\(\int e^{3 \coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx\) [712]

Optimal result

Integrand size = 27, antiderivative size = 136 \[ \int e^{3 \coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\frac {3 x^m \sqrt {c-a^2 c x^2}}{a (1+m) \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x^{1+m} \sqrt {c-a^2 c x^2}}{(2+m) \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {4 x^m \sqrt {c-a^2 c x^2} \operatorname {Hypergeometric2F1}(1,1+m,2+m,a x)}{a (1+m) \sqrt {1-\frac {1}{a^2 x^2}}} \] Output:

3*x^m*(-a^2*c*x^2+c)^(1/2)/a/(1+m)/(1-1/a^2/x^2)^(1/2)+x^(1+m)*(-a^2*c*x^2 
+c)^(1/2)/(2+m)/(1-1/a^2/x^2)^(1/2)-4*x^m*(-a^2*c*x^2+c)^(1/2)*hypergeom([ 
1, 1+m],[2+m],a*x)/a/(1+m)/(1-1/a^2/x^2)^(1/2)
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.54 \[ \int e^{3 \coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\frac {x^m \sqrt {c-a^2 c x^2} (6+a x+m (3+a x)-4 (2+m) \operatorname {Hypergeometric2F1}(1,1+m,2+m,a x))}{a (1+m) (2+m) \sqrt {1-\frac {1}{a^2 x^2}}} \] Input:

Integrate[E^(3*ArcCoth[a*x])*x^m*Sqrt[c - a^2*c*x^2],x]
 

Output:

(x^m*Sqrt[c - a^2*c*x^2]*(6 + a*x + m*(3 + a*x) - 4*(2 + m)*Hypergeometric 
2F1[1, 1 + m, 2 + m, a*x]))/(a*(1 + m)*(2 + m)*Sqrt[1 - 1/(a^2*x^2)])
 

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.63, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6746, 6747, 25, 99, 2009}

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.

\(\displaystyle \int x^m \sqrt {c-a^2 c x^2} e^{3 \coth ^{-1}(a x)} \, dx\)

\(\Big \downarrow \) 6746

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

\(\Big \downarrow \) 6747

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 99

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\sqrt {c-a^2 c x^2} \left (\frac {4 x^{m+1} \operatorname {Hypergeometric2F1}(1,m+1,m+2,a x)}{m+1}-\frac {a x^{m+2}}{m+2}-\frac {3 x^{m+1}}{m+1}\right )}{a x \sqrt {1-\frac {1}{a^2 x^2}}}\)

Input:

Int[E^(3*ArcCoth[a*x])*x^m*Sqrt[c - a^2*c*x^2],x]
 

Output:

-((Sqrt[c - a^2*c*x^2]*((-3*x^(1 + m))/(1 + m) - (a*x^(2 + m))/(2 + m) + ( 
4*x^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, a*x])/(1 + m)))/(a*Sqrt[1 - 
 1/(a^2*x^2)]*x))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | 
| (GtQ[m, 0] && GeQ[n, -1]))
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6746

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbo 
l] :> Simp[(c + d*x^2)^p/(x^(2*p)*(1 - 1/(a^2*x^2))^p)   Int[u*x^(2*p)*(1 - 
 1/(a^2*x^2))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && 
EqQ[a^2*c + d, 0] &&  !IntegerQ[n/2] &&  !IntegerQ[p]
 

rule 6747

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_.), x_Symb 
ol] :> Simp[c^p/a^(2*p)   Int[(u/x^(2*p))*(-1 + a*x)^(p - n/2)*(1 + a*x)^(p 
 + n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] &&  !Inte 
gerQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) && IntegersQ[2*p, p + n/2]
 

Maple [F]

\[\int \frac {x^{m} \sqrt {-a^{2} c \,x^{2}+c}}{\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

Timed out. \[ \int e^{3 \coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int e^{3 \coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\text {Exception raised: TypeError} \] Input:

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

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

Output:

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

Reduce [F]

\[ \int e^{3 \coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\frac {\sqrt {c}\, \left (x^{m} \sqrt {a x -1}\, \sqrt {-a x +1}\, a^{2} m^{2} x^{2}+x^{m} \sqrt {a x -1}\, \sqrt {-a x +1}\, a^{2} m \,x^{2}+3 x^{m} \sqrt {a x -1}\, \sqrt {-a x +1}\, a \,m^{2} x +6 x^{m} \sqrt {a x -1}\, \sqrt {-a x +1}\, a m x +4 x^{m} \sqrt {a x -1}\, \sqrt {-a x +1}\, m^{2}+12 x^{m} \sqrt {a x -1}\, \sqrt {-a x +1}\, m +8 x^{m} \sqrt {a x -1}\, \sqrt {-a x +1}+4 \left (\int \frac {x^{m} \sqrt {a x -1}\, \sqrt {-a x +1}}{a^{2} x^{3}-2 a \,x^{2}+x}d x \right ) a \,m^{3} x +12 \left (\int \frac {x^{m} \sqrt {a x -1}\, \sqrt {-a x +1}}{a^{2} x^{3}-2 a \,x^{2}+x}d x \right ) a \,m^{2} x +8 \left (\int \frac {x^{m} \sqrt {a x -1}\, \sqrt {-a x +1}}{a^{2} x^{3}-2 a \,x^{2}+x}d x \right ) a m x -4 \left (\int \frac {x^{m} \sqrt {a x -1}\, \sqrt {-a x +1}}{a^{2} x^{3}-2 a \,x^{2}+x}d x \right ) m^{3}-12 \left (\int \frac {x^{m} \sqrt {a x -1}\, \sqrt {-a x +1}}{a^{2} x^{3}-2 a \,x^{2}+x}d x \right ) m^{2}-8 \left (\int \frac {x^{m} \sqrt {a x -1}\, \sqrt {-a x +1}}{a^{2} x^{3}-2 a \,x^{2}+x}d x \right ) m \right )}{a m \left (a \,m^{2} x +3 a m x +2 a x -m^{2}-3 m -2\right )} \] Input:

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

Output:

(sqrt(c)*(x**m*sqrt(a*x - 1)*sqrt( - a*x + 1)*a**2*m**2*x**2 + x**m*sqrt(a 
*x - 1)*sqrt( - a*x + 1)*a**2*m*x**2 + 3*x**m*sqrt(a*x - 1)*sqrt( - a*x + 
1)*a*m**2*x + 6*x**m*sqrt(a*x - 1)*sqrt( - a*x + 1)*a*m*x + 4*x**m*sqrt(a* 
x - 1)*sqrt( - a*x + 1)*m**2 + 12*x**m*sqrt(a*x - 1)*sqrt( - a*x + 1)*m + 
8*x**m*sqrt(a*x - 1)*sqrt( - a*x + 1) + 4*int((x**m*sqrt(a*x - 1)*sqrt( - 
a*x + 1))/(a**2*x**3 - 2*a*x**2 + x),x)*a*m**3*x + 12*int((x**m*sqrt(a*x - 
 1)*sqrt( - a*x + 1))/(a**2*x**3 - 2*a*x**2 + x),x)*a*m**2*x + 8*int((x**m 
*sqrt(a*x - 1)*sqrt( - a*x + 1))/(a**2*x**3 - 2*a*x**2 + x),x)*a*m*x - 4*i 
nt((x**m*sqrt(a*x - 1)*sqrt( - a*x + 1))/(a**2*x**3 - 2*a*x**2 + x),x)*m** 
3 - 12*int((x**m*sqrt(a*x - 1)*sqrt( - a*x + 1))/(a**2*x**3 - 2*a*x**2 + x 
),x)*m**2 - 8*int((x**m*sqrt(a*x - 1)*sqrt( - a*x + 1))/(a**2*x**3 - 2*a*x 
**2 + x),x)*m))/(a*m*(a*m**2*x + 3*a*m*x + 2*a*x - m**2 - 3*m - 2))