\(\int e^{-3 \coth ^{-1}(a x)} x^3 \sqrt {c-a c x} \, dx\) [372]

Optimal result

Integrand size = 23, antiderivative size = 281 \[ \int e^{-3 \coth ^{-1}(a x)} x^3 \sqrt {c-a c x} \, dx=\frac {1312 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{45 a^4 \sqrt {1-\frac {1}{a x}}}-\frac {656 \sqrt {1+\frac {1}{a x}} x \sqrt {c-a c x}}{45 a^3 \sqrt {1-\frac {1}{a x}}}-\frac {82 x^2 \sqrt {c-a c x}}{9 a^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {164 \sqrt {1+\frac {1}{a x}} x^2 \sqrt {c-a c x}}{15 a^2 \sqrt {1-\frac {1}{a x}}}-\frac {8 x^3 \sqrt {c-a c x}}{9 a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {2 x^4 \sqrt {c-a c x}}{9 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}} \] Output:

1312/45*(1+1/a/x)^(1/2)*(-a*c*x+c)^(1/2)/a^4/(1-1/a/x)^(1/2)-656/45*(1+1/a 
/x)^(1/2)*x*(-a*c*x+c)^(1/2)/a^3/(1-1/a/x)^(1/2)-82/9*x^2*(-a*c*x+c)^(1/2) 
/a^2/(1-1/a/x)^(1/2)/(1+1/a/x)^(1/2)+164/15*(1+1/a/x)^(1/2)*x^2*(-a*c*x+c) 
^(1/2)/a^2/(1-1/a/x)^(1/2)-8/9*x^3*(-a*c*x+c)^(1/2)/a/(1-1/a/x)^(1/2)/(1+1 
/a/x)^(1/2)+2/9*x^4*(-a*c*x+c)^(1/2)/(1-1/a/x)^(1/2)/(1+1/a/x)^(1/2)
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.26 \[ \int e^{-3 \coth ^{-1}(a x)} x^3 \sqrt {c-a c x} \, dx=\frac {2 \sqrt {c-a c x} \left (656+328 a x-82 a^2 x^2+41 a^3 x^3-20 a^4 x^4+5 a^5 x^5\right )}{45 a^5 \sqrt {1-\frac {1}{a^2 x^2}} x} \] Input:

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

Output:

(2*Sqrt[c - a*c*x]*(656 + 328*a*x - 82*a^2*x^2 + 41*a^3*x^3 - 20*a^4*x^4 + 
 5*a^5*x^5))/(45*a^5*Sqrt[1 - 1/(a^2*x^2)]*x)
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.73, 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, 87, 55, 55, 55, 48}

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.

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

\(\Big \downarrow \) 6730

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 100

\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {2}{9} \int -\frac {28 a-\frac {9}{x}}{2 \left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{9/2}}d\frac {1}{x}-\frac {2 a^2}{9 \left (\frac {1}{x}\right )^{9/2} \sqrt {\frac {1}{a x}+1}}\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (-\frac {1}{9} \int \frac {28 a-\frac {9}{x}}{\left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{9/2}}d\frac {1}{x}-\frac {2 a^2}{9 \left (\frac {1}{x}\right )^{9/2} \sqrt {\frac {1}{a x}+1}}\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 87

\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {1}{9} \left (41 \int \frac {1}{\left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{7/2}}d\frac {1}{x}+\frac {8 a}{\left (\frac {1}{x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 a^2}{9 \left (\frac {1}{x}\right )^{9/2} \sqrt {\frac {1}{a x}+1}}\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 55

\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {1}{9} \left (41 \left (6 \int \frac {1}{\sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{7/2}}d\frac {1}{x}+\frac {2}{\left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )+\frac {8 a}{\left (\frac {1}{x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 a^2}{9 \left (\frac {1}{x}\right )^{9/2} \sqrt {\frac {1}{a x}+1}}\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 55

\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {1}{9} \left (41 \left (6 \left (-\frac {4 \int \frac {1}{\sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{5/2}}d\frac {1}{x}}{5 a}-\frac {2 \sqrt {\frac {1}{a x}+1}}{5 \left (\frac {1}{x}\right )^{5/2}}\right )+\frac {2}{\left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )+\frac {8 a}{\left (\frac {1}{x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 a^2}{9 \left (\frac {1}{x}\right )^{9/2} \sqrt {\frac {1}{a x}+1}}\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 55

\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {1}{9} \left (41 \left (6 \left (-\frac {4 \left (-\frac {2 \int \frac {1}{\sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{3/2}}d\frac {1}{x}}{3 a}-\frac {2 \sqrt {\frac {1}{a x}+1}}{3 \left (\frac {1}{x}\right )^{3/2}}\right )}{5 a}-\frac {2 \sqrt {\frac {1}{a x}+1}}{5 \left (\frac {1}{x}\right )^{5/2}}\right )+\frac {2}{\left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )+\frac {8 a}{\left (\frac {1}{x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 a^2}{9 \left (\frac {1}{x}\right )^{9/2} \sqrt {\frac {1}{a x}+1}}\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 48

\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \left (\frac {1}{9} \left (\frac {8 a}{\left (\frac {1}{x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}+41 \left (6 \left (-\frac {4 \left (\frac {4 \sqrt {\frac {1}{a x}+1}}{3 a \sqrt {\frac {1}{x}}}-\frac {2 \sqrt {\frac {1}{a x}+1}}{3 \left (\frac {1}{x}\right )^{3/2}}\right )}{5 a}-\frac {2 \sqrt {\frac {1}{a x}+1}}{5 \left (\frac {1}{x}\right )^{5/2}}\right )+\frac {2}{\left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )\right )-\frac {2 a^2}{9 \left (\frac {1}{x}\right )^{9/2} \sqrt {\frac {1}{a x}+1}}\right ) \sqrt {c-a c x}}{a^2 \sqrt {1-\frac {1}{a x}}}\)

Input:

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

Output:

-((((41*(6*((-4*((-2*Sqrt[1 + 1/(a*x)])/(3*(x^(-1))^(3/2)) + (4*Sqrt[1 + 1 
/(a*x)])/(3*a*Sqrt[x^(-1)])))/(5*a) - (2*Sqrt[1 + 1/(a*x)])/(5*(x^(-1))^(5 
/2))) + 2/(Sqrt[1 + 1/(a*x)]*(x^(-1))^(5/2))) + (8*a)/(Sqrt[1 + 1/(a*x)]*( 
x^(-1))^(7/2)))/9 - (2*a^2)/(9*Sqrt[1 + 1/(a*x)]*(x^(-1))^(9/2)))*Sqrt[x^( 
-1)]*Sqrt[c - a*c*x])/(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 48

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp 
[(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ 
a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
 

rule 55

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S 
implify[m + n + 2]/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^Simplify[m + 1]*( 
c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 
 2], 0] && NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ 
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] ||  !SumSimp 
lerQ[n, 1])
 

rule 87

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

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 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 = 0.17 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.28

method	result	size
gosper	\(\frac {2 \left (a x +1\right ) \left (5 a^{5} x^{5}-20 a^{4} x^{4}+41 a^{3} x^{3}-82 a^{2} x^{2}+328 a x +656\right ) \sqrt {-a c x +c}\, \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{45 a^{4} \left (a x -1\right )^{2}}\)	\(80\)
orering	\(\frac {2 \left (a x +1\right ) \left (5 a^{5} x^{5}-20 a^{4} x^{4}+41 a^{3} x^{3}-82 a^{2} x^{2}+328 a x +656\right ) \sqrt {-a c x +c}\, \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{45 a^{4} \left (a x -1\right )^{2}}\)	\(80\)
default	\(\frac {2 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, \left (5 a^{5} x^{5}-20 a^{4} x^{4}+41 a^{3} x^{3}-82 a^{2} x^{2}+328 a x +656\right )}{45 \left (a x -1\right )^{2} a^{4}}\)	\(81\)
risch	\(-\frac {2 \left (5 a^{4} x^{4}-25 a^{3} x^{3}+66 a^{2} x^{2}-148 a x +476\right ) \left (a x +1\right ) c \sqrt {\frac {a x -1}{a x +1}}}{45 a^{4} \sqrt {-c \left (a x -1\right )}}-\frac {8 c \sqrt {\frac {a x -1}{a x +1}}}{a^{4} \sqrt {-c \left (a x -1\right )}}\)	\(99\)

Input:

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

Output:

2/45*(a*x+1)*(5*a^5*x^5-20*a^4*x^4+41*a^3*x^3-82*a^2*x^2+328*a*x+656)*(-a* 
c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/a^4/(a*x-1)^2
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.27 \[ \int e^{-3 \coth ^{-1}(a x)} x^3 \sqrt {c-a c x} \, dx=\frac {2 \, {\left (5 \, a^{5} x^{5} - 20 \, a^{4} x^{4} + 41 \, a^{3} x^{3} - 82 \, a^{2} x^{2} + 328 \, a x + 656\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{45 \, {\left (a^{5} x - a^{4}\right )}} \] Input:

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

Output:

2/45*(5*a^5*x^5 - 20*a^4*x^4 + 41*a^3*x^3 - 82*a^2*x^2 + 328*a*x + 656)*sq 
rt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1))/(a^5*x - a^4)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.42 \[ \int e^{-3 \coth ^{-1}(a x)} x^3 \sqrt {c-a c x} \, dx=\frac {2 \, {\left (5 \, a^{6} \sqrt {-c} x^{6} - 15 \, a^{5} \sqrt {-c} x^{5} + 21 \, a^{4} \sqrt {-c} x^{4} - 41 \, a^{3} \sqrt {-c} x^{3} + 246 \, a^{2} \sqrt {-c} x^{2} + 984 \, a \sqrt {-c} x + 656 \, \sqrt {-c}\right )} {\left (a x - 1\right )}^{2}}{45 \, {\left (a^{6} x^{2} - 2 \, a^{5} x + a^{4}\right )} {\left (a x + 1\right )}^{\frac {3}{2}}} \] Input:

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

Output:

2/45*(5*a^6*sqrt(-c)*x^6 - 15*a^5*sqrt(-c)*x^5 + 21*a^4*sqrt(-c)*x^4 - 41* 
a^3*sqrt(-c)*x^3 + 246*a^2*sqrt(-c)*x^2 + 984*a*sqrt(-c)*x + 656*sqrt(-c)) 
*(a*x - 1)^2/((a^6*x^2 - 2*a^5*x + a^4)*(a*x + 1)^(3/2))
 

Giac [F(-2)]

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

integrate(x^3*(-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(3/2),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 [B] (verification not implemented)

Time = 13.72 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.26 \[ \int e^{-3 \coth ^{-1}(a x)} x^3 \sqrt {c-a c x} \, dx=\frac {2\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (5\,a^5\,x^5-20\,a^4\,x^4+41\,a^3\,x^3-82\,a^2\,x^2+328\,a\,x+656\right )}{45\,a^4\,\left (a\,x-1\right )} \] Input:

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

Output:

(2*(c - a*c*x)^(1/2)*((a*x - 1)/(a*x + 1))^(1/2)*(328*a*x - 82*a^2*x^2 + 4 
1*a^3*x^3 - 20*a^4*x^4 + 5*a^5*x^5 + 656))/(45*a^4*(a*x - 1))
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.19 \[ \int e^{-3 \coth ^{-1}(a x)} x^3 \sqrt {c-a c x} \, dx=\frac {2 \sqrt {c}\, i \left (-5 a^{5} x^{5}+20 a^{4} x^{4}-41 a^{3} x^{3}+82 a^{2} x^{2}-328 a x -656\right )}{45 \sqrt {a x +1}\, a^{4}} \] Input:

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

Output:

(2*sqrt(c)*i*( - 5*a**5*x**5 + 20*a**4*x**4 - 41*a**3*x**3 + 82*a**2*x**2 
- 328*a*x - 656))/(45*sqrt(a*x + 1)*a**4)