\(\int e^{n \text {arctanh}(a+b x)} (c x)^m \, dx\) [905]

Optimal result

Integrand size = 16, antiderivative size = 114 \[ \int e^{n \text {arctanh}(a+b x)} (c x)^m \, dx=\frac {(c x)^{1+m} (1-a-b x)^{-n/2} (1+a+b x)^{n/2} \left (1-\frac {b x}{1-a}\right )^{n/2} \left (1+\frac {b x}{1+a}\right )^{-n/2} \operatorname {AppellF1}\left (1+m,\frac {n}{2},-\frac {n}{2},2+m,\frac {b x}{1-a},-\frac {b x}{1+a}\right )}{c (1+m)} \] Output:

(c*x)^(1+m)*(b*x+a+1)^(1/2*n)*(1-b*x/(1-a))^(1/2*n)*AppellF1(1+m,1/2*n,-1/ 
2*n,2+m,b*x/(1-a),-b*x/(1+a))/c/(1+m)/((-b*x-a+1)^(1/2*n))/((1+b*x/(1+a))^ 
(1/2*n))
 

Mathematica [F]

\[ \int e^{n \text {arctanh}(a+b x)} (c x)^m \, dx=\int e^{n \text {arctanh}(a+b x)} (c x)^m \, dx \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6713, 152, 152, 150}

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.

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

\(\Big \downarrow \) 6713

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

\(\Big \downarrow \) 152

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

\(\Big \downarrow \) 152

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

\(\Big \downarrow \) 150

\(\displaystyle \frac {(c x)^{m+1} (-a-b x+1)^{-n/2} (a+b x+1)^{n/2} \left (1-\frac {b x}{1-a}\right )^{n/2} \left (\frac {b x}{a+1}+1\right )^{-n/2} \operatorname {AppellF1}\left (m+1,\frac {n}{2},-\frac {n}{2},m+2,\frac {b x}{1-a},-\frac {b x}{a+1}\right )}{c (m+1)}\)

Input:

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

Output:

((c*x)^(1 + m)*(1 + a + b*x)^(n/2)*(1 - (b*x)/(1 - a))^(n/2)*AppellF1[1 + 
m, n/2, -1/2*n, 2 + m, (b*x)/(1 - a), -((b*x)/(1 + a))])/(c*(1 + m)*(1 - a 
 - b*x)^(n/2)*(1 + (b*x)/(1 + a))^(n/2))
 

Defintions of rubi rules used

rule 150

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 
, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] &&  !In 
tegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
 

rule 152

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^IntPart[n]*((c + d*x)^FracPart[n]/(1 + d*(x/c))^FracPart[n]) 
Int[(b*x)^m*(1 + d*(x/c))^n*(e + f*x)^p, x], x] /; FreeQ[{b, c, d, e, f, m, 
 n, p}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] &&  !GtQ[c, 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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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