\(\int e^{-4 \text {arctanh}(a x)} (c-\frac {c}{a x})^p \, dx\) [584]

Optimal result

Integrand size = 22, antiderivative size = 164 \[ \int e^{-4 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\frac {3 \left (c-\frac {c}{a x}\right )^{3+p}}{2 c^3 \left (a+\frac {1}{x}\right )}+\frac {a \left (c-\frac {c}{a x}\right )^{3+p} x}{c^3 \left (a+\frac {1}{x}\right )}+\frac {(2-p) \left (c-\frac {c}{a x}\right )^{3+p} \operatorname {Hypergeometric2F1}\left (1,3+p,4+p,\frac {a-\frac {1}{x}}{2 a}\right )}{4 a c^3 (3+p)}-\frac {(4+p) \left (c-\frac {c}{a x}\right )^{3+p} \operatorname {Hypergeometric2F1}\left (1,3+p,4+p,1-\frac {1}{a x}\right )}{a c^3 (3+p)} \] Output:

3/2*(c-c/a/x)^(3+p)/c^3/(a+1/x)+a*(c-c/a/x)^(3+p)*x/c^3/(a+1/x)+1/4*(2-p)* 
(c-c/a/x)^(3+p)*hypergeom([1, 3+p],[4+p],1/2*(a-1/x)/a)/a/c^3/(3+p)-(4+p)* 
(c-c/a/x)^(3+p)*hypergeom([1, 3+p],[4+p],1-1/a/x)/a/c^3/(3+p)
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.71 \[ \int e^{-4 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\frac {\left (c-\frac {c}{a x}\right )^p (-1+a x)^3 \left (2 a (3+p) x (3+2 a x)-(-2+p) (1+a x) \operatorname {Hypergeometric2F1}\left (1,3+p,4+p,\frac {a-\frac {1}{x}}{2 a}\right )-4 (4+p) (1+a x) \operatorname {Hypergeometric2F1}\left (1,3+p,4+p,1-\frac {1}{a x}\right )\right )}{4 a^4 (3+p) x^3 (1+a x)} \] Input:

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

Output:

((c - c/(a*x))^p*(-1 + a*x)^3*(2*a*(3 + p)*x*(3 + 2*a*x) - (-2 + p)*(1 + a 
*x)*Hypergeometric2F1[1, 3 + p, 4 + p, (a - x^(-1))/(2*a)] - 4*(4 + p)*(1 
+ a*x)*Hypergeometric2F1[1, 3 + p, 4 + p, 1 - 1/(a*x)]))/(4*a^4*(3 + p)*x^ 
3*(1 + a*x))
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.12, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6683, 1035, 281, 899, 114, 27, 168, 27, 174, 75, 78}

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

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

\(\Big \downarrow \) 1035

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

\(\Big \downarrow \) 281

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

\(\Big \downarrow \) 899

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

\(\Big \downarrow \) 114

\(\displaystyle -\frac {a^2 \left (-\frac {\int \frac {c \left (c-\frac {c}{a x}\right )^{p+2} \left (\frac {p+1}{x}+a (p+4)\right ) x}{a \left (a+\frac {1}{x}\right )^2}d\frac {1}{x}}{a c}-\frac {x \left (c-\frac {c}{a x}\right )^{p+3}}{a c \left (a+\frac {1}{x}\right )}\right )}{c^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {a^2 \left (-\frac {\int \frac {\left (c-\frac {c}{a x}\right )^{p+2} \left (\frac {p+1}{x}+a (p+4)\right ) x}{\left (a+\frac {1}{x}\right )^2}d\frac {1}{x}}{a^2}-\frac {x \left (c-\frac {c}{a x}\right )^{p+3}}{a c \left (a+\frac {1}{x}\right )}\right )}{c^2}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {a^2 \left (-\frac {\frac {\int \frac {c \left (c-\frac {c}{a x}\right )^{p+2} \left (\frac {3 (p+2)}{x}+2 a (p+4)\right ) x}{a+\frac {1}{x}}d\frac {1}{x}}{2 a c}+\frac {3 \left (c-\frac {c}{a x}\right )^{p+3}}{2 c \left (a+\frac {1}{x}\right )}}{a^2}-\frac {x \left (c-\frac {c}{a x}\right )^{p+3}}{a c \left (a+\frac {1}{x}\right )}\right )}{c^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {a^2 \left (-\frac {\frac {\int \frac {\left (c-\frac {c}{a x}\right )^{p+2} \left (\frac {3 (p+2)}{x}+2 a (p+4)\right ) x}{a+\frac {1}{x}}d\frac {1}{x}}{2 a}+\frac {3 \left (c-\frac {c}{a x}\right )^{p+3}}{2 c \left (a+\frac {1}{x}\right )}}{a^2}-\frac {x \left (c-\frac {c}{a x}\right )^{p+3}}{a c \left (a+\frac {1}{x}\right )}\right )}{c^2}\)

\(\Big \downarrow \) 174

\(\displaystyle -\frac {a^2 \left (-\frac {\frac {2 (p+4) \int \left (c-\frac {c}{a x}\right )^{p+2} xd\frac {1}{x}-(2-p) \int \frac {\left (c-\frac {c}{a x}\right )^{p+2}}{a+\frac {1}{x}}d\frac {1}{x}}{2 a}+\frac {3 \left (c-\frac {c}{a x}\right )^{p+3}}{2 c \left (a+\frac {1}{x}\right )}}{a^2}-\frac {x \left (c-\frac {c}{a x}\right )^{p+3}}{a c \left (a+\frac {1}{x}\right )}\right )}{c^2}\)

\(\Big \downarrow \) 75

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

\(\Big \downarrow \) 78

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

Input:

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

Output:

-((a^2*(-(((c - c/(a*x))^(3 + p)*x)/(a*c*(a + x^(-1)))) - ((3*(c - c/(a*x) 
)^(3 + p))/(2*c*(a + x^(-1))) + (((2 - p)*(c - c/(a*x))^(3 + p)*Hypergeome 
tric2F1[1, 3 + p, 4 + p, (a - x^(-1))/(2*a)])/(2*c*(3 + p)) - (2*(4 + p)*( 
c - c/(a*x))^(3 + p)*Hypergeometric2F1[1, 3 + p, 4 + p, 1 - 1/(a*x)])/(c*( 
3 + p)))/(2*a))/a^2))/c^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 78

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

rule 114

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 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e 
 - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
 

rule 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
 

rule 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d)   Int[(e + f*x)^p/(c + d 
*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
 

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]

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

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)
 

Fricas [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int e^{-4 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx =\text {Too large to display} \] Input:

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

Output:

((a*c*x - c)**p*a**2*p**2*x**2 + 3*(a*c*x - c)**p*a**2*p*x**2 - (a*c*x - c 
)**p*a*p**2*x + 4*(a*c*x - c)**p*a*p*x + 4*(a*c*x - c)**p*a*x + (a*c*x - c 
)**p*p**2 - 7*(a*c*x - c)**p*p + 4*(a*c*x - c)**p - x**p*int((a*c*x - c)** 
p/(x**p*a**3*p*x**4 + 3*x**p*a**3*x**4 + x**p*a**2*p*x**3 + 3*x**p*a**2*x* 
*3 - x**p*a*p*x**2 - 3*x**p*a*x**2 - x**p*p*x - 3*x**p*x),x)*a*p**4*x + 4* 
x**p*int((a*c*x - c)**p/(x**p*a**3*p*x**4 + 3*x**p*a**3*x**4 + x**p*a**2*p 
*x**3 + 3*x**p*a**2*x**3 - x**p*a*p*x**2 - 3*x**p*a*x**2 - x**p*p*x - 3*x* 
*p*x),x)*a*p**3*x + 17*x**p*int((a*c*x - c)**p/(x**p*a**3*p*x**4 + 3*x**p* 
a**3*x**4 + x**p*a**2*p*x**3 + 3*x**p*a**2*x**3 - x**p*a*p*x**2 - 3*x**p*a 
*x**2 - x**p*p*x - 3*x**p*x),x)*a*p**2*x - 12*x**p*int((a*c*x - c)**p/(x** 
p*a**3*p*x**4 + 3*x**p*a**3*x**4 + x**p*a**2*p*x**3 + 3*x**p*a**2*x**3 - x 
**p*a*p*x**2 - 3*x**p*a*x**2 - x**p*p*x - 3*x**p*x),x)*a*p*x - x**p*int((a 
*c*x - c)**p/(x**p*a**3*p*x**4 + 3*x**p*a**3*x**4 + x**p*a**2*p*x**3 + 3*x 
**p*a**2*x**3 - x**p*a*p*x**2 - 3*x**p*a*x**2 - x**p*p*x - 3*x**p*x),x)*p* 
*4 + 4*x**p*int((a*c*x - c)**p/(x**p*a**3*p*x**4 + 3*x**p*a**3*x**4 + x**p 
*a**2*p*x**3 + 3*x**p*a**2*x**3 - x**p*a*p*x**2 - 3*x**p*a*x**2 - x**p*p*x 
 - 3*x**p*x),x)*p**3 + 17*x**p*int((a*c*x - c)**p/(x**p*a**3*p*x**4 + 3*x* 
*p*a**3*x**4 + x**p*a**2*p*x**3 + 3*x**p*a**2*x**3 - x**p*a*p*x**2 - 3*x** 
p*a*x**2 - x**p*p*x - 3*x**p*x),x)*p**2 - 12*x**p*int((a*c*x - c)**p/(x**p 
*a**3*p*x**4 + 3*x**p*a**3*x**4 + x**p*a**2*p*x**3 + 3*x**p*a**2*x**3 -...