\(\int e^{-4 \text {arctanh}(a x)} (c x)^m \, dx\) [135]

Optimal result

Integrand size = 14, antiderivative size = 60 \[ \int e^{-4 \text {arctanh}(a x)} (c x)^m \, dx=\frac {(c x)^{1+m}}{c (1+m)}+\frac {4 (c x)^{1+m}}{c (1+a x)}-\frac {4 (c x)^{1+m} \operatorname {Hypergeometric2F1}(1,1+m,2+m,-a x)}{c} \] Output:

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.85 \[ \int e^{-4 \text {arctanh}(a x)} (c x)^m \, dx=-\frac {x (c x)^m (-5-4 m-a x+4 (1+m) (1+a x) \operatorname {Hypergeometric2F1}(1,1+m,2+m,-a x))}{(1+m) (1+a x)} \] Input:

Integrate[(c*x)^m/E^(4*ArcTanh[a*x]),x]
 

Output:

-((x*(c*x)^m*(-5 - 4*m - a*x + 4*(1 + m)*(1 + a*x)*Hypergeometric2F1[1, 1 
+ m, 2 + m, -(a*x)]))/((1 + m)*(1 + a*x)))
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6676, 100, 27, 90, 74}

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.

\(\displaystyle \int e^{-4 \text {arctanh}(a x)} (c x)^m \, dx\)

\(\Big \downarrow \) 6676

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

\(\Big \downarrow \) 100

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 90

\(\displaystyle -4 (m+1) \int \frac {(c x)^m}{a x+1}dx+\frac {4 (c x)^{m+1}}{c (a x+1)}+\frac {(c x)^{m+1}}{c (m+1)}\)

\(\Big \downarrow \) 74

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

Input:

Int[(c*x)^m/E^(4*ArcTanh[a*x]),x]
 

Output:

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

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 74

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x 
)^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] 
/; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] 
 &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))
 

rule 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 100

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d 
*e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1))   Int[(c + d*x)^ 
(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( 
p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x 
, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n 
 + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))
 

rule 6676

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_.)*(x_))^(m_.), x_Symbol] :> Int[(c*x) 
^m*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] /; FreeQ[{a, c, m, n}, x] &&  !Int 
egerQ[(n - 1)/2]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 5.

Time = 0.19 (sec) , antiderivative size = 403, normalized size of antiderivative = 6.72

method	result	size
meijerg	\(\frac {a^{-1-m} \left (x c \right )^{m} x^{-m} \left (\frac {x^{m} a^{m} \left (6 a^{4} x^{4} m -a^{2} x^{2} m^{4}-6 a^{3} m \,x^{3}-11 a^{2} x^{2} m^{3}-24 a^{3} x^{3}-46 a^{2} m^{2} x^{2}-2 a \,m^{4} x -90 a^{2} m \,x^{2}-21 a \,m^{3} x -72 a^{2} x^{2}-79 a x \,m^{2}-m^{4}-126 a m x -10 m^{3}-72 a x -35 m^{2}-50 m -24\right )}{\left (1+m \right ) m \left (a x +1\right )^{3}}+x^{m} a^{m} \left (m^{3}+9 m^{2}+26 m +24\right ) \operatorname {LerchPhi}\left (-a x , 1, m\right )\right )}{6}-\frac {a^{-1-m} \left (x c \right )^{m} x^{-m} \left (-\frac {x^{m} a^{m} \left (a^{2} m^{2} x^{2}+4 a^{2} m \,x^{2}+6 a^{2} x^{2}+2 a x \,m^{2}+7 a m x +6 a x +m^{2}+3 m +2\right )}{\left (a x +1\right )^{3}}+x^{m} a^{m} m \left (m^{2}+3 m +2\right ) \operatorname {LerchPhi}\left (-a x , 1, m\right )\right )}{3}+\frac {a^{-1-m} \left (x c \right )^{m} x^{-m} \left (-\frac {x^{m} a^{m} \left (a^{2} m^{2} x^{2}-2 a^{2} m \,x^{2}+2 a x \,m^{2}-5 a m x +m^{2}-3 m +2\right )}{\left (a x +1\right )^{3}}+x^{m} a^{m} \left (m^{2}-3 m +2\right ) m \operatorname {LerchPhi}\left (-a x , 1, m\right )\right )}{6}\)	\(403\)

Input:

int((x*c)^m/(a*x+1)^4*(-a^2*x^2+1)^2,x,method=_RETURNVERBOSE)
 

Output:

1/6*a^(-1-m)*(x*c)^m*x^(-m)*(x^m*a^m*(6*a^4*m*x^4-a^2*m^4*x^2-6*a^3*m*x^3- 
11*a^2*m^3*x^2-24*a^3*x^3-46*a^2*m^2*x^2-2*a*m^4*x-90*a^2*m*x^2-21*a*m^3*x 
-72*a^2*x^2-79*a*m^2*x-m^4-126*a*m*x-10*m^3-72*a*x-35*m^2-50*m-24)/(1+m)/m 
/(a*x+1)^3+x^m*a^m*(m^3+9*m^2+26*m+24)*LerchPhi(-a*x,1,m))-1/3*a^(-1-m)*(x 
*c)^m*x^(-m)*(-x^m*a^m*(a^2*m^2*x^2+4*a^2*m*x^2+6*a^2*x^2+2*a*m^2*x+7*a*m* 
x+6*a*x+m^2+3*m+2)/(a*x+1)^3+x^m*a^m*m*(m^2+3*m+2)*LerchPhi(-a*x,1,m))+1/6 
*a^(-1-m)*(x*c)^m*x^(-m)*(-x^m*a^m*(a^2*m^2*x^2-2*a^2*m*x^2+2*a*m^2*x-5*a* 
m*x+m^2-3*m+2)/(a*x+1)^3+x^m*a^m*(m^2-3*m+2)*m*LerchPhi(-a*x,1,m))
 

Fricas [F]

\[ \int e^{-4 \text {arctanh}(a x)} (c x)^m \, dx=\int { \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \left (c x\right )^{m}}{{\left (a x + 1\right )}^{4}} \,d x } \] Input:

integrate((c*x)^m/(a*x+1)^4*(-a^2*x^2+1)^2,x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int e^{-4 \text {arctanh}(a x)} (c x)^m \, dx=\int \frac {\left (c x\right )^{m} \left (a x - 1\right )^{2}}{\left (a x + 1\right )^{2}}\, dx \] Input:

integrate((c*x)**m/(a*x+1)**4*(-a**2*x**2+1)**2,x)
 

Output:

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

Maxima [F]

\[ \int e^{-4 \text {arctanh}(a x)} (c x)^m \, dx=\int { \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \left (c x\right )^{m}}{{\left (a x + 1\right )}^{4}} \,d x } \] Input:

integrate((c*x)^m/(a*x+1)^4*(-a^2*x^2+1)^2,x, algorithm="maxima")
 

Output:

integrate((a^2*x^2 - 1)^2*(c*x)^m/(a*x + 1)^4, x)
 

Giac [F]

\[ \int e^{-4 \text {arctanh}(a x)} (c x)^m \, dx=\int { \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \left (c x\right )^{m}}{{\left (a x + 1\right )}^{4}} \,d x } \] Input:

integrate((c*x)^m/(a*x+1)^4*(-a^2*x^2+1)^2,x, algorithm="giac")
 

Output:

integrate((a^2*x^2 - 1)^2*(c*x)^m/(a*x + 1)^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int e^{-4 \text {arctanh}(a x)} (c x)^m \, dx=\int \frac {{\left (c\,x\right )}^m\,{\left (a^2\,x^2-1\right )}^2}{{\left (a\,x+1\right )}^4} \,d x \] Input:

int(((c*x)^m*(a^2*x^2 - 1)^2)/(a*x + 1)^4,x)
 

Output:

int(((c*x)^m*(a^2*x^2 - 1)^2)/(a*x + 1)^4, x)
 

Reduce [F]

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

int((c*x)^m/(a*x+1)^4*(-a^2*x^2+1)^2,x)
 

Output:

(c**m*(x**m*a**2*m**2*x**2 - x**m*a**2*m*x**2 - 3*x**m*a*m**2*x - x**m*a*m 
*x + 4*x**m*a*x + 4*x**m*m**2 + 8*x**m*m + 4*x**m - 4*int(x**m/(a**2*m*x** 
3 - a**2*x**3 + 2*a*m*x**2 - 2*a*x**2 + m*x - x),x)*a*m**4*x - 4*int(x**m/ 
(a**2*m*x**3 - a**2*x**3 + 2*a*m*x**2 - 2*a*x**2 + m*x - x),x)*a*m**3*x + 
4*int(x**m/(a**2*m*x**3 - a**2*x**3 + 2*a*m*x**2 - 2*a*x**2 + m*x - x),x)* 
a*m**2*x + 4*int(x**m/(a**2*m*x**3 - a**2*x**3 + 2*a*m*x**2 - 2*a*x**2 + m 
*x - x),x)*a*m*x - 4*int(x**m/(a**2*m*x**3 - a**2*x**3 + 2*a*m*x**2 - 2*a* 
x**2 + m*x - x),x)*m**4 - 4*int(x**m/(a**2*m*x**3 - a**2*x**3 + 2*a*m*x**2 
 - 2*a*x**2 + m*x - x),x)*m**3 + 4*int(x**m/(a**2*m*x**3 - a**2*x**3 + 2*a 
*m*x**2 - 2*a*x**2 + m*x - x),x)*m**2 + 4*int(x**m/(a**2*m*x**3 - a**2*x** 
3 + 2*a*m*x**2 - 2*a*x**2 + m*x - x),x)*m))/(a*m*(a*m**2*x - a*x + m**2 - 
1))