Integrand size = 19, antiderivative size = 58 \[ \int e^{-2 p \text {arctanh}(a x)} (c-a c x)^p \, dx=-\frac {2^{-p} (1-a x)^{1+p} (c-a c x)^p \operatorname {Hypergeometric2F1}\left (p,1+2 p,2 (1+p),\frac {1}{2} (1-a x)\right )}{a (1+2 p)} \] Output:
-(-a*x+1)^(p+1)*(-a*c*x+c)^p*hypergeom([p, 1+2*p],[2*p+2],-1/2*a*x+1/2)/(2 ^p)/a/(1+2*p)
Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.97 \[ \int e^{-2 p \text {arctanh}(a x)} (c-a c x)^p \, dx=-\frac {2^{-p} (1-a x)^{1+p} (c-a c x)^p \operatorname {Hypergeometric2F1}\left (p,1+2 p,2+2 p,\frac {1}{2}-\frac {a x}{2}\right )}{a+2 a p} \] Input:
Integrate[(c - a*c*x)^p/E^(2*p*ArcTanh[a*x]),x]
Output:
-(((1 - a*x)^(1 + p)*(c - a*c*x)^p*Hypergeometric2F1[p, 1 + 2*p, 2 + 2*p, 1/2 - (a*x)/2])/(2^p*(a + 2*a*p)))
Time = 0.25 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, 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.
\(\displaystyle \int e^{-2 p \text {arctanh}(a x)} (c-a c x)^p \, dx\) |
\(\Big \downarrow \) 6680 |
\(\displaystyle \int (1-a x)^p (a x+1)^{-p} (c-a c x)^pdx\) |
\(\Big \downarrow \) 37 |
\(\displaystyle (1-a x)^{-p} (c-a c x)^p \int (1-a x)^{2 p} (a x+1)^{-p}dx\) |
\(\Big \downarrow \) 79 |
\(\displaystyle -\frac {2^{-p} (1-a x)^{p+1} (c-a c x)^p \operatorname {Hypergeometric2F1}\left (p,2 p+1,2 (p+1),\frac {1}{2} (1-a x)\right )}{a (2 p+1)}\) |
Input:
Int[(c - a*c*x)^p/E^(2*p*ArcTanh[a*x]),x]
Output:
-(((1 - a*x)^(1 + p)*(c - a*c*x)^p*Hypergeometric2F1[p, 1 + 2*p, 2*(1 + p) , (1 - a*x)/2])/(2^p*a*(1 + 2*p)))
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]
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]))
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])
\[\int \left (-a c x +c \right )^{p} {\mathrm e}^{-2 p \,\operatorname {arctanh}\left (a x \right )}d x\]
Input:
int((-a*c*x+c)^p/exp(2*p*arctanh(a*x)),x)
Output:
int((-a*c*x+c)^p/exp(2*p*arctanh(a*x)),x)
\[ \int e^{-2 p \text {arctanh}(a x)} (c-a c x)^p \, dx=\int { \frac {{\left (-a c x + c\right )}^{p}}{\left (-\frac {a x + 1}{a x - 1}\right )^{p}} \,d x } \] Input:
integrate((-a*c*x+c)^p/exp(2*p*arctanh(a*x)),x, algorithm="fricas")
Output:
integral((-a*c*x + c)^p/(-(a*x + 1)/(a*x - 1))^p, x)
\[ \int e^{-2 p \text {arctanh}(a x)} (c-a c x)^p \, dx=\int \left (- c \left (a x - 1\right )\right )^{p} e^{- 2 p \operatorname {atanh}{\left (a x \right )}}\, dx \] Input:
integrate((-a*c*x+c)**p/exp(2*p*atanh(a*x)),x)
Output:
Integral((-c*(a*x - 1))**p*exp(-2*p*atanh(a*x)), x)
\[ \int e^{-2 p \text {arctanh}(a x)} (c-a c x)^p \, dx=\int { \frac {{\left (-a c x + c\right )}^{p}}{\left (-\frac {a x + 1}{a x - 1}\right )^{p}} \,d x } \] Input:
integrate((-a*c*x+c)^p/exp(2*p*arctanh(a*x)),x, algorithm="maxima")
Output:
integrate((-a*c*x + c)^p/(-(a*x + 1)/(a*x - 1))^p, x)
\[ \int e^{-2 p \text {arctanh}(a x)} (c-a c x)^p \, dx=\int { \frac {{\left (-a c x + c\right )}^{p}}{\left (-\frac {a x + 1}{a x - 1}\right )^{p}} \,d x } \] Input:
integrate((-a*c*x+c)^p/exp(2*p*arctanh(a*x)),x, algorithm="giac")
Output:
integrate((-a*c*x + c)^p/(-(a*x + 1)/(a*x - 1))^p, x)
Timed out. \[ \int e^{-2 p \text {arctanh}(a x)} (c-a c x)^p \, dx=\int {\mathrm {e}}^{-2\,p\,\mathrm {atanh}\left (a\,x\right )}\,{\left (c-a\,c\,x\right )}^p \,d x \] Input:
int(exp(-2*p*atanh(a*x))*(c - a*c*x)^p,x)
Output:
int(exp(-2*p*atanh(a*x))*(c - a*c*x)^p, x)
\[ \int e^{-2 p \text {arctanh}(a x)} (c-a c x)^p \, dx=\int \frac {\left (-a c x +c \right )^{p}}{e^{2 \mathit {atanh} \left (a x \right ) p}}d x \] Input:
int((-a*c*x+c)^p/exp(2*p*atanh(a*x)),x)
Output:
int(( - a*c*x + c)**p/e**(2*atanh(a*x)*p),x)