\(\int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx\) [503]

Optimal result

Integrand size = 25, antiderivative size = 164 \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=-\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x}{8 a^2 \sqrt {c-\frac {c}{a x}}}+\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^2}{12 a \sqrt {c-\frac {c}{a x}}}+\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^3}{3 \sqrt {c-\frac {c}{a x}}}+\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{8 a^3} \] Output:

-1/8*c*(1-1/a^2/x^2)^(1/2)*x/a^2/(c-c/a/x)^(1/2)+1/12*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)+1/8* 
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 = 0.93 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.90 \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\frac {\frac {2 a^2 \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} x^2 \left (-3+2 a x+8 a^2 x^2\right )}{-1+a x}-3 \sqrt {c} \log (1-a x)+3 \sqrt {c} \log \left (2 a^2 \sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} x^2+c \left (-1-a x+2 a^2 x^2\right )\right )}{48 a^3} \] Input:

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

Output:

((2*a^2*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2*(-3 + 2*a*x + 8*a^2*x^ 
2))/(-1 + a*x) - 3*Sqrt[c]*Log[1 - a*x] + 3*Sqrt[c]*Log[2*a^2*Sqrt[c]*Sqrt 
[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2 + c*(-1 - a*x + 2*a^2*x^2)])/(48*a 
^3)
 

Rubi [A] (verified)

Time = 0.88 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {6733, 575, 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

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

\(\Big \downarrow \) 575

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

\(\Big \downarrow \) 579

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

\(\Big \downarrow \) 579

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

\(\Big \downarrow \) 573

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

-(c*(-1/3*(Sqrt[1 - 1/(a^2*x^2)]*x^3)/Sqrt[c - c/(a*x)] + (-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 575

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), 
x_Symbol] :> Simp[(e*x)^(m + 1)*(c + d*x)^n*((a + b*x^2)^p/(e*(m + 1))), x] 
 + Simp[b*(n/(d*e*(m + 1)))   Int[(e*x)^(m + 1)*(c + d*x)^(n + 1)*(a + b*x^ 
2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[b*c^2 + a*d^2, 0] && 
 EqQ[n + p, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !(IntegerQ[m + p] && LeQ[m + 
p + 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 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 = 0.07 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.74

method	result	size
default	\(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (16 a^{\frac {5}{2}} x^{2} \sqrt {x \left (a x +1\right )}+4 a^{\frac {3}{2}} x \sqrt {x \left (a x +1\right )}-6 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+3 \ln \left (\frac {2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right )\right )}{48 \sqrt {\frac {a x -1}{a x +1}}\, a^{\frac {5}{2}} \sqrt {x \left (a x +1\right )}}\)	\(121\)
risch	\(\frac {\left (8 a^{2} x^{2}+2 a x -3\right ) x \sqrt {\frac {c \left (a x -1\right )}{a x}}}{24 a^{2} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\left (a x +1\right ) a c x}}{16 a^{2} \sqrt {a^{2} c}\, \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\)	\(148\)

Input:

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

Output:

1/48/((a*x-1)/(a*x+1))^(1/2)*(c*(a*x-1)/a/x)^(1/2)*x/a^(5/2)*(16*a^(5/2)*x 
^2*(x*(a*x+1))^(1/2)+4*a^(3/2)*x*(x*(a*x+1))^(1/2)-6*(x*(a*x+1))^(1/2)*a^( 
1/2)+3*ln(1/2*(2*(x*(a*x+1))^(1/2)*a^(1/2)+2*a*x+1)/a^(1/2)))/(x*(a*x+1))^ 
(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 337, normalized size of antiderivative = 2.05 \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\left [\frac {3 \, {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (8 \, a^{4} x^{4} + 10 \, a^{3} x^{3} - a^{2} x^{2} - 3 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{96 \, {\left (a^{4} x - a^{3}\right )}}, -\frac {3 \, {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \, {\left (8 \, a^{4} x^{4} + 10 \, a^{3} x^{3} - a^{2} x^{2} - 3 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{48 \, {\left (a^{4} x - a^{3}\right )}}\right ] \] Input:

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

Output:

[1/96*(3*(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 + 10*a^3*x^3 - a^2*x^2 - 3*a*x)*sqrt((a*x - 
1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^4*x - a^3), -1/48*(3*(a*x - 1)*s 
qrt(-c)*arctan(2*(a^2*x^2 + a*x)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1))*sqrt(( 
a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*(8*a^4*x^4 + 10*a^3*x^3 - 
 a^2*x^2 - 3*a*x)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^4* 
x - a^3)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\int \frac {x^2\,\sqrt {c-\frac {c}{a\,x}}}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \] 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 \[ \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\frac {\sqrt {c}\, \left (8 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{2} x^{2}+2 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a x -3 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}+3 \,\mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}\right )\right )}{24 a^{3}} \] Input:

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

Output:

(sqrt(c)*(8*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a**2*x**2 + 2*sqrt(x)*sqrt(a)*sq 
rt(a*x + 1)*a*x - 3*sqrt(x)*sqrt(a)*sqrt(a*x + 1) + 3*log(sqrt(a*x + 1) + 
sqrt(x)*sqrt(a))))/(24*a**3)