\(\int e^{n \text {arctanh}(a+b x)} x \, dx\) [908]

Optimal result

Integrand size = 12, antiderivative size = 114 \[ \int e^{n \text {arctanh}(a+b x)} x \, dx=-\frac {(1-a-b x)^{1-\frac {n}{2}} (1+a+b x)^{\frac {2+n}{2}}}{2 b^2}+\frac {2^{n/2} (2 a-n) (1-a-b x)^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-a-b x)\right )}{b^2 (2-n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.84 \[ \int e^{n \text {arctanh}(a+b x)} x \, dx=\frac {(1-a-b x)^{1-\frac {n}{2}} \left (-b (1+a+b x)^{1+\frac {n}{2}}+\frac {2^{1+\frac {n}{2}} b (-2 a+n) \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-a-b x)\right )}{-2+n}\right )}{2 b^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6713, 90, 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.

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

\(\Big \downarrow \) 6713

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

\(\Big \downarrow \) 90

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

\(\Big \downarrow \) 79

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

Input:

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

Output:

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

Defintions of rubi rules used

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 90

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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