\(\int e^{-3 \coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx\) [717]

Optimal result

Integrand size = 27, antiderivative size = 137 \[ \int e^{-3 \coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=-\frac {3 x^m \sqrt {c-a^2 c x^2}}{a (1+m) \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x^{1+m} \sqrt {c-a^2 c x^2}}{(2+m) \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 x^m \sqrt {c-a^2 c x^2} \operatorname {Hypergeometric2F1}(1,1+m,2+m,-a x)}{a (1+m) \sqrt {1-\frac {1}{a^2 x^2}}} \] 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 = 75, normalized size of antiderivative = 0.55 \[ \int e^{-3 \coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\frac {x^m \sqrt {c-a^2 c x^2} (-6+a x+m (-3+a x)+4 (2+m) \operatorname {Hypergeometric2F1}(1,1+m,2+m,-a x))}{a (1+m) (2+m) \sqrt {1-\frac {1}{a^2 x^2}}} \] Input:

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

Output:

(x^m*Sqrt[c - a^2*c*x^2]*(-6 + a*x + m*(-3 + a*x) + 4*(2 + m)*Hypergeometr 
ic2F1[1, 1 + m, 2 + m, -(a*x)]))/(a*(1 + m)*(2 + m)*Sqrt[1 - 1/(a^2*x^2)])
 

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.62, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6746, 6747, 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

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

\(\Big \downarrow \) 99

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

\(\Big \downarrow \) 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[(x^m*Sqrt[c - a^2*c*x^2])/E^(3*ArcCoth[a*x]),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 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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

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

Output:

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