\(\int e^{-\coth ^{-1}(a x)} (c x)^m \, dx\) [146]

Optimal result

Integrand size = 14, antiderivative size = 78 \[ \int e^{-\coth ^{-1}(a x)} (c x)^m \, dx=\frac {x (c x)^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-1-m),\frac {1-m}{2},\frac {1}{a^2 x^2}\right )}{1+m}-\frac {(c x)^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {m}{2},1-\frac {m}{2},\frac {1}{a^2 x^2}\right )}{a m} \] Output:

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

Mathematica [C] (warning: unable to verify)

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

Time = 0.30 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.49 \[ \int e^{-\coth ^{-1}(a x)} (c x)^m \, dx=x (c x)^m \left (-\frac {\sqrt {1-\frac {1}{a^2 x^2}} \sqrt {\frac {-1+a x}{a^2}} \operatorname {AppellF1}\left (m,-\frac {1}{2},\frac {1}{2},1+m,a x,-a x\right )}{m \sqrt {1-a x} \sqrt {-\frac {1}{a^2}+x^2}}+\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {1}{2}-\frac {m}{2},\frac {1}{2}-\frac {m}{2},\frac {1}{a^2 x^2}\right )}{1+m}\right ) \] Input:

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

Output:

x*(c*x)^m*(-((Sqrt[1 - 1/(a^2*x^2)]*Sqrt[(-1 + a*x)/a^2]*AppellF1[m, -1/2, 
 1/2, 1 + m, a*x, -(a*x)])/(m*Sqrt[1 - a*x]*Sqrt[-a^(-2) + x^2])) + Hyperg 
eometric2F1[-1/2, -1/2 - m/2, 1/2 - m/2, 1/(a^2*x^2)]/(1 + m))
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.23, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6722, 27, 557, 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.

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

\(\Big \downarrow \) 6722

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 557

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

\(\Big \downarrow \) 278

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

Input:

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

Output:

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

Defintions of rubi rules used

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 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 6722

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_.)*(x_))^(m_), x_Symbol] :> Simp[(-(c 
*x)^m)*(1/x)^m   Subst[Int[(1 + x/a)^((n + 1)/2)/(x^(m + 2)*(1 - x/a)^((n - 
 1)/2)*Sqrt[1 - x^2/a^2]), x], x, 1/x], x] /; FreeQ[{a, c, m}, x] && Intege 
rQ[(n - 1)/2] &&  !IntegerQ[m]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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