3.7.23 \(\int \frac {e^{n \text {arctanh}(a x)}}{(c-\frac {c}{a x})^2} \, dx\) [623]

3.7.23.1 Optimal result

Integrand size = 22, antiderivative size = 139 \[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\frac {(3+n) (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {2+n}{2}}}{a c^2 (2+n)}-\frac {x (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {2+n}{2}}}{c^2}-\frac {2^{1+\frac {n}{2}} (2+n) (1-a x)^{-n/2} \operatorname {Hypergeometric2F1}\left (-\frac {n}{2},-\frac {n}{2},1-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a c^2 n} \]

output

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

3.7.23.2 Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.70 \[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=-\frac {(1-a x)^{-1-\frac {n}{2}} \left ((1+a x)^{1+\frac {n}{2}} (1+n+2 a x+a n x)-2^{2+\frac {n}{2}} (2+n) \operatorname {Hypergeometric2F1}\left (-1-\frac {n}{2},-1-\frac {n}{2},-\frac {n}{2},\frac {1}{2} (1-a x)\right )\right )}{a c^2 (2+n)} \]

input

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

output

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

3.7.23.3 Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6681, 6679, 101, 25, 88, 79}

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.

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

\(\Big \downarrow \) 6681

\(\displaystyle \frac {a^2 \int \frac {e^{n \text {arctanh}(a x)} x^2}{(1-a x)^2}dx}{c^2}\)

\(\Big \downarrow \) 6679

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

\(\Big \downarrow \) 101

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 88

\(\displaystyle \frac {a^2 \left (\frac {\frac {(n+3) (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n+2}{2}}}{a (n+2)}-(n+2) \int (1-a x)^{-\frac {n}{2}-1} (a x+1)^{n/2}dx}{a^2}-\frac {x (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n+2}{2}}}{a^2}\right )}{c^2}\)

\(\Big \downarrow \) 79

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

input

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

output

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

3.7.23.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 79

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( 
a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 
, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] 
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] 
 ||  !(RationalQ[n] && GtQ[-d/(b*c - a*d), 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 101

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + 
 p + 3))), x] + Simp[1/(d*f*(n + p + 3))   Int[(c + d*x)^n*(e + f*x)^p*Simp 
[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f 
*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, 
 c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
 

rule 6679

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

rule 6681

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

3.7.23.4 Maple [F]

\[\int \frac {{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )}}{\left (c -\frac {c}{a x}\right )^{2}}d x\]

input

int(exp(n*arctanh(a*x))/(c-c/a/x)^2,x)
 

output

int(exp(n*arctanh(a*x))/(c-c/a/x)^2,x)
 

3.7.23.5 Fricas [F]

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

input

integrate(exp(n*arctanh(a*x))/(c-c/a/x)^2,x, algorithm="fricas")
 

output

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

3.7.23.6 Sympy [F]

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

input

integrate(exp(n*atanh(a*x))/(c-c/a/x)**2,x)
 

output

a**2*Integral(x**2*exp(n*atanh(a*x))/(a**2*x**2 - 2*a*x + 1), x)/c**2
 

3.7.23.7 Maxima [F]

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

input

integrate(exp(n*arctanh(a*x))/(c-c/a/x)^2,x, algorithm="maxima")
 

output

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

3.7.23.8 Giac [F]

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

input

integrate(exp(n*arctanh(a*x))/(c-c/a/x)^2,x, algorithm="giac")
 

output

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

3.7.23.9 Mupad [F(-1)]

Timed out. \[ \int \frac {e^{n \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{{\left (c-\frac {c}{a\,x}\right )}^2} \,d x \]

input

int(exp(n*atanh(a*x))/(c - c/(a*x))^2,x)
 

output

int(exp(n*atanh(a*x))/(c - c/(a*x))^2, x)