\(\int e^{-2 \text {arctanh}(a x)} (c-a c x)^p \, dx\) [305]

Optimal result

Integrand size = 18, antiderivative size = 44 \[ \int e^{-2 \text {arctanh}(a x)} (c-a c x)^p \, dx=-\frac {(c-a c x)^{2+p} \operatorname {Hypergeometric2F1}\left (1,2+p,3+p,\frac {1}{2} (1-a x)\right )}{2 a c^2 (2+p)} \] Output:

-1/2*(-a*c*x+c)^(2+p)*hypergeom([1, 2+p],[3+p],-1/2*a*x+1/2)/a/c^2/(2+p)
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.98 \[ \int e^{-2 \text {arctanh}(a x)} (c-a c x)^p \, dx=\frac {(-1+a x) (c-a c x)^p \left (-1+\operatorname {Hypergeometric2F1}\left (1,1+p,2+p,\frac {1}{2} (1-a x)\right )\right )}{a (1+p)} \] Input:

Integrate[(c - a*c*x)^p/E^(2*ArcTanh[a*x]),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 44, 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, 35, 78}

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.

\(\displaystyle \int e^{-2 \text {arctanh}(a x)} (c-a c x)^p \, dx\)

\(\Big \downarrow \) 6680

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

\(\Big \downarrow \) 35

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

\(\Big \downarrow \) 78

\(\displaystyle -\frac {(c-a c x)^{p+2} \operatorname {Hypergeometric2F1}\left (1,p+2,p+3,\frac {1}{2} (1-a x)\right )}{2 a c^2 (p+2)}\)

Input:

Int[(c - a*c*x)^p/E^(2*ArcTanh[a*x]),x]
 

Output:

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

Defintions of rubi rules used

rule 35

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

rule 78

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b 
*c - a*d)^n*((a + b*x)^(m + 1)/(b^(n + 1)*(m + 1)))*Hypergeometric2F1[-n, m 
 + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m}, x] 
 &&  !IntegerQ[m] && IntegerQ[n]
 

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]

\[\int \frac {\left (-a c x +c \right )^{p} \left (-a^{2} x^{2}+1\right )}{\left (a x +1\right )^{2}}d x\]

Input:

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

Output:

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

Fricas [F]

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

integrate((-a*c*x+c)^p/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

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

Output:

-Integral(-(-a*c*x + c)**p/(a*x + 1), x) - Integral(a*x*(-a*c*x + c)**p/(a 
*x + 1), x)
 

Maxima [F]

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

integrate((-a*c*x+c)^p/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((-a*c*x+c)^p/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int e^{-2 \text {arctanh}(a x)} (c-a c x)^p \, dx=\frac {-\left (-a c x +c \right )^{p} a p x -\left (-a c x +c \right )^{p} p -2 \left (-a c x +c \right )^{p}+4 \left (\int \frac {\left (-a c x +c \right )^{p} x}{a^{2} x^{2}-1}d x \right ) a^{2} p^{2}+4 \left (\int \frac {\left (-a c x +c \right )^{p} x}{a^{2} x^{2}-1}d x \right ) a^{2} p}{a p \left (p +1\right )} \] Input:

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

Output:

( - ( - a*c*x + c)**p*a*p*x - ( - a*c*x + c)**p*p - 2*( - a*c*x + c)**p + 
4*int((( - a*c*x + c)**p*x)/(a**2*x**2 - 1),x)*a**2*p**2 + 4*int((( - a*c* 
x + c)**p*x)/(a**2*x**2 - 1),x)*a**2*p)/(a*p*(p + 1))