Integrand size = 22, antiderivative size = 116 \[ \int e^{n \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=-\frac {2^{1-\frac {n}{2}+p} \left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (1+\frac {1}{a x}\right )^{1+\frac {n}{2}+p} \operatorname {AppellF1}\left (1+\frac {n}{2}+p,\frac {1}{2} (n-2 p),2,2+\frac {n}{2}+p,\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right )}{a (2+n+2 p)} \]
-2^(1-1/2*n+p)*(c-c/a^2/x^2)^p*(1+1/a/x)^(1+1/2*n+p)*AppellF1(1+1/2*n+p,1/ 2*n-p,2,2+1/2*n+p,1/2*(a+1/x)/a,1+1/a/x)/a/(2+n+2*p)/((1-1/a^2/x^2)^p)
\[ \int e^{n \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int e^{n \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx \]
Time = 0.38 (sec) , antiderivative size = 116, 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 = {6751, 6748, 153}
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.
\(\displaystyle \int \left (c-\frac {c}{a^2 x^2}\right )^p e^{n \coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6751 |
\(\displaystyle \left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \int e^{n \coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^pdx\) |
\(\Big \downarrow \) 6748 |
\(\displaystyle -\left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \int \left (1-\frac {1}{a x}\right )^{p-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {n}{2}+p} x^2d\frac {1}{x}\) |
\(\Big \downarrow \) 153 |
\(\displaystyle -\frac {2^{-\frac {n}{2}+p+1} \left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (\frac {1}{a x}+1\right )^{\frac {n}{2}+p+1} \operatorname {AppellF1}\left (\frac {n}{2}+p+1,\frac {1}{2} (n-2 p),2,\frac {n}{2}+p+2,\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right )}{a (n+2 p+2)}\) |
-((2^(1 - n/2 + p)*(c - c/(a^2*x^2))^p*(1 + 1/(a*x))^(1 + n/2 + p)*AppellF 1[1 + n/2 + p, (n - 2*p)/2, 2, 2 + n/2 + p, (a + x^(-1))/(2*a), 1 + 1/(a*x )])/(a*(2 + n + 2*p)*(1 - 1/(a^2*x^2))^p))
3.10.33.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) ^(p_), x_] :> Simp[(b*e - a*f)^p*((a + b*x)^(m + 1)/(b^(p + 1)*(m + 1)*Simp lify[b/(b*c - a*d)]^n))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/(b*c - a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && IntegerQ[p] && GtQ[Simplify[b/( b*c - a*d)], 0] && !(GtQ[Simplify[d/(d*a - c*b)], 0] && SimplerQ[c + d*x, a + b*x])
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_)^2)^(p_.), x_Symbol] :> Simp[-c^p Subst[Int[(1 - x/a)^(p - n/2)*((1 + x/a)^(p + n/2)/x^2), x], x , 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && !IntegerQ[ n/2] && (IntegerQ[p] || GtQ[c, 0]) && !IntegersQ[2*p, p + n/2]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_), x_Symbo l] :> Simp[c^IntPart[p]*((c + d/x^2)^FracPart[p]/(1 - 1/(a^2*x^2))^FracPart [p]) Int[u*(1 - 1/(a^2*x^2))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && !IntegerQ[n/2] && !(IntegerQ[p] || GtQ[c, 0])
\[\int {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} \left (c -\frac {c}{a^{2} x^{2}}\right )^{p}d x\]
\[ \int e^{n \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int { {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
\[ \int e^{n \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int \left (- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )\right )^{p} e^{n \operatorname {acoth}{\left (a x \right )}}\, dx \]
\[ \int e^{n \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int { {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
\[ \int e^{n \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int { {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
Timed out. \[ \int e^{n \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int {\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}\,{\left (c-\frac {c}{a^2\,x^2}\right )}^p \,d x \]