\(\int e^{6 \text {arctanh}(a x)} (c-\frac {c}{a x})^p \, dx\) [595]

Optimal result

Integrand size = 22, antiderivative size = 163 \[ \int e^{6 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\frac {c^2 \left (4-8 p+p^2\right ) \left (c-\frac {c}{a x}\right )^{-2+p}}{a (1-p) (2-p)}-\frac {c^2 (2-p) \left (c-\frac {c}{a x}\right )^{-2+p} x}{1-p}+\frac {c^2 \left (a+\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{-2+p} x}{a^2 (1-p)}+\frac {c (6-p) \left (c-\frac {c}{a x}\right )^{-1+p} \operatorname {Hypergeometric2F1}\left (1,-1+p,p,1-\frac {1}{a x}\right )}{a (1-p)} \] Output:

c^2*(p^2-8*p+4)*(c-c/a/x)^(-2+p)/a/(1-p)/(2-p)-c^2*(2-p)*(c-c/a/x)^(-2+p)* 
x/(1-p)+c^2*(a+1/x)^2*(c-c/a/x)^(-2+p)*x/a^2/(1-p)+c*(6-p)*(c-c/a/x)^(-1+p 
)*hypergeom([1, -1+p],[p],1-1/a/x)/a/(1-p)
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.63 \[ \int e^{6 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=-\frac {\left (c-\frac {c}{a x}\right )^p x \left (-2+p-2 a x+3 a p x+2 a^2 x^2-3 a^2 p x^2+a^2 p^2 x^2-a \left (6-7 p+p^2\right ) x \operatorname {Hypergeometric2F1}\left (1,-2+p,-1+p,1-\frac {1}{a x}\right )\right )}{(-2+p) (-1+p) (-1+a x)^2} \] Input:

Integrate[E^(6*ArcTanh[a*x])*(c - c/(a*x))^p,x]
 

Output:

-(((c - c/(a*x))^p*x*(-2 + p - 2*a*x + 3*a*p*x + 2*a^2*x^2 - 3*a^2*p*x^2 + 
 a^2*p^2*x^2 - a*(6 - 7*p + p^2)*x*Hypergeometric2F1[1, -2 + p, -1 + p, 1 
- 1/(a*x)]))/((-2 + p)*(-1 + p)*(-1 + a*x)^2))
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.94, 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, 111, 27, 163, 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.

\(\displaystyle \int e^{6 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx\)

\(\Big \downarrow \) 6683

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

\(\Big \downarrow \) 1035

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

\(\Big \downarrow \) 281

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

\(\Big \downarrow \) 899

\(\displaystyle \frac {c^3 \int \left (a+\frac {1}{x}\right )^3 \left (c-\frac {c}{a x}\right )^{p-3} x^2d\frac {1}{x}}{a^3}\)

\(\Big \downarrow \) 111

\(\displaystyle \frac {c^3 \left (\frac {a \int c \left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{p-3} \left (a (2-p)-\frac {p+2}{x}\right ) x^2d\frac {1}{x}}{c (1-p)}+\frac {a x \left (a+\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{p-2}}{c (1-p)}\right )}{a^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {c^3 \left (\frac {a \int \left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{p-3} \left (a (2-p)-\frac {p+2}{x}\right ) x^2d\frac {1}{x}}{1-p}+\frac {a x \left (a+\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{p-2}}{c (1-p)}\right )}{a^3}\)

\(\Big \downarrow \) 163

\(\displaystyle \frac {c^3 \left (\frac {a \left (a (1-p) (6-p) \int \left (c-\frac {c}{a x}\right )^{p-3} xd\frac {1}{x}-\frac {a x \left (a (2-p)^2+\frac {p+2}{x}\right ) \left (c-\frac {c}{a x}\right )^{p-2}}{c (2-p)}\right )}{1-p}+\frac {a x \left (a+\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{p-2}}{c (1-p)}\right )}{a^3}\)

\(\Big \downarrow \) 75

\(\displaystyle \frac {c^3 \left (\frac {a \left (\frac {a (1-p) (6-p) \left (c-\frac {c}{a x}\right )^{p-2} \operatorname {Hypergeometric2F1}\left (1,p-2,p-1,1-\frac {1}{a x}\right )}{c (2-p)}-\frac {a x \left (a (2-p)^2+\frac {p+2}{x}\right ) \left (c-\frac {c}{a x}\right )^{p-2}}{c (2-p)}\right )}{1-p}+\frac {a x \left (a+\frac {1}{x}\right )^2 \left (c-\frac {c}{a x}\right )^{p-2}}{c (1-p)}\right )}{a^3}\)

Input:

Int[E^(6*ArcTanh[a*x])*(c - c/(a*x))^p,x]
 

Output:

(c^3*((a*(a + x^(-1))^2*(c - c/(a*x))^(-2 + p)*x)/(c*(1 - p)) + (a*(-((a*( 
c - c/(a*x))^(-2 + p)*(a*(2 - p)^2 + (2 + p)/x)*x)/(c*(2 - p))) + (a*(1 - 
p)*(6 - p)*(c - c/(a*x))^(-2 + p)*Hypergeometric2F1[1, -2 + p, -1 + p, 1 - 
 1/(a*x)])/(c*(2 - p))))/(1 - p)))/a^3
 

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 111

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

rule 163

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

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]

\[\int \frac {\left (a x +1\right )^{6} \left (c -\frac {c}{a x}\right )^{p}}{\left (-a^{2} x^{2}+1\right )^{3}}d x\]

Input:

int((a*x+1)^6/(-a^2*x^2+1)^3*(c-c/a/x)^p,x)
 

Output:

int((a*x+1)^6/(-a^2*x^2+1)^3*(c-c/a/x)^p,x)
 

Fricas [F]

\[ \int e^{6 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\int { -\frac {{\left (a x + 1\right )}^{6} {\left (c - \frac {c}{a x}\right )}^{p}}{{\left (a^{2} x^{2} - 1\right )}^{3}} \,d x } \] Input:

integrate((a*x+1)^6/(-a^2*x^2+1)^3*(c-c/a/x)^p,x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int e^{6 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=- \int \frac {\left (c - \frac {c}{a x}\right )^{p}}{a^{3} x^{3} - 3 a^{2} x^{2} + 3 a x - 1}\, dx - \int \frac {3 a x \left (c - \frac {c}{a x}\right )^{p}}{a^{3} x^{3} - 3 a^{2} x^{2} + 3 a x - 1}\, dx - \int \frac {3 a^{2} x^{2} \left (c - \frac {c}{a x}\right )^{p}}{a^{3} x^{3} - 3 a^{2} x^{2} + 3 a x - 1}\, dx - \int \frac {a^{3} x^{3} \left (c - \frac {c}{a x}\right )^{p}}{a^{3} x^{3} - 3 a^{2} x^{2} + 3 a x - 1}\, dx \] Input:

integrate((a*x+1)**6/(-a**2*x**2+1)**3*(c-c/a/x)**p,x)
 

Output:

-Integral((c - c/(a*x))**p/(a**3*x**3 - 3*a**2*x**2 + 3*a*x - 1), x) - Int 
egral(3*a*x*(c - c/(a*x))**p/(a**3*x**3 - 3*a**2*x**2 + 3*a*x - 1), x) - I 
ntegral(3*a**2*x**2*(c - c/(a*x))**p/(a**3*x**3 - 3*a**2*x**2 + 3*a*x - 1) 
, x) - Integral(a**3*x**3*(c - c/(a*x))**p/(a**3*x**3 - 3*a**2*x**2 + 3*a* 
x - 1), x)
 

Maxima [F]

\[ \int e^{6 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\int { -\frac {{\left (a x + 1\right )}^{6} {\left (c - \frac {c}{a x}\right )}^{p}}{{\left (a^{2} x^{2} - 1\right )}^{3}} \,d x } \] Input:

integrate((a*x+1)^6/(-a^2*x^2+1)^3*(c-c/a/x)^p,x, algorithm="maxima")
 

Output:

-integrate((a*x + 1)^6*(c - c/(a*x))^p/(a^2*x^2 - 1)^3, x)
                                                                                    
                                                                                    
 

Giac [F]

\[ \int e^{6 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\int { -\frac {{\left (a x + 1\right )}^{6} {\left (c - \frac {c}{a x}\right )}^{p}}{{\left (a^{2} x^{2} - 1\right )}^{3}} \,d x } \] Input:

integrate((a*x+1)^6/(-a^2*x^2+1)^3*(c-c/a/x)^p,x, algorithm="giac")
 

Output:

integrate(-(a*x + 1)^6*(c - c/(a*x))^p/(a^2*x^2 - 1)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int e^{6 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\int -\frac {{\left (c-\frac {c}{a\,x}\right )}^p\,{\left (a\,x+1\right )}^6}{{\left (a^2\,x^2-1\right )}^3} \,d x \] Input:

int(-((c - c/(a*x))^p*(a*x + 1)^6)/(a^2*x^2 - 1)^3,x)
 

Output:

int(-((c - c/(a*x))^p*(a*x + 1)^6)/(a^2*x^2 - 1)^3, x)
 

Reduce [F]

\[ \int e^{6 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx =\text {Too large to display} \] Input:

int((a*x+1)^6/(-a^2*x^2+1)^3*(c-c/a/x)^p,x)
 

Output:

( - (a*c*x - c)**p*a**2*p**2*x**3 + 3*(a*c*x - c)**p*a**2*p*x**3 - 2*(a*c* 
x - c)**p*a**2*x**3 - 3*(a*c*x - c)**p*a*p*x**2 + 2*(a*c*x - c)**p*a*x**2 
- (a*c*x - c)**p*p*x + 2*(a*c*x - c)**p*x + x**p*int(((a*c*x - c)**p*x**2) 
/(x**p*a**3*x**3 - 3*x**p*a**2*x**2 + 3*x**p*a*x - x**p),x)*a**4*p**3*x**2 
 - 9*x**p*int(((a*c*x - c)**p*x**2)/(x**p*a**3*x**3 - 3*x**p*a**2*x**2 + 3 
*x**p*a*x - x**p),x)*a**4*p**2*x**2 + 20*x**p*int(((a*c*x - c)**p*x**2)/(x 
**p*a**3*x**3 - 3*x**p*a**2*x**2 + 3*x**p*a*x - x**p),x)*a**4*p*x**2 - 12* 
x**p*int(((a*c*x - c)**p*x**2)/(x**p*a**3*x**3 - 3*x**p*a**2*x**2 + 3*x**p 
*a*x - x**p),x)*a**4*x**2 - 2*x**p*int(((a*c*x - c)**p*x**2)/(x**p*a**3*x* 
*3 - 3*x**p*a**2*x**2 + 3*x**p*a*x - x**p),x)*a**3*p**3*x + 18*x**p*int((( 
a*c*x - c)**p*x**2)/(x**p*a**3*x**3 - 3*x**p*a**2*x**2 + 3*x**p*a*x - x**p 
),x)*a**3*p**2*x - 40*x**p*int(((a*c*x - c)**p*x**2)/(x**p*a**3*x**3 - 3*x 
**p*a**2*x**2 + 3*x**p*a*x - x**p),x)*a**3*p*x + 24*x**p*int(((a*c*x - c)* 
*p*x**2)/(x**p*a**3*x**3 - 3*x**p*a**2*x**2 + 3*x**p*a*x - x**p),x)*a**3*x 
 + x**p*int(((a*c*x - c)**p*x**2)/(x**p*a**3*x**3 - 3*x**p*a**2*x**2 + 3*x 
**p*a*x - x**p),x)*a**2*p**3 - 9*x**p*int(((a*c*x - c)**p*x**2)/(x**p*a**3 
*x**3 - 3*x**p*a**2*x**2 + 3*x**p*a*x - x**p),x)*a**2*p**2 + 20*x**p*int(( 
(a*c*x - c)**p*x**2)/(x**p*a**3*x**3 - 3*x**p*a**2*x**2 + 3*x**p*a*x - x** 
p),x)*a**2*p - 12*x**p*int(((a*c*x - c)**p*x**2)/(x**p*a**3*x**3 - 3*x**p* 
a**2*x**2 + 3*x**p*a*x - x**p),x)*a**2)/(x**p*a**p*(a**2*p**2*x**2 - 3*...