\(\int e^{n \text {arctanh}(a x)} (c-a c x)^p \, dx\) [312]

Optimal result

Integrand size = 18, antiderivative size = 79 \[ \int e^{n \text {arctanh}(a x)} (c-a c x)^p \, dx=-\frac {2^{1+\frac {n}{2}} (1-a x)^{1-\frac {n}{2}} (c-a c x)^p \operatorname {Hypergeometric2F1}\left (-\frac {n}{2},1-\frac {n}{2}+p,2-\frac {n}{2}+p,\frac {1}{2} (1-a x)\right )}{a (2-n+2 p)} \] Output:

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.97 \[ \int e^{n \text {arctanh}(a x)} (c-a c x)^p \, dx=\frac {2^{1+\frac {n}{2}} (1-a x)^{1-\frac {n}{2}} (c-a c x)^p \operatorname {Hypergeometric2F1}\left (-\frac {n}{2},1-\frac {n}{2}+p,2-\frac {n}{2}+p,\frac {1}{2}-\frac {a x}{2}\right )}{a (n-2 (1+p))} \] Input:

Integrate[E^(n*ArcTanh[a*x])*(c - a*c*x)^p,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6680, 37, 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.

\(\displaystyle \int e^{n \text {arctanh}(a x)} (c-a c x)^p \, dx\)

\(\Big \downarrow \) 6680

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

\(\Big \downarrow \) 37

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

\(\Big \downarrow \) 79

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

Input:

Int[E^(n*ArcTanh[a*x])*(c - a*c*x)^p,x]
 

Output:

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

Defintions of rubi rules used

rule 37

Int[(u_.)*((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> S 
imp[(a + b*x)^m/(c + d*x)^m   Int[u*(c + d*x)^(m + n), x], x] /; FreeQ[{a, 
b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] &&  !SimplerQ[a + b*x, c + d*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 6680

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol 
] :> Int[u*(c + d*x)^p*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] /; FreeQ[{a, c 
, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] &&  !(IntegerQ[p] || GtQ[c, 0])
 

Maple [F]

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

Input:

int(exp(n*arctanh(a*x))*(-a*c*x+c)^p,x)
 

Output:

int(exp(n*arctanh(a*x))*(-a*c*x+c)^p,x)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

\[ \int e^{n \text {arctanh}(a x)} (c-a c x)^p \, dx=\int \left (- c \left (a x - 1\right )\right )^{p} e^{n \operatorname {atanh}{\left (a x \right )}}\, dx \] Input:

integrate(exp(n*atanh(a*x))*(-a*c*x+c)**p,x)
 

Output:

Integral((-c*(a*x - 1))**p*exp(n*atanh(a*x)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

int(exp(n*atanh(a*x))*(c - a*c*x)^p,x)
 

Output:

int(exp(n*atanh(a*x))*(c - a*c*x)^p, x)
 

Reduce [F]

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

int(exp(n*atanh(a*x))*(-a*c*x+c)^p,x)
 

Output:

int(e**(atanh(a*x)*n)*( - a*c*x + c)**p,x)