\(\int \frac {e^{n \text {arctanh}(a+b x)}}{x^3} \, dx\) [897]

Optimal result

Integrand size = 14, antiderivative size = 152 \[ \int \frac {e^{n \text {arctanh}(a+b x)}}{x^3} \, dx=-\frac {(1-a-b x)^{1-\frac {n}{2}} (1+a+b x)^{\frac {2+n}{2}}}{2 \left (1-a^2\right ) x^2}-\frac {2 b^2 (2 a+n) (1-a-b x)^{1-\frac {n}{2}} (1+a+b x)^{\frac {1}{2} (-2+n)} \operatorname {Hypergeometric2F1}\left (2,1-\frac {n}{2},2-\frac {n}{2},\frac {(1+a) (1-a-b x)}{(1-a) (1+a+b x)}\right )}{(1-a)^3 (1+a) (2-n)} \] Output:

-1/2*(-b*x-a+1)^(1-1/2*n)*(b*x+a+1)^(1+1/2*n)/(-a^2+1)/x^2-2*b^2*(2*a+n)*( 
-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)^3/(1+a)/(2-n)
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.81 \[ \int \frac {e^{n \text {arctanh}(a+b x)}}{x^3} \, dx=\frac {(1-a-b x)^{1-\frac {n}{2}} (1+a+b x)^{-1+\frac {n}{2}} \left ((-1+a)^2 (-2+n) (1+a+b x)^2-4 b^2 (2 a+n) x^2 \operatorname {Hypergeometric2F1}\left (2,1-\frac {n}{2},2-\frac {n}{2},\frac {(1+a) (-1+a+b x)}{(-1+a) (1+a+b x)}\right )\right )}{2 (-1+a)^3 (1+a) (-2+n) x^2} \] Input:

Integrate[E^(n*ArcTanh[a + b*x])/x^3,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6713, 107, 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.

\(\displaystyle \int \frac {e^{n \text {arctanh}(a+b x)}}{x^3} \, dx\)

\(\Big \downarrow \) 6713

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

\(\Big \downarrow \) 107

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

\(\Big \downarrow \) 141

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

Input:

Int[E^(n*ArcTanh[a + b*x])/x^3,x]
 

Output:

-1/2*((1 - a - b*x)^(1 - n/2)*(1 + a + b*x)^((2 + n)/2))/((1 - a^2)*x^2) - 
 (2*b^2*(2*a + n)*(1 - a - b*x)^(1 - n/2)*(1 + a + b*x)^((-2 + n)/2)*Hyper 
geometric2F1[2, 1 - n/2, 2 - n/2, ((1 + a)*(1 - a - b*x))/((1 - a)*(1 + a 
+ b*x))])/((1 - a)^2*(1 - a^2)*(2 - n))
 

Defintions of rubi rules used

rule 107

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[(a*d*f*(m + 1) + b*c*f*(n + 
 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x 
] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 1])
 

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]

\[\int \frac {{\mathrm e}^{n \,\operatorname {arctanh}\left (b x +a \right )}}{x^{3}}d x\]

Input:

int(exp(n*arctanh(b*x+a))/x^3,x)
 

Output:

int(exp(n*arctanh(b*x+a))/x^3,x)
 

Fricas [F]

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

integrate(exp(n*arctanh(b*x+a))/x^3,x, algorithm="fricas")
 

Output:

integral((-(b*x + a + 1)/(b*x + a - 1))^(1/2*n)/x^3, x)
 

Sympy [F]

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

integrate(exp(n*atanh(b*x+a))/x**3,x)
 

Output:

Integral(exp(n*atanh(a + b*x))/x**3, x)
 

Maxima [F]

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

integrate(exp(n*arctanh(b*x+a))/x^3,x, algorithm="maxima")
 

Output:

integrate((-(b*x + a + 1)/(b*x + a - 1))^(1/2*n)/x^3, x)
 

Giac [F]

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

integrate(exp(n*arctanh(b*x+a))/x^3,x, algorithm="giac")
 

Output:

integrate((-(b*x + a + 1)/(b*x + a - 1))^(1/2*n)/x^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{n \text {arctanh}(a+b x)}}{x^3} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a+b\,x\right )}}{x^3} \,d x \] Input:

int(exp(n*atanh(a + b*x))/x^3,x)
 

Output:

int(exp(n*atanh(a + b*x))/x^3, x)
 

Reduce [F]

\[ \int \frac {e^{n \text {arctanh}(a+b x)}}{x^3} \, dx=\frac {-e^{\mathit {atanh} \left (b x +a \right ) n}-2 \left (\int \frac {e^{\mathit {atanh} \left (b x +a \right ) n}}{2 a^{3} b^{2} x^{4}+a^{2} b^{2} n \,x^{4}+4 a^{4} b \,x^{3}+2 a^{3} b n \,x^{3}+2 a^{5} x^{2}+a^{4} n \,x^{2}-2 a \,b^{2} x^{4}-b^{2} n \,x^{4}-4 a^{2} b \,x^{3}-2 a b n \,x^{3}-4 a^{3} x^{2}-2 a^{2} n \,x^{2}+2 a \,x^{2}+n \,x^{2}}d x \right ) a^{3} b n \,x^{2}-\left (\int \frac {e^{\mathit {atanh} \left (b x +a \right ) n}}{2 a^{3} b^{2} x^{4}+a^{2} b^{2} n \,x^{4}+4 a^{4} b \,x^{3}+2 a^{3} b n \,x^{3}+2 a^{5} x^{2}+a^{4} n \,x^{2}-2 a \,b^{2} x^{4}-b^{2} n \,x^{4}-4 a^{2} b \,x^{3}-2 a b n \,x^{3}-4 a^{3} x^{2}-2 a^{2} n \,x^{2}+2 a \,x^{2}+n \,x^{2}}d x \right ) a^{2} b \,n^{2} x^{2}+2 \left (\int \frac {e^{\mathit {atanh} \left (b x +a \right ) n}}{2 a^{3} b^{2} x^{4}+a^{2} b^{2} n \,x^{4}+4 a^{4} b \,x^{3}+2 a^{3} b n \,x^{3}+2 a^{5} x^{2}+a^{4} n \,x^{2}-2 a \,b^{2} x^{4}-b^{2} n \,x^{4}-4 a^{2} b \,x^{3}-2 a b n \,x^{3}-4 a^{3} x^{2}-2 a^{2} n \,x^{2}+2 a \,x^{2}+n \,x^{2}}d x \right ) a b n \,x^{2}+\left (\int \frac {e^{\mathit {atanh} \left (b x +a \right ) n}}{2 a^{3} b^{2} x^{4}+a^{2} b^{2} n \,x^{4}+4 a^{4} b \,x^{3}+2 a^{3} b n \,x^{3}+2 a^{5} x^{2}+a^{4} n \,x^{2}-2 a \,b^{2} x^{4}-b^{2} n \,x^{4}-4 a^{2} b \,x^{3}-2 a b n \,x^{3}-4 a^{3} x^{2}-2 a^{2} n \,x^{2}+2 a \,x^{2}+n \,x^{2}}d x \right ) b \,n^{2} x^{2}}{2 x^{2}} \] Input:

int(exp(n*atanh(b*x+a))/x^3,x)
 

Output:

( - e**(atanh(a + b*x)*n) - 2*int(e**(atanh(a + b*x)*n)/(2*a**5*x**2 + 4*a 
**4*b*x**3 + a**4*n*x**2 + 2*a**3*b**2*x**4 + 2*a**3*b*n*x**3 - 4*a**3*x** 
2 + a**2*b**2*n*x**4 - 4*a**2*b*x**3 - 2*a**2*n*x**2 - 2*a*b**2*x**4 - 2*a 
*b*n*x**3 + 2*a*x**2 - b**2*n*x**4 + n*x**2),x)*a**3*b*n*x**2 - int(e**(at 
anh(a + b*x)*n)/(2*a**5*x**2 + 4*a**4*b*x**3 + a**4*n*x**2 + 2*a**3*b**2*x 
**4 + 2*a**3*b*n*x**3 - 4*a**3*x**2 + a**2*b**2*n*x**4 - 4*a**2*b*x**3 - 2 
*a**2*n*x**2 - 2*a*b**2*x**4 - 2*a*b*n*x**3 + 2*a*x**2 - b**2*n*x**4 + n*x 
**2),x)*a**2*b*n**2*x**2 + 2*int(e**(atanh(a + b*x)*n)/(2*a**5*x**2 + 4*a* 
*4*b*x**3 + a**4*n*x**2 + 2*a**3*b**2*x**4 + 2*a**3*b*n*x**3 - 4*a**3*x**2 
 + a**2*b**2*n*x**4 - 4*a**2*b*x**3 - 2*a**2*n*x**2 - 2*a*b**2*x**4 - 2*a* 
b*n*x**3 + 2*a*x**2 - b**2*n*x**4 + n*x**2),x)*a*b*n*x**2 + int(e**(atanh( 
a + b*x)*n)/(2*a**5*x**2 + 4*a**4*b*x**3 + a**4*n*x**2 + 2*a**3*b**2*x**4 
+ 2*a**3*b*n*x**3 - 4*a**3*x**2 + a**2*b**2*n*x**4 - 4*a**2*b*x**3 - 2*a** 
2*n*x**2 - 2*a*b**2*x**4 - 2*a*b*n*x**3 + 2*a*x**2 - b**2*n*x**4 + n*x**2) 
,x)*b*n**2*x**2)/(2*x**2)