Integrand size = 22, antiderivative size = 127 \[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\frac {\left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {1}{2} (n-2 p)} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}+p} \left (1+\frac {1}{a x}\right )^{1+\frac {n}{2}+p} x \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-1-2 p,\frac {1}{2} (n-2 p),-2 p,\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{1+2 p} \]
((a-1/x)/(a+1/x))^(1/2*n-p)*(1-1/a/x)^(-1/2*n+p)*(1+1/a/x)^(1+1/2*n+p)*x*( -a^2*c*x^2+c)^p*hypergeom([-1-2*p, 1/2*n-p],[-2*p],2/(a+1/x)/x)/(1+2*p)/(( 1-1/a^2/x^2)^p)
Time = 0.21 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.65 \[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=-\frac {e^{(-2+n) \coth ^{-1}(a x)} \left (-1+e^{2 \coth ^{-1}(a x)}\right ) \left (-1+a^2 x^2\right ) \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (1,-\frac {n}{2}-p,2-\frac {n}{2}+p,e^{-2 \coth ^{-1}(a x)}\right )}{a (n-2 (1+p))} \]
-((E^((-2 + n)*ArcCoth[a*x])*(-1 + E^(2*ArcCoth[a*x]))*(-1 + a^2*x^2)*(c - a^2*c*x^2)^p*Hypergeometric2F1[1, -1/2*n - p, 2 - n/2 + p, E^(-2*ArcCoth[ a*x])])/(a*(n - 2*(1 + p))))
Time = 0.41 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6746, 6750, 142}
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.
\(\displaystyle \int \left (c-a^2 c x^2\right )^p e^{n \coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6746 |
\(\displaystyle x^{-2 p} \left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (c-a^2 c x^2\right )^p \int e^{n \coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^p x^{2 p}dx\) |
\(\Big \downarrow \) 6750 |
\(\displaystyle \left (\frac {1}{x}\right )^{2 p} \left (-\left (1-\frac {1}{a^2 x^2}\right )^{-p}\right ) \left (c-a^2 c x^2\right )^p \int \left (1-\frac {1}{a x}\right )^{p-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {n}{2}+p} \left (\frac {1}{x}\right )^{-2 (p+1)}d\frac {1}{x}\) |
\(\Big \downarrow \) 142 |
\(\displaystyle \frac {x \left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {1}{2} (n-2 p)} \left (1-\frac {1}{a x}\right )^{p-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n}{2}+p+1} \operatorname {Hypergeometric2F1}\left (-2 p-1,\frac {1}{2} (n-2 p),-2 p,\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{2 p+1}\) |
(((a - x^(-1))/(a + x^(-1)))^((n - 2*p)/2)*(1 - 1/(a*x))^(-1/2*n + p)*(1 + 1/(a*x))^(1 + n/2 + p)*x*(c - a^2*c*x^2)^p*Hypergeometric2F1[-1 - 2*p, (n - 2*p)/2, -2*p, 2/((a + x^(-1))*x)])/((1 + 2*p)*(1 - 1/(a^2*x^2))^p)
3.8.62.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((b*e - a*f)*(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2, (-(d*e - c*f))*((a + b*x)/((b*c - a*d)*(e + f*x)))])/((b*e - a*f)*((c + d*x)/((b*c - a*d)*(e + f *x))))^n, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[m + n + p + 2, 0] && !IntegerQ[n]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbo l] :> Simp[(c + d*x^2)^p/(x^(2*p)*(1 - 1/(a^2*x^2))^p) Int[u*x^(2*p)*(1 - 1/(a^2*x^2))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && !IntegerQ[n/2] && !IntegerQ[p]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_)^2)^(p_.)*(x_)^(m_), x_ Symbol] :> Simp[(-c^p)*x^m*(1/x)^m Subst[Int[(1 - x/a)^(p - n/2)*((1 + x/ a)^(p + n/2)/x^(m + 2)), x], x, 1/x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[c + a^2*d, 0] && !IntegerQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) && !In tegersQ[2*p, p + n/2] && !IntegerQ[m]
\[\int {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} \left (-a^{2} c \,x^{2}+c \right )^{p}d x\]
\[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\int { {\left (-a^{2} c x^{2} + c\right )}^{p} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
\[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\int \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{p} e^{n \operatorname {acoth}{\left (a x \right )}}\, dx \]
\[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\int { {\left (-a^{2} c x^{2} + c\right )}^{p} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
\[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\int { {\left (-a^{2} c x^{2} + c\right )}^{p} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
Timed out. \[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\int {\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}\,{\left (c-a^2\,c\,x^2\right )}^p \,d x \]