3.8.34 \(\int e^{-2 \coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx\) [734]

3.8.34.1 Optimal result

Integrand size = 27, antiderivative size = 172 \[ \int e^{-2 \coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\frac {x^{1+m} \sqrt {c-a^2 c x^2}}{2+m}-\frac {c (3+2 m) x^{1+m} \sqrt {1-a^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},a^2 x^2\right )}{(1+m) (2+m) \sqrt {c-a^2 c x^2}}+\frac {2 a c x^{2+m} \sqrt {1-a^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},a^2 x^2\right )}{(2+m) \sqrt {c-a^2 c x^2}} \]

output

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

3.8.34.2 Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 0.26 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.64 \[ \int e^{-2 \coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx=\frac {x^{1+m} \left (-\frac {2 \sqrt {c-a c x} \operatorname {AppellF1}\left (1+m,\frac {1}{2},-\frac {1}{2},2+m,-a x,a x\right )}{\sqrt {1-a x}}+\frac {\sqrt {c-a^2 c x^2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},a^2 x^2\right )}{\sqrt {1-a^2 x^2}}\right )}{1+m} \]

input

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

output

(x^(1 + m)*((-2*Sqrt[c - a*c*x]*AppellF1[1 + m, 1/2, -1/2, 2 + m, -(a*x), 
a*x])/Sqrt[1 - a*x] + (Sqrt[c - a^2*c*x^2]*Hypergeometric2F1[-1/2, (1 + m) 
/2, (3 + m)/2, a^2*x^2])/Sqrt[1 - a^2*x^2]))/(1 + m)
 

3.8.34.3 Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {6717, 6702, 559, 25, 27, 557, 279, 278}

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.

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

\(\Big \downarrow \) 6717

\(\displaystyle -\int e^{-2 \text {arctanh}(a x)} x^m \sqrt {c-a^2 c x^2}dx\)

\(\Big \downarrow \) 6702

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

\(\Big \downarrow \) 559

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 557

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

\(\Big \downarrow \) 279

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

\(\Big \downarrow \) 278

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

input

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

output

-(c*(-((x^(1 + m)*Sqrt[c - a^2*c*x^2])/(c*(2 + m))) + (((3 + 2*m)*x^(1 + m 
)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, a^2*x^2]) 
/((1 + m)*Sqrt[c - a^2*c*x^2]) - (2*a*x^(2 + m)*Sqrt[1 - a^2*x^2]*Hypergeo 
metric2F1[1/2, (2 + m)/2, (4 + m)/2, a^2*x^2])/Sqrt[c - a^2*c*x^2])/(2 + m 
)))
 

3.8.34.3.1 Defintions of rubi rules used

rule 25

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

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 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

rule 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntP 
art[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p])   Int[(c*x)^m* 
(1 + b*(x^2/a))^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && 
!(ILtQ[p, 0] || GtQ[a, 0])
 

rule 557

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym 
bol] :> Simp[c   Int[(e*x)^m*(a + b*x^2)^p, x], x] + Simp[d/e   Int[(e*x)^( 
m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x]
 

rule 559

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

rule 6702

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_ 
Symbol] :> Simp[1/c^(n/2)   Int[x^m*((c + d*x^2)^(p + n/2)/(1 - a*x)^n), x] 
, x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] &&  !(IntegerQ[p] || 
 GtQ[c, 0]) && ILtQ[n/2, 0]
 

rule 6717

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Simp[(-1)^(n/2)   Int[ 
u*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[a, x] && IntegerQ[n/2]
 

3.8.34.4 Maple [F]

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

input

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

output

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

3.8.34.5 Fricas [F]

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

input

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

output

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

3.8.34.6 Sympy [F]

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

input

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

output

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

3.8.34.7 Maxima [F]

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

input

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

output

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

3.8.34.8 Giac [F(-2)]

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

input

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

3.8.34.9 Mupad [F(-1)]

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

input

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

output

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