\(\int \frac {e^{n \text {arctanh}(a+b x)}}{x^2} \, dx\) [896]

Optimal result

Integrand size = 14, antiderivative size = 92 \[ \int \frac {e^{n \text {arctanh}(a+b x)}}{x^2} \, dx=-\frac {4 b (1-a-b x)^{1-\frac {n}{2}} (1+a+b x)^{\frac {1}{2} (-2+n)} \operatorname {Hypergeometric2F1}\left (2,1-\frac {n}{2},2-\frac {n}{2},\frac {(1+a) (1-a-b x)}{(1-a) (1+a+b x)}\right )}{(1-a)^2 (2-n)} \] Output:

-4*b*(-b*x-a+1)^(1-1/2*n)*(b*x+a+1)^(-1+1/2*n)*hypergeom([2, 1-1/2*n],[2-1 
/2*n],(1+a)*(-b*x-a+1)/(1-a)/(b*x+a+1))/(1-a)^2/(2-n)
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.90 \[ \int \frac {e^{n \text {arctanh}(a+b x)}}{x^2} \, dx=\frac {4 b (1-a-b x)^{1-\frac {n}{2}} (1+a+b x)^{-1+\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (2,1-\frac {n}{2},2-\frac {n}{2},\frac {(1+a) (-1+a+b x)}{(-1+a) (1+a+b x)}\right )}{(-1+a)^2 (-2+n)} \] Input:

Integrate[E^(n*ArcTanh[a + b*x])/x^2,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6713, 141}

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.

\(\displaystyle \int \frac {e^{n \text {arctanh}(a+b x)}}{x^2} \, dx\)

\(\Big \downarrow \) 6713

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

\(\Big \downarrow \) 141

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

Input:

Int[E^(n*ArcTanh[a + b*x])/x^2,x]
 

Output:

(-4*b*(1 - a - b*x)^(1 - n/2)*(1 + a + b*x)^((-2 + n)/2)*Hypergeometric2F1 
[2, 1 - n/2, 2 - n/2, ((1 + a)*(1 - a - b*x))/((1 - a)*(1 + a + b*x))])/(( 
1 - a)^2*(2 - n))
 

Defintions of rubi rules used

rule 141

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(b*c - a*d)^n*((a + b*x)^(m + 1)/((m + 1)*(b*e - a*f)^( 
n + 1)*(e + f*x)^(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2, (-(d*e - c*f 
))*((a + b*x)/((b*c - a*d)*(e + f*x)))], x] /; FreeQ[{a, b, c, d, e, f, m, 
p}, x] && EqQ[m + n + p + 2, 0] && ILtQ[n, 0] && (SumSimplerQ[m, 1] ||  !Su 
mSimplerQ[p, 1]) &&  !ILtQ[m, 0]
 

rule 6713

Int[E^(ArcTanh[(c_.)*((a_) + (b_.)*(x_))]*(n_.))*((d_.) + (e_.)*(x_))^(m_.) 
, x_Symbol] :> Int[(d + e*x)^m*((1 + a*c + b*c*x)^(n/2)/(1 - a*c - b*c*x)^( 
n/2)), x] /; FreeQ[{a, b, c, d, e, m, n}, x]
 

Maple [F]

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

Input:

int(exp(n*arctanh(b*x+a))/x^2,x)
 

Output:

int(exp(n*arctanh(b*x+a))/x^2,x)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

integrate(exp(n*atanh(b*x+a))/x**2,x)
 

Output:

Integral(exp(n*atanh(a + b*x))/x**2, x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

int(exp(n*atanh(a + b*x))/x^2,x)
 

Output:

int(exp(n*atanh(a + b*x))/x^2, x)
 

Reduce [F]

\[ \int \frac {e^{n \text {arctanh}(a+b x)}}{x^2} \, dx=\frac {-e^{\mathit {atanh} \left (b x +a \right ) n}-\left (\int \frac {e^{\mathit {atanh} \left (b x +a \right ) n}}{a^{2} b^{2} x^{3}+2 a^{3} b \,x^{2}+a^{4} x -b^{2} x^{3}-2 a b \,x^{2}-2 a^{2} x +x}d x \right ) a^{2} b n x +\left (\int \frac {e^{\mathit {atanh} \left (b x +a \right ) n}}{a^{2} b^{2} x^{3}+2 a^{3} b \,x^{2}+a^{4} x -b^{2} x^{3}-2 a b \,x^{2}-2 a^{2} x +x}d x \right ) b n x}{x} \] Input:

int(exp(n*atanh(b*x+a))/x^2,x)
 

Output:

( - e**(atanh(a + b*x)*n) - int(e**(atanh(a + b*x)*n)/(a**4*x + 2*a**3*b*x 
**2 + a**2*b**2*x**3 - 2*a**2*x - 2*a*b*x**2 - b**2*x**3 + x),x)*a**2*b*n* 
x + int(e**(atanh(a + b*x)*n)/(a**4*x + 2*a**3*b*x**2 + a**2*b**2*x**3 - 2 
*a**2*x - 2*a*b*x**2 - b**2*x**3 + x),x)*b*n*x)/x