Integrand size = 18, antiderivative size = 71 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-a c x} \, dx=\frac {2 \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2} \operatorname {Hypergeometric2F1}\left (1,-\frac {n}{2},1-\frac {n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a c n} \] Output:
2*(1+1/a/x)^(1/2*n)*hypergeom([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 = 0.33 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.23 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-a c x} \, dx=-\frac {e^{n \coth ^{-1}(a x)} \left (e^{2 \coth ^{-1}(a x)} n \operatorname {Hypergeometric2F1}\left (1,1+\frac {n}{2},2+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )+(2+n) \left (-1+\operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},1+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )\right )\right )}{a c n (2+n)} \] Input:
Integrate[E^(n*ArcCoth[a*x])/(c - a*c*x),x]
Output:
-((E^(n*ArcCoth[a*x])*(E^(2*ArcCoth[a*x])*n*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, E^(2*ArcCoth[a*x])] + (2 + n)*(-1 + Hypergeometric2F1[1, n/2, 1 + n/2, E^(2*ArcCoth[a*x])])))/(a*c*n*(2 + n)))
Time = 0.41 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6725, 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.
\(\displaystyle \int \frac {e^{n \coth ^{-1}(a x)}}{c-a c x} \, dx\) |
\(\Big \downarrow \) 6725 |
\(\displaystyle \frac {\int \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1} \left (1+\frac {1}{a x}\right )^{n/2} xd\frac {1}{x}}{a c}\) |
\(\Big \downarrow \) 141 |
\(\displaystyle \frac {2 \left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{n/2} \operatorname {Hypergeometric2F1}\left (1,-\frac {n}{2},1-\frac {n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a c n}\) |
Input:
Int[E^(n*ArcCoth[a*x])/(c - a*c*x),x]
Output:
(2*(1 + 1/(a*x))^(n/2)*Hypergeometric2F1[1, -1/2*n, 1 - n/2, (a - x^(-1))/ (a + x^(-1))])/(a*c*n*(1 - 1/(a*x))^(n/2))
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[-d^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]
\[\int \frac {{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}}{-a c x +c}d x\]
Input:
int(exp(n*arccoth(a*x))/(-a*c*x+c),x)
Output:
int(exp(n*arccoth(a*x))/(-a*c*x+c),x)
\[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-a c x} \, dx=\int { -\frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a c x - c} \,d x } \] Input:
integrate(exp(n*arccoth(a*x))/(-a*c*x+c),x, algorithm="fricas")
Output:
integral(-((a*x + 1)/(a*x - 1))^(1/2*n)/(a*c*x - c), x)
\[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-a c x} \, dx=- \frac {\int \frac {e^{n \operatorname {acoth}{\left (a x \right )}}}{a x - 1}\, dx}{c} \] Input:
integrate(exp(n*acoth(a*x))/(-a*c*x+c),x)
Output:
-Integral(exp(n*acoth(a*x))/(a*x - 1), x)/c
\[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-a c x} \, dx=\int { -\frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a c x - c} \,d x } \] Input:
integrate(exp(n*arccoth(a*x))/(-a*c*x+c),x, algorithm="maxima")
Output:
-integrate(((a*x + 1)/(a*x - 1))^(1/2*n)/(a*c*x - c), x)
\[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-a c x} \, dx=\int { -\frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a c x - c} \,d x } \] Input:
integrate(exp(n*arccoth(a*x))/(-a*c*x+c),x, algorithm="giac")
Output:
integrate(-((a*x + 1)/(a*x - 1))^(1/2*n)/(a*c*x - c), x)
Timed out. \[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-a c x} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}}{c-a\,c\,x} \,d x \] Input:
int(exp(n*acoth(a*x))/(c - a*c*x),x)
Output:
int(exp(n*acoth(a*x))/(c - a*c*x), x)
\[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-a c x} \, dx=-\frac {\int \frac {e^{\mathit {acoth} \left (a x \right ) n}}{a x -1}d x}{c} \] Input:
int(exp(n*acoth(a*x))/(-a*c*x+c),x)
Output:
( - int(e**(acoth(a*x)*n)/(a*x - 1),x))/c