\(\int \frac {e^{n \text {arctanh}(a+b x)}}{x} \, dx\) [910]

Optimal result

Integrand size = 14, antiderivative size = 135 \[ \int \frac {e^{n \text {arctanh}(a+b x)}}{x} \, dx=\frac {2 (1-a-b x)^{-n/2} (1+a+b x)^{n/2} \operatorname {Hypergeometric2F1}\left (1,-\frac {n}{2},1-\frac {n}{2},\frac {(1+a) (1-a-b x)}{(1-a) (1+a+b x)}\right )}{n}-\frac {2^{1+\frac {n}{2}} (1-a-b x)^{-n/2} \operatorname {Hypergeometric2F1}\left (-\frac {n}{2},-\frac {n}{2},1-\frac {n}{2},\frac {1}{2} (1-a-b x)\right )}{n} \] Output:

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.82 \[ \int \frac {e^{n \text {arctanh}(a+b x)}}{x} \, dx=\frac {2 (1-a-b x)^{-n/2} \left ((1+a+b x)^{n/2} \operatorname {Hypergeometric2F1}\left (1,-\frac {n}{2},1-\frac {n}{2},\frac {(1+a) (-1+a+b x)}{(-1+a) (1+a+b x)}\right )-2^{n/2} \operatorname {Hypergeometric2F1}\left (-\frac {n}{2},-\frac {n}{2},1-\frac {n}{2},\frac {1}{2} (1-a-b x)\right )\right )}{n} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6713, 140, 27, 79, 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.

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

\(\Big \downarrow \) 6713

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

\(\Big \downarrow \) 140

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 79

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

\(\Big \downarrow \) 141

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

Input:

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

Output:

(2*(1 + a + b*x)^(n/2)*Hypergeometric2F1[1, -1/2*n, 1 - n/2, ((1 + a)*(1 - 
 a - b*x))/((1 - a)*(1 + a + b*x))])/(n*(1 - a - b*x)^(n/2)) - (2^(1 + n/2 
)*Hypergeometric2F1[-1/2*n, -1/2*n, 1 - n/2, (1 - a - b*x)/2])/(n*(1 - a - 
 b*x)^(n/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 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 140

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(e**(atanh(a + b*x)*n)/x,x)