\(\int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx\) [529]

Optimal result

Integrand size = 27, antiderivative size = 164 \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\frac {11 c \sqrt {1-\frac {1}{a^2 x^2}} x}{8 a^2 \sqrt {c-\frac {c}{a x}}}-\frac {11 c \sqrt {1-\frac {1}{a^2 x^2}} x^2}{12 a \sqrt {c-\frac {c}{a x}}}+\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^3}{3 \sqrt {c-\frac {c}{a x}}}-\frac {11 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{8 a^3} \] Output:

11/8*c*(1-1/a^2/x^2)^(1/2)*x/a^2/(c-c/a/x)^(1/2)-11/12*c*(1-1/a^2/x^2)^(1/ 
2)*x^2/a/(c-c/a/x)^(1/2)+1/3*c*(1-1/a^2/x^2)^(1/2)*x^3/(c-c/a/x)^(1/2)-11/ 
8*c^(1/2)*arctanh(c^(1/2)*(1-1/a^2/x^2)^(1/2)/(c-c/a/x)^(1/2))/a^3
 

Mathematica [A] (verified)

Time = 0.95 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.90 \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\frac {\frac {2 a^2 \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} x^2 \left (33-22 a x+8 a^2 x^2\right )}{-1+a x}+33 \sqrt {c} \log (1-a x)-33 \sqrt {c} \log \left (2 a^2 \sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} x^2+c \left (-1-a x+2 a^2 x^2\right )\right )}{48 a^3} \] Input:

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

Output:

((2*a^2*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2*(33 - 22*a*x + 8*a^2*x 
^2))/(-1 + a*x) + 33*Sqrt[c]*Log[1 - a*x] - 33*Sqrt[c]*Log[2*a^2*Sqrt[c]*S 
qrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2 + c*(-1 - a*x + 2*a^2*x^2)])/(4 
8*a^3)
 

Rubi [A] (verified)

Time = 0.90 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6733, 580, 579, 579, 573, 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.

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

\(\Big \downarrow \) 6733

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

\(\Big \downarrow \) 580

\(\displaystyle -\frac {-\frac {11 c \int \frac {\sqrt {c-\frac {c}{a x}} x^3}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}}{6 a}-\frac {c^2 x^3 \sqrt {1-\frac {1}{a^2 x^2}}}{3 \sqrt {c-\frac {c}{a x}}}}{c}\)

\(\Big \downarrow \) 579

\(\displaystyle -\frac {-\frac {11 c \left (-\frac {3 \int \frac {\sqrt {c-\frac {c}{a x}} x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}}{4 a}-\frac {c x^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 \sqrt {c-\frac {c}{a x}}}\right )}{6 a}-\frac {c^2 x^3 \sqrt {1-\frac {1}{a^2 x^2}}}{3 \sqrt {c-\frac {c}{a x}}}}{c}\)

\(\Big \downarrow \) 579

\(\displaystyle -\frac {-\frac {11 c \left (-\frac {3 \left (-\frac {\int \frac {\sqrt {c-\frac {c}{a x}} x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}}{2 a}-\frac {c x \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{4 a}-\frac {c x^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 \sqrt {c-\frac {c}{a x}}}\right )}{6 a}-\frac {c^2 x^3 \sqrt {1-\frac {1}{a^2 x^2}}}{3 \sqrt {c-\frac {c}{a x}}}}{c}\)

\(\Big \downarrow \) 573

\(\displaystyle -\frac {-\frac {11 c \left (-\frac {3 \left (\frac {c \int \frac {1}{1-\frac {c}{x^2}}d\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}}{a}-\frac {c x \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{4 a}-\frac {c x^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 \sqrt {c-\frac {c}{a x}}}\right )}{6 a}-\frac {c^2 x^3 \sqrt {1-\frac {1}{a^2 x^2}}}{3 \sqrt {c-\frac {c}{a x}}}}{c}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {-\frac {11 c \left (-\frac {3 \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a}-\frac {c x \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{4 a}-\frac {c x^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 \sqrt {c-\frac {c}{a x}}}\right )}{6 a}-\frac {c^2 x^3 \sqrt {1-\frac {1}{a^2 x^2}}}{3 \sqrt {c-\frac {c}{a x}}}}{c}\)

Input:

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

Output:

-((-1/3*(c^2*Sqrt[1 - 1/(a^2*x^2)]*x^3)/Sqrt[c - c/(a*x)] - (11*c*(-1/2*(c 
*Sqrt[1 - 1/(a^2*x^2)]*x^2)/Sqrt[c - c/(a*x)] - (3*(-((c*Sqrt[1 - 1/(a^2*x 
^2)]*x)/Sqrt[c - c/(a*x)]) + (Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[1 - 1/(a^2*x^2 
)])/Sqrt[c - c/(a*x)]])/a))/(4*a)))/(6*a))/c)
 

Defintions of rubi rules used

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 573

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

rule 579

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

rule 580

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

rule 6733

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

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.81

method	result	size
default	\(\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (16 a^{\frac {5}{2}} x^{2} \sqrt {x \left (a x +1\right )}-44 a^{\frac {3}{2}} x \sqrt {x \left (a x +1\right )}+66 \sqrt {x \left (a x +1\right )}\, \sqrt {a}-33 \ln \left (\frac {2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right )\right )}{48 a^{\frac {5}{2}} \left (a x -1\right ) \sqrt {x \left (a x +1\right )}}\)	\(133\)
risch	\(\frac {\left (8 a^{2} x^{2}-22 a x +33\right ) \left (a x +1\right ) x \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}{24 a^{2} \left (a x -1\right )}-\frac {11 \ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\left (a x +1\right ) a c x}}{16 a^{2} \sqrt {a^{2} c}\, \left (a x -1\right )}\)	\(160\)

Input:

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

Output:

1/48*((a*x-1)/(a*x+1))^(1/2)*(a*x+1)*(c*(a*x-1)/a/x)^(1/2)*x*(16*a^(5/2)*x 
^2*(x*(a*x+1))^(1/2)-44*a^(3/2)*x*(x*(a*x+1))^(1/2)+66*(x*(a*x+1))^(1/2)*a 
^(1/2)-33*ln(1/2*(2*(x*(a*x+1))^(1/2)*a^(1/2)+2*a*x+1)/a^(1/2)))/a^(5/2)/( 
a*x-1)/(x*(a*x+1))^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 337, normalized size of antiderivative = 2.05 \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\left [\frac {33 \, {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (8 \, a^{4} x^{4} - 14 \, a^{3} x^{3} + 11 \, a^{2} x^{2} + 33 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{96 \, {\left (a^{4} x - a^{3}\right )}}, \frac {33 \, {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (8 \, a^{4} x^{4} - 14 \, a^{3} x^{3} + 11 \, a^{2} x^{2} + 33 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{48 \, {\left (a^{4} x - a^{3}\right )}}\right ] \] Input:

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

Output:

[1/96*(33*(a*x - 1)*sqrt(c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^3*x^3 + 3 
*a^2*x^2 + a*x)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)) 
- c)/(a*x - 1)) + 4*(8*a^4*x^4 - 14*a^3*x^3 + 11*a^2*x^2 + 33*a*x)*sqrt((a 
*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^4*x - a^3), 1/48*(33*(a*x - 
 1)*sqrt(-c)*arctan(2*(a^2*x^2 + a*x)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1))*s 
qrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) + 2*(8*a^4*x^4 - 14*a^3* 
x^3 + 11*a^2*x^2 + 33*a*x)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x 
)))/(a^4*x - a^3)]
 

Sympy [F(-1)]

Timed out. \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\text {Exception raised: TypeError} \] Input:

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.41 \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\frac {\sqrt {c}\, \left (8 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{2} x^{2}-22 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a x +33 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}-33 \,\mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}\right )\right )}{24 a^{3}} \] Input:

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

Output:

(sqrt(c)*(8*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a**2*x**2 - 22*sqrt(x)*sqrt(a)*s 
qrt(a*x + 1)*a*x + 33*sqrt(x)*sqrt(a)*sqrt(a*x + 1) - 33*log(sqrt(a*x + 1) 
 + sqrt(x)*sqrt(a))))/(24*a**3)