\(\int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx\) [379]

Optimal result

Integrand size = 23, antiderivative size = 238 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx=-\frac {8 \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x^3}-\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}} x^3}+\frac {119 a \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{12 \sqrt {1-\frac {1}{a x}} x^2}-\frac {119 a^2 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{8 \sqrt {1-\frac {1}{a x}} x}+\frac {119 a^3 \sqrt {\frac {1}{a x}} \sqrt {c-a c x} \text {arcsinh}\left (\sqrt {\frac {1}{a x}}\right )}{8 \sqrt {1-\frac {1}{a x}}} \] Output:

-8*(-a*c*x+c)^(1/2)/(1-1/a/x)^(1/2)/(1+1/a/x)^(1/2)/x^3-1/3*(1+1/a/x)^(1/2 
)*(-a*c*x+c)^(1/2)/(1-1/a/x)^(1/2)/x^3+119/12*a*(1+1/a/x)^(1/2)*(-a*c*x+c) 
^(1/2)/(1-1/a/x)^(1/2)/x^2-119/8*a^2*(1+1/a/x)^(1/2)*(-a*c*x+c)^(1/2)/(1-1 
/a/x)^(1/2)/x+119/8*a^3*(1/a/x)^(1/2)*(-a*c*x+c)^(1/2)*arcsinh((1/a/x)^(1/ 
2))/(1-1/a/x)^(1/2)
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.41 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\frac {\sqrt {c-a c x} \left (-8+38 a x-119 a^2 x^2-357 a^3 x^3+\frac {357 a^{7/2} \sqrt {1+\frac {1}{a x}} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{\left (\frac {1}{x}\right )^{7/2}}\right )}{24 a \sqrt {1-\frac {1}{a^2 x^2}} x^4} \] Input:

Integrate[Sqrt[c - a*c*x]/(E^(3*ArcCoth[a*x])*x^4),x]
 

Output:

(Sqrt[c - a*c*x]*(-8 + 38*a*x - 119*a^2*x^2 - 357*a^3*x^3 + (357*a^(7/2)*S 
qrt[1 + 1/(a*x)]*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]])/(x^(-1))^(7/2)))/(24*a*Sqr 
t[1 - 1/(a^2*x^2)]*x^4)
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.72, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {6730, 27, 100, 27, 90, 60, 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.

\(\displaystyle \int \frac {\sqrt {c-a c x} e^{-3 \coth ^{-1}(a x)}}{x^4} \, dx\)

\(\Big \downarrow \) 6730

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 100

\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {8 a^2 \left (\frac {1}{x}\right )^{5/2}}{\sqrt {\frac {1}{a x}+1}}-2 a^2 \int \frac {\left (19 a-\frac {1}{x}\right ) \left (\frac {1}{x}\right )^{3/2}}{2 a \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {8 a^2 \left (\frac {1}{x}\right )^{5/2}}{\sqrt {\frac {1}{a x}+1}}-a \int \frac {\left (19 a-\frac {1}{x}\right ) \left (\frac {1}{x}\right )^{3/2}}{\sqrt {1+\frac {1}{a x}}}d\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 90

\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {8 a^2 \left (\frac {1}{x}\right )^{5/2}}{\sqrt {\frac {1}{a x}+1}}-a \left (\frac {119}{6} a \int \frac {\left (\frac {1}{x}\right )^{3/2}}{\sqrt {1+\frac {1}{a x}}}d\frac {1}{x}-\frac {1}{3} a \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}\right )\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 60

\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {8 a^2 \left (\frac {1}{x}\right )^{5/2}}{\sqrt {\frac {1}{a x}+1}}-a \left (\frac {119}{6} a \left (\frac {1}{2} a \left (\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}-\frac {3}{4} a \int \frac {\sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{a x}}}d\frac {1}{x}\right )-\frac {1}{3} a \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}\right )\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 60

\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {8 a^2 \left (\frac {1}{x}\right )^{5/2}}{\sqrt {\frac {1}{a x}+1}}-a \left (\frac {119}{6} a \left (\frac {1}{2} a \left (\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}-\frac {3}{4} a \left (a \sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}-\frac {1}{2} a \int \frac {1}{\sqrt {1+\frac {1}{a x}} \sqrt {\frac {1}{x}}}d\frac {1}{x}\right )\right )-\frac {1}{3} a \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}\right )\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 63

\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {8 a^2 \left (\frac {1}{x}\right )^{5/2}}{\sqrt {\frac {1}{a x}+1}}-a \left (\frac {119}{6} a \left (\frac {1}{2} a \left (\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}-\frac {3}{4} a \left (a \sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}-a \int \frac {1}{\sqrt {1+\frac {1}{x^2 a}}}d\sqrt {\frac {1}{x}}\right )\right )-\frac {1}{3} a \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}\right )\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 222

\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \left (\frac {8 a^2 \left (\frac {1}{x}\right )^{5/2}}{\sqrt {\frac {1}{a x}+1}}-a \left (\frac {119}{6} a \left (\frac {1}{2} a \left (\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}-\frac {3}{4} a \left (a \sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}-a^{3/2} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )\right )\right )-\frac {1}{3} a \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}\right )\right ) \sqrt {c-a c x}}{a^2 \sqrt {1-\frac {1}{a x}}}\)

Input:

Int[Sqrt[c - a*c*x]/(E^(3*ArcCoth[a*x])*x^4),x]
 

Output:

-((Sqrt[x^(-1)]*Sqrt[c - a*c*x]*((8*a^2*(x^(-1))^(5/2))/Sqrt[1 + 1/(a*x)] 
- a*(-1/3*(a*Sqrt[1 + 1/(a*x)]*(x^(-1))^(5/2)) + (119*a*((a*Sqrt[1 + 1/(a* 
x)]*(x^(-1))^(3/2))/2 - (3*a*(a*Sqrt[1 + 1/(a*x)]*Sqrt[x^(-1)] - a^(3/2)*A 
rcSinh[Sqrt[x^(-1)]/Sqrt[a]]))/4))/6)))/(a^2*Sqrt[1 - 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 6730

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

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.48

method	result	size
default	\(-\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, \left (357 \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}}{\sqrt {c}}\right ) a^{3} x^{3} \sqrt {-c \left (a x +1\right )}+357 a^{3} x^{3} \sqrt {c}+119 \sqrt {c}\, a^{2} x^{2}-38 \sqrt {c}\, a x +8 \sqrt {c}\right )}{24 \left (a x -1\right )^{2} \sqrt {c}\, x^{3}}\)	\(114\)
risch	\(\frac {\left (165 a^{3} x^{3}+119 a^{2} x^{2}-38 a x +8\right ) c \sqrt {\frac {a x -1}{a x +1}}}{24 x^{3} \sqrt {-c \left (a x -1\right )}}-\frac {\left (-\frac {8 a^{3}}{\sqrt {-a c x -c}}-\frac {119 a^{3} \arctan \left (\frac {\sqrt {-a c x -c}}{\sqrt {c}}\right )}{8 \sqrt {c}}\right ) c \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x +1\right )}}{\sqrt {-c \left (a x -1\right )}}\)	\(134\)

Input:

int((-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^4,x,method=_RETURNVERBOSE)
 

Output:

-1/24*((a*x-1)/(a*x+1))^(3/2)/(a*x-1)^2*(a*x+1)*(-c*(a*x-1))^(1/2)*(357*ar 
ctan((-c*(a*x+1))^(1/2)/c^(1/2))*a^3*x^3*(-c*(a*x+1))^(1/2)+357*a^3*x^3*c^ 
(1/2)+119*c^(1/2)*a^2*x^2-38*c^(1/2)*a*x+8*c^(1/2))/c^(1/2)/x^3
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.19 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\left [\frac {357 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + a c x - 2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 2 \, c}{a x^{2} - x}\right ) - 2 \, {\left (357 \, a^{3} x^{3} + 119 \, a^{2} x^{2} - 38 \, a x + 8\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{48 \, {\left (a x^{4} - x^{3}\right )}}, \frac {357 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) - {\left (357 \, a^{3} x^{3} + 119 \, a^{2} x^{2} - 38 \, a x + 8\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{24 \, {\left (a x^{4} - x^{3}\right )}}\right ] \] Input:

integrate((-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^4,x, algorithm="frica 
s")
 

Output:

[1/48*(357*(a^4*x^4 - a^3*x^3)*sqrt(-c)*log(-(a^2*c*x^2 + a*c*x - 2*sqrt(- 
a*c*x + c)*(a*x + 1)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1)) - 2*c)/(a*x^2 - x) 
) - 2*(357*a^3*x^3 + 119*a^2*x^2 - 38*a*x + 8)*sqrt(-a*c*x + c)*sqrt((a*x 
- 1)/(a*x + 1)))/(a*x^4 - x^3), 1/24*(357*(a^4*x^4 - a^3*x^3)*sqrt(c)*arct 
an(sqrt(-a*c*x + c)*(a*x + 1)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))/(a*c*x - c 
)) - (357*a^3*x^3 + 119*a^2*x^2 - 38*a*x + 8)*sqrt(-a*c*x + c)*sqrt((a*x - 
 1)/(a*x + 1)))/(a*x^4 - x^3)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\text {Timed out} \] Input:

integrate((-a*c*x+c)**(1/2)*((a*x-1)/(a*x+1))**(3/2)/x**4,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\int { \frac {\sqrt {-a c x + c} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x^{4}} \,d x } \] Input:

integrate((-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^4,x, algorithm="maxim 
a")
 

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^4,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. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\int \frac {\sqrt {c-a\,c\,x}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{x^4} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.41 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\frac {\sqrt {c}\, i \left (357 \sqrt {a x +1}\, \mathrm {log}\left (\frac {2 \sqrt {a x +1}-2}{\sqrt {2}}\right ) a^{3} x^{3}-357 \sqrt {a x +1}\, \mathrm {log}\left (\frac {2 \sqrt {a x +1}+2}{\sqrt {2}}\right ) a^{3} x^{3}+714 a^{3} x^{3}+238 a^{2} x^{2}-76 a x +16\right )}{48 \sqrt {a x +1}\, x^{3}} \] Input:

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

Output:

(sqrt(c)*i*(357*sqrt(a*x + 1)*log((2*sqrt(a*x + 1) - 2)/sqrt(2))*a**3*x**3 
 - 357*sqrt(a*x + 1)*log((2*sqrt(a*x + 1) + 2)/sqrt(2))*a**3*x**3 + 714*a* 
*3*x**3 + 238*a**2*x**2 - 76*a*x + 16))/(48*sqrt(a*x + 1)*x**3)