Integrand size = 24, antiderivative size = 88 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{-2+\frac {n}{2}} \, dx=-\frac {2 \left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-2+n)} x (c-a c x)^{\frac {1}{2} (-4+n)} \operatorname {Hypergeometric2F1}\left (2,1-\frac {n}{2},2-\frac {n}{2},\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{2-n} \]
-2*(1-1/a/x)^(2-1/2*n)*(1+1/a/x)^(-1+1/2*n)*x*(-a*c*x+c)^(-2+1/2*n)*hyperg eom([2, 1-1/2*n],[2-1/2*n],2/(a+1/x)/x)/(2-n)
Time = 0.05 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.01 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{-2+\frac {n}{2}} \, dx=\frac {2 \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2} (c-a c x)^{n/2} \operatorname {Hypergeometric2F1}\left (2,1-\frac {n}{2},2-\frac {n}{2},\frac {2}{1+a x}\right )}{a c^2 (-2+n) (1+a x)} \]
(2*(1 + 1/(a*x))^(n/2)*(c - a*c*x)^(n/2)*Hypergeometric2F1[2, 1 - n/2, 2 - n/2, 2/(1 + a*x)])/(a*c^2*(-2 + n)*(1 - 1/(a*x))^(n/2)*(1 + a*x))
Time = 0.28 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.19, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6727, 27, 141}
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.
\(\displaystyle \int (c-a c x)^{\frac {n}{2}-2} e^{n \coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6727 |
\(\displaystyle \left (\frac {1}{x}\right )^{\frac {n-4}{2}} \left (-\left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}}\right ) (c-a c x)^{\frac {n-4}{2}} \int \frac {a^2 \left (1+\frac {1}{a x}\right )^{n/2} \left (\frac {1}{x}\right )^{-n/2}}{\left (a-\frac {1}{x}\right )^2}d\frac {1}{x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -a^2 \left (\frac {1}{x}\right )^{\frac {n-4}{2}} \left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} (c-a c x)^{\frac {n-4}{2}} \int \frac {\left (1+\frac {1}{a x}\right )^{n/2} \left (\frac {1}{x}\right )^{-n/2}}{\left (a-\frac {1}{x}\right )^2}d\frac {1}{x}\) |
\(\Big \downarrow \) 141 |
\(\displaystyle -\frac {2 \left (\frac {1}{x}\right )^{\frac {n-4}{2}-\frac {n}{2}+1} \left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-2}{2}} (c-a c x)^{\frac {n-4}{2}} \operatorname {Hypergeometric2F1}\left (2,1-\frac {n}{2},2-\frac {n}{2},\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{2-n}\) |
(-2*(1 - 1/(a*x))^(2 - n/2)*(1 + 1/(a*x))^((-2 + n)/2)*(x^(-1))^(1 + (-4 + n)/2 - n/2)*(c - a*c*x)^((-4 + n)/2)*Hypergeometric2F1[2, 1 - n/2, 2 - n/ 2, 2/((a + x^(-1))*x)])/(2 - n)
3.4.63.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(b*c - a*d)^n*((a + b*x)^(m + 1)/((m + 1)*(b*e - a*f)^( n + 1)*(e + f*x)^(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2, (-(d*e - c*f ))*((a + b*x)/((b*c - a*d)*(e + f*x)))], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && ILtQ[n, 0] && (SumSimplerQ[m, 1] || !Su mSimplerQ[p, 1]) && !ILtQ[m, 0]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_), x_Symbol] :> Si mp[(-(1/x)^p)*((c + d*x)^p/(1 + c/(d*x))^p) Subst[Int[((1 + c*(x/d))^p*(( 1 + x/a)^(n/2)/x^(p + 2)))/(1 - x/a)^(n/2), x], x, 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && !IntegerQ[p]
\[\int {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} \left (-a c x +c \right )^{-2+\frac {n}{2}}d x\]
\[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{-2+\frac {n}{2}} \, dx=\int { {\left (-a c x + c\right )}^{\frac {1}{2} \, n - 2} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
\[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{-2+\frac {n}{2}} \, dx=\int \left (- c \left (a x - 1\right )\right )^{\frac {n}{2} - 2} e^{n \operatorname {acoth}{\left (a x \right )}}\, dx \]
\[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{-2+\frac {n}{2}} \, dx=\int { {\left (-a c x + c\right )}^{\frac {1}{2} \, n - 2} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
\[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{-2+\frac {n}{2}} \, dx=\int { {\left (-a c x + c\right )}^{\frac {1}{2} \, n - 2} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
Timed out. \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{-2+\frac {n}{2}} \, dx=\int {\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}\,{\left (c-a\,c\,x\right )}^{\frac {n}{2}-2} \,d x \]