\(\int e^{-2 \text {arctanh}(a x)} (c-\frac {c}{a x})^p \, dx\) [583]

Optimal result

Integrand size = 22, antiderivative size = 114 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=-\frac {\left (c-\frac {c}{a x}\right )^{2+p} x}{c^2}-\frac {\left (c-\frac {c}{a x}\right )^{2+p} \operatorname {Hypergeometric2F1}\left (1,2+p,3+p,\frac {a-\frac {1}{x}}{2 a}\right )}{2 a c^2 (2+p)}+\frac {\left (c-\frac {c}{a x}\right )^{2+p} \operatorname {Hypergeometric2F1}\left (1,2+p,3+p,1-\frac {1}{a x}\right )}{a c^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.76 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=-\frac {\left (c-\frac {c}{a x}\right )^p (-1+a x)^2 \left (\operatorname {Hypergeometric2F1}\left (1,2+p,3+p,\frac {a-\frac {1}{x}}{2 a}\right )+2 (2+p) \left (a x-\operatorname {Hypergeometric2F1}\left (1,2+p,3+p,1-\frac {1}{a x}\right )\right )\right )}{2 a^3 (2+p) x^2} \] Input:

Integrate[(c - c/(a*x))^p/E^(2*ArcTanh[a*x]),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {6683, 1035, 281, 899, 114, 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.

\(\displaystyle \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx\)

\(\Big \downarrow \) 6683

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

\(\Big \downarrow \) 1035

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

\(\Big \downarrow \) 281

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

\(\Big \downarrow \) 899

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 75

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

\(\Big \downarrow \) 78

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

Input:

Int[(c - c/(a*x))^p/E^(2*ArcTanh[a*x]),x]
 

Output:

(a*(-(((c - c/(a*x))^(2 + p)*x)/(a*c)) - (((c - c/(a*x))^(2 + p)*Hypergeom 
etric2F1[1, 2 + p, 3 + p, (a - x^(-1))/(2*a)])/(2*c*(2 + p)) - ((c - c/(a* 
x))^(2 + p)*Hypergeometric2F1[1, 2 + p, 3 + p, 1 - 1/(a*x)])/c)/a^2))/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 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 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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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