Integrand size = 22, antiderivative size = 113 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=\frac {\left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x}{c}-\frac {2 (1+n) \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2} \operatorname {Hypergeometric2F1}\left (1,-\frac {n}{2},1-\frac {n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a c n} \]
(1+1/a/x)^(1+1/2*n)*x/c/((1-1/a/x)^(1/2*n))-2*(1+n)*(1+1/a/x)^(1/2*n)*hype rgeom([1, -1/2*n],[1-1/2*n],(a-1/x)/(a+1/x))/a/c/n/((1-1/a/x)^(1/2*n))
Time = 0.47 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.86 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=\frac {e^{n \coth ^{-1}(a x)} \left (e^{2 \coth ^{-1}(a x)} n (1+n) \operatorname {Hypergeometric2F1}\left (1,1+\frac {n}{2},2+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )+(2+n) \left (-1+a n x+(1+n) \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},1+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )\right )\right )}{a c n (2+n)} \]
(E^(n*ArcCoth[a*x])*(E^(2*ArcCoth[a*x])*n*(1 + n)*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, E^(2*ArcCoth[a*x])] + (2 + n)*(-1 + a*n*x + (1 + n)*Hyperge ometric2F1[1, n/2, 1 + n/2, E^(2*ArcCoth[a*x])])))/(a*c*n*(2 + n))
Time = 0.29 (sec) , antiderivative size = 113, 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 = {6732, 107, 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.
\(\displaystyle \int \frac {e^{n \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx\) |
\(\Big \downarrow \) 6732 |
\(\displaystyle -\frac {\int \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1} \left (1+\frac {1}{a x}\right )^{n/2} x^2d\frac {1}{x}}{c}\) |
\(\Big \downarrow \) 107 |
\(\displaystyle -\frac {\frac {(n+1) \int \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1} \left (1+\frac {1}{a x}\right )^{n/2} xd\frac {1}{x}}{a}-x \left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}}{c}\) |
\(\Big \downarrow \) 141 |
\(\displaystyle -\frac {\frac {2 (n+1) \left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{n/2} \operatorname {Hypergeometric2F1}\left (1,-\frac {n}{2},1-\frac {n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a n}-x \left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}}{c}\) |
-((-(((1 + 1/(a*x))^((2 + n)/2)*x)/(1 - 1/(a*x))^(n/2)) + (2*(1 + n)*(1 + 1/(a*x))^(n/2)*Hypergeometric2F1[1, -1/2*n, 1 - n/2, (a - x^(-1))/(a + x^( -1))])/(a*n*(1 - 1/(a*x))^(n/2)))/c)
3.6.43.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[(a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x ] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 1])
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] :> S imp[-c^p Subst[Int[(1 + d*(x/c))^p*((1 + x/a)^(n/2)/(x^2*(1 - x/a)^(n/2)) ), x], x, 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 - a^2*d^2, 0] && !IntegerQ[n/2] && (IntegerQ[p] || GtQ[c, 0])
\[\int \frac {{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}}{c -\frac {c}{a x}}d x\]
\[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{c - \frac {c}{a x}} \,d x } \]
\[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=\frac {a \int \frac {x e^{n \operatorname {acoth}{\left (a x \right )}}}{a x - 1}\, dx}{c} \]
\[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{c - \frac {c}{a x}} \,d x } \]
\[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{c - \frac {c}{a x}} \,d x } \]
Timed out. \[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}}{c-\frac {c}{a\,x}} \,d x \]