a^2*x^2+1)^(1/2)/a^(3/2)/(a*x+1)/(-x*(a*x+1))^(1/2)/(a*x-1)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.75 \[ \int e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx=\left [\frac {47 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) - 4 \, {\left (2 \, a^{3} x^{3} - 13 \, a^{2} x^{2} - 47 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{16 \, {\left (a^{4} x^{2} - a^{2}\right )}}, -\frac {47 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (2 \, a^{3} x^{3} - 13 \, a^{2} x^{2} - 47 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{8 \, {\left (a^{4} x^{2} - a^{2}\right )}}\right ] \] Input:

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

Output:

[1/16*(47*(a^2*x^2 - 1)*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*(2*a^3*x^3 - 13*a^2*x^2 - 47*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - 
c)/(a*x)))/(a^4*x^2 - a^2), -1/8*(47*(a^2*x^2 - 1)*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*(2*a^3*x^3 - 13*a^2*x^2 - 47*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c 
)/(a*x)))/(a^4*x^2 - a^2)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.43 \[ \int e^{-3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx=\frac {\sqrt {c}\, i \left (-47 \sqrt {a x +1}\, \mathrm {log}\left (\sqrt {a x +1}\, i +\sqrt {x}\, \sqrt {a}\, i \right )+36 \sqrt {a x +1}-2 \sqrt {x}\, \sqrt {a}\, a^{2} x^{2}+13 \sqrt {x}\, \sqrt {a}\, a x +47 \sqrt {x}\, \sqrt {a}\right )}{4 \sqrt {a x +1}\, a^{2}} \] Input:

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

Output:

(sqrt(c)*i*( - 47*sqrt(a*x + 1)*log(sqrt(a*x + 1)*i + sqrt(x)*sqrt(a)*i) + 
 36*sqrt(a*x + 1) - 2*sqrt(x)*sqrt(a)*a**2*x**2 + 13*sqrt(x)*sqrt(a)*a*x + 
 47*sqrt(x)*sqrt(a)))/(4*sqrt(a*x + 1)*a**2)