Integrand size = 18, antiderivative size = 44 \[ \int e^{-2 \coth ^{-1}(a x)} (c-a c x)^p \, dx=\frac {(c-a c x)^{2+p} \operatorname {Hypergeometric2F1}\left (1,2+p,3+p,\frac {1}{2} (1-a x)\right )}{2 a c^2 (2+p)} \] 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)
Time = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00 \[ \int e^{-2 \coth ^{-1}(a x)} (c-a c x)^p \, dx=-\frac {(-1+a x) (c-a c x)^p \left (-1+\operatorname {Hypergeometric2F1}\left (1,1+p,2+p,\frac {1}{2} (1-a x)\right )\right )}{a (1+p)} \] Input:
Integrate[(c - a*c*x)^p/E^(2*ArcCoth[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)))
Time = 0.47 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6717, 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.
\(\displaystyle \int e^{-2 \coth ^{-1}(a x)} (c-a c x)^p \, dx\) |
\(\Big \downarrow \) 6717 |
\(\displaystyle -\int e^{-2 \text {arctanh}(a x)} (c-a c x)^pdx\) |
\(\Big \downarrow \) 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*ArcCoth[a*x]),x]
Output:
((c - a*c*x)^(2 + p)*Hypergeometric2F1[1, 2 + p, 3 + p, (1 - a*x)/2])/(2*a *c^2*(2 + p))
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])
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]
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])
Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Simp[(-1)^(n/2) Int[ u*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[a, x] && IntegerQ[n/2]
\[\int \frac {\left (-a c x +c \right )^{p} \left (a x -1\right )}{a x +1}d x\]
Input:
int((-a*c*x+c)^p*(a*x-1)/(a*x+1),x)
Output:
int((-a*c*x+c)^p*(a*x-1)/(a*x+1),x)
\[ \int e^{-2 \coth ^{-1}(a x)} (c-a c x)^p \, dx=\int { \frac {{\left (a x - 1\right )} {\left (-a c x + c\right )}^{p}}{a x + 1} \,d x } \] Input:
integrate((-a*c*x+c)^p*(a*x-1)/(a*x+1),x, algorithm="fricas")
Output:
integral((a*x - 1)*(-a*c*x + c)^p/(a*x + 1), x)
\[ \int e^{-2 \coth ^{-1}(a x)} (c-a c x)^p \, dx=\int \frac {\left (- c \left (a x - 1\right )\right )^{p} \left (a x - 1\right )}{a x + 1}\, dx \] Input:
integrate((-a*c*x+c)**p*(a*x-1)/(a*x+1),x)
Output:
Integral((-c*(a*x - 1))**p*(a*x - 1)/(a*x + 1), x)
\[ \int e^{-2 \coth ^{-1}(a x)} (c-a c x)^p \, dx=\int { \frac {{\left (a x - 1\right )} {\left (-a c x + c\right )}^{p}}{a x + 1} \,d x } \] Input:
integrate((-a*c*x+c)^p*(a*x-1)/(a*x+1),x, algorithm="maxima")
Output:
integrate((a*x - 1)*(-a*c*x + c)^p/(a*x + 1), x)
\[ \int e^{-2 \coth ^{-1}(a x)} (c-a c x)^p \, dx=\int { \frac {{\left (a x - 1\right )} {\left (-a c x + c\right )}^{p}}{a x + 1} \,d x } \] Input:
integrate((-a*c*x+c)^p*(a*x-1)/(a*x+1),x, algorithm="giac")
Output:
integrate((a*x - 1)*(-a*c*x + c)^p/(a*x + 1), x)
Timed out. \[ \int e^{-2 \coth ^{-1}(a x)} (c-a c x)^p \, dx=\int \frac {{\left (c-a\,c\,x\right )}^p\,\left (a\,x-1\right )}{a\,x+1} \,d x \] Input:
int(((c - a*c*x)^p*(a*x - 1))/(a*x + 1),x)
Output:
int(((c - a*c*x)^p*(a*x - 1))/(a*x + 1), x)
\[ \int e^{-2 \coth ^{-1}(a x)} (c-a c x)^p \, dx=\frac {\left (-a c x +c \right )^{p} a p x +\left (-a c x +c \right )^{p} p +2 \left (-a c x +c \right )^{p}-4 \left (\int \frac {\left (-a c x +c \right )^{p} x}{a^{2} x^{2}-1}d x \right ) a^{2} p^{2}-4 \left (\int \frac {\left (-a c x +c \right )^{p} x}{a^{2} x^{2}-1}d x \right ) a^{2} p}{a p \left (p +1\right )} \] Input:
int((-a*c*x+c)^p*(a*x-1)/(a*x+1),x)
Output:
(( - a*c*x + c)**p*a*p*x + ( - a*c*x + c)**p*p + 2*( - a*c*x + c)**p - 4*i nt((( - 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))