\(\int e^{4 \text {arctanh}(a x)} (c-\frac {c}{a x})^p \, dx\) [581]

Optimal result

Integrand size = 22, antiderivative size = 93 \[ \int e^{4 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=-\frac {c (5-p) \left (c-\frac {c}{a x}\right )^{-1+p}}{a (1-p)}+c \left (c-\frac {c}{a x}\right )^{-1+p} x+\frac {(4-p) \left (c-\frac {c}{a x}\right )^p \operatorname {Hypergeometric2F1}\left (1,p,1+p,1-\frac {1}{a x}\right )}{a p} \] Output:

-c*(5-p)*(c-c/a/x)^(-1+p)/a/(1-p)+c*(c-c/a/x)^(-1+p)*x+(4-p)*(c-c/a/x)^p*h 
ypergeom([1, p],[p+1],1-1/a/x)/a/p
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.87 \[ \int e^{4 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\frac {\left (c-\frac {c}{a x}\right )^p \left (a p x (5-a x+p (-1+a x))-\left (4-5 p+p^2\right ) (-1+a x) \operatorname {Hypergeometric2F1}\left (1,p,1+p,1-\frac {1}{a x}\right )\right )}{a (-1+p) p (-1+a x)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.16, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6683, 1035, 281, 899, 100, 27, 88, 75}

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.

\(\displaystyle \int e^{4 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx\)

\(\Big \downarrow \) 6683

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

\(\Big \downarrow \) 1035

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

\(\Big \downarrow \) 281

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

\(\Big \downarrow \) 899

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

\(\Big \downarrow \) 100

\(\displaystyle -\frac {c^2 \left (\frac {\int c \left (a (4-p)+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{p-2} xd\frac {1}{x}}{c}-\frac {a^2 x \left (c-\frac {c}{a x}\right )^{p-1}}{c}\right )}{a^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {c^2 \left (\int \left (a (4-p)+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{p-2} xd\frac {1}{x}-\frac {a^2 x \left (c-\frac {c}{a x}\right )^{p-1}}{c}\right )}{a^2}\)

\(\Big \downarrow \) 88

\(\displaystyle -\frac {c^2 \left (\frac {a (4-p) \int \left (c-\frac {c}{a x}\right )^{p-1} xd\frac {1}{x}}{c}-\frac {a^2 x \left (c-\frac {c}{a x}\right )^{p-1}}{c}+\frac {a (5-p) \left (c-\frac {c}{a x}\right )^{p-1}}{c (1-p)}\right )}{a^2}\)

\(\Big \downarrow \) 75

\(\displaystyle -\frac {c^2 \left (-\frac {a^2 x \left (c-\frac {c}{a x}\right )^{p-1}}{c}-\frac {a (4-p) \left (c-\frac {c}{a x}\right )^p \operatorname {Hypergeometric2F1}\left (1,p,p+1,1-\frac {1}{a x}\right )}{c^2 p}+\frac {a (5-p) \left (c-\frac {c}{a x}\right )^{p-1}}{c (1-p)}\right )}{a^2}\)

Input:

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

Output:

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

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 75

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

rule 88

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

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 281

Int[(u_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_ 
Symbol] :> Simp[(b/d)^p   Int[u*(c + d*x^n)^(p + q), x], x] /; FreeQ[{a, b, 
 c, d, n, p, q}, x] && EqQ[b*c - a*d, 0] && IntegerQ[p] &&  !(IntegerQ[q] & 
& SimplerQ[a + b*x^n, c + d*x^n])
 

rule 899

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol 
] :> -Subst[Int[(a + b/x^n)^p*((c + d/x^n)^q/x^2), x], x, 1/x] /; FreeQ[{a, 
 b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0]
 

rule 1035

Int[((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_.) + (b_.)*(x_)^(n_.))^(p_.)*((e_) 
 + (f_.)*(x_)^(n_.))^(r_.), x_Symbol] :> Int[x^(n*(p + r))*(b + a/x^n)^p*(c 
 + d/x^n)^q*(f + e/x^n)^r, x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && EqQ[ 
mn, -n] && IntegerQ[p] && IntegerQ[r]
 

rule 6683

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

Maple [F]

\[\int \frac {\left (a x +1\right )^{4} \left (c -\frac {c}{a x}\right )^{p}}{\left (-a^{2} x^{2}+1\right )^{2}}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int e^{4 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\frac {\left (a c x -c \right )^{p} a p \,x^{2}-\left (a c x -c \right )^{p} a \,x^{2}+\left (a c x -c \right )^{p} x -x^{p} \left (\int \frac {\left (a c x -c \right )^{p} x}{x^{p} a^{2} x^{2}-2 x^{p} a x +x^{p}}d x \right ) a^{2} p^{2} x +5 x^{p} \left (\int \frac {\left (a c x -c \right )^{p} x}{x^{p} a^{2} x^{2}-2 x^{p} a x +x^{p}}d x \right ) a^{2} p x -4 x^{p} \left (\int \frac {\left (a c x -c \right )^{p} x}{x^{p} a^{2} x^{2}-2 x^{p} a x +x^{p}}d x \right ) a^{2} x +x^{p} \left (\int \frac {\left (a c x -c \right )^{p} x}{x^{p} a^{2} x^{2}-2 x^{p} a x +x^{p}}d x \right ) a \,p^{2}-5 x^{p} \left (\int \frac {\left (a c x -c \right )^{p} x}{x^{p} a^{2} x^{2}-2 x^{p} a x +x^{p}}d x \right ) a p +4 x^{p} \left (\int \frac {\left (a c x -c \right )^{p} x}{x^{p} a^{2} x^{2}-2 x^{p} a x +x^{p}}d x \right ) a}{x^{p} a^{p} \left (a p x -a x -p +1\right )} \] Input:

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

Output:

((a*c*x - c)**p*a*p*x**2 - (a*c*x - c)**p*a*x**2 + (a*c*x - c)**p*x - x**p 
*int(((a*c*x - c)**p*x)/(x**p*a**2*x**2 - 2*x**p*a*x + x**p),x)*a**2*p**2* 
x + 5*x**p*int(((a*c*x - c)**p*x)/(x**p*a**2*x**2 - 2*x**p*a*x + x**p),x)* 
a**2*p*x - 4*x**p*int(((a*c*x - c)**p*x)/(x**p*a**2*x**2 - 2*x**p*a*x + x* 
*p),x)*a**2*x + x**p*int(((a*c*x - c)**p*x)/(x**p*a**2*x**2 - 2*x**p*a*x + 
 x**p),x)*a*p**2 - 5*x**p*int(((a*c*x - c)**p*x)/(x**p*a**2*x**2 - 2*x**p* 
a*x + x**p),x)*a*p + 4*x**p*int(((a*c*x - c)**p*x)/(x**p*a**2*x**2 - 2*x** 
p*a*x + x**p),x)*a)/(x**p*a**p*(a*p*x - a*x - p + 1))