Integrand size = 23, antiderivative size = 76 \[ \int e^{-2 p \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\frac {\left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (1-\frac {1}{a x}\right )^{1+2 p} \operatorname {Hypergeometric2F1}\left (2,1+2 p,2 (1+p),1-\frac {1}{a x}\right )}{a (1+2 p)} \]
(c-c/a^2/x^2)^p*(1-1/a/x)^(1+2*p)*hypergeom([2, 1+2*p],[2*p+2],1-1/a/x)/a/ (1+2*p)/((1-1/a^2/x^2)^p)
\[ \int e^{-2 p \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int e^{-2 p \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx \]
Time = 0.34 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6751, 6748, 75}
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.
\(\displaystyle \int \left (c-\frac {c}{a^2 x^2}\right )^p e^{-2 p \coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6751 |
\(\displaystyle \left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \int e^{-2 p \coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^pdx\) |
\(\Big \downarrow \) 6748 |
\(\displaystyle -\left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \int \left (1-\frac {1}{a x}\right )^{2 p} x^2d\frac {1}{x}\) |
\(\Big \downarrow \) 75 |
\(\displaystyle \frac {\left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (1-\frac {1}{a x}\right )^{2 p+1} \left (c-\frac {c}{a^2 x^2}\right )^p \operatorname {Hypergeometric2F1}\left (2,2 p+1,2 (p+1),1-\frac {1}{a x}\right )}{a (2 p+1)}\) |
((c - c/(a^2*x^2))^p*(1 - 1/(a*x))^(1 + 2*p)*Hypergeometric2F1[2, 1 + 2*p, 2*(1 + p), 1 - 1/(a*x)])/(a*(1 + 2*p)*(1 - 1/(a^2*x^2))^p)
3.10.34.3.1 Defintions of rubi rules used
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x )^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))*Hypergeometric2F1[-m, n + 1, n + 2, 1 + d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[n] && (IntegerQ[m] || GtQ[-d/(b*c), 0])
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])
\[\int \left (c -\frac {c}{a^{2} x^{2}}\right )^{p} {\mathrm e}^{-2 p \,\operatorname {arccoth}\left (a x \right )}d x\]
\[ \int e^{-2 p \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int { \frac {{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p}}{\left (\frac {a x + 1}{a x - 1}\right )^{p}} \,d x } \]
\[ \int e^{-2 p \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int \left (- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )\right )^{p} e^{- 2 p \operatorname {acoth}{\left (a x \right )}}\, dx \]
\[ \int e^{-2 p \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int { \frac {{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p}}{\left (\frac {a x + 1}{a x - 1}\right )^{p}} \,d x } \]
\[ \int e^{-2 p \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int { \frac {{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p}}{\left (\frac {a x + 1}{a x - 1}\right )^{p}} \,d x } \]
Timed out. \[ \int e^{-2 p \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int {\mathrm {e}}^{-2\,p\,\mathrm {acoth}\left (a\,x\right )}\,{\left (c-\frac {c}{a^2\,x^2}\right )}^p \,d x \]