\(\int \frac {e^{n \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx\) [543]

Optimal result

Integrand size = 22, antiderivative size = 113 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=\frac {\left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x}{c}-\frac {2 (1+n) \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2} \operatorname {Hypergeometric2F1}\left (1,-\frac {n}{2},1-\frac {n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a c n} \]

[Out]

(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)*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))

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6314, 98, 133} \[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=\frac {x \left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}}{c}-\frac {2 (n+1) \left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{n/2} \operatorname {Hypergeometric2F1}\left (1,-\frac {n}{2},1-\frac {n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a c n} \]

[In]

Int[E^(n*ArcCoth[a*x])/(c - c/(a*x)),x]

[Out]

((1 + 1/(a*x))^((2 + n)/2)*x)/(c*(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*c*n*(1 - 1/(a*x))^(n/2))

Rule 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> 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] + Dist[(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] || Sum
SimplerQ[m, 1])

Rule 133

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> 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] ||  !SumSimplerQ[p, 1]) &&  !ILtQ[m, 0]

Rule 6314

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.), x_Symbol] :> Dist[-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])

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{-1-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2}}{x^2} \, dx,x,\frac {1}{x}\right )}{c} \\ & = \frac {\left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x}{c}-\frac {(1+n) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{-1-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2}}{x} \, dx,x,\frac {1}{x}\right )}{a c} \\ & = \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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.47 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.86 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=\frac {e^{n \coth ^{-1}(a x)} \left (e^{2 \coth ^{-1}(a x)} n (1+n) \operatorname {Hypergeometric2F1}\left (1,1+\frac {n}{2},2+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )+(2+n) \left (-1+a n x+(1+n) \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},1+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )\right )\right )}{a c n (2+n)} \]

[In]

Integrate[E^(n*ArcCoth[a*x])/(c - c/(a*x)),x]

[Out]

(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)*Hypergeometric2F1[1, n/2, 1 + n/2, E^(2*ArcCoth[a*x])])))/(a*c*n*(2 + n))

Maple [F]

\[\int \frac {{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}}{c -\frac {c}{a x}}d x\]

[In]

int(exp(n*arccoth(a*x))/(c-c/a/x),x)

[Out]

int(exp(n*arccoth(a*x))/(c-c/a/x),x)

Fricas [F]

\[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{c - \frac {c}{a x}} \,d x } \]

[In]

integrate(exp(n*arccoth(a*x))/(c-c/a/x),x, algorithm="fricas")

[Out]

integral(a*x*((a*x + 1)/(a*x - 1))^(1/2*n)/(a*c*x - c), x)

Sympy [F]

\[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=\frac {a \int \frac {x e^{n \operatorname {acoth}{\left (a x \right )}}}{a x - 1}\, dx}{c} \]

[In]

integrate(exp(n*acoth(a*x))/(c-c/a/x),x)

[Out]

a*Integral(x*exp(n*acoth(a*x))/(a*x - 1), x)/c

Maxima [F]

\[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{c - \frac {c}{a x}} \,d x } \]

[In]

integrate(exp(n*arccoth(a*x))/(c-c/a/x),x, algorithm="maxima")

[Out]

integrate(((a*x + 1)/(a*x - 1))^(1/2*n)/(c - c/(a*x)), x)

Giac [F]

\[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{c - \frac {c}{a x}} \,d x } \]

[In]

integrate(exp(n*arccoth(a*x))/(c-c/a/x),x, algorithm="giac")

[Out]

integrate(((a*x + 1)/(a*x - 1))^(1/2*n)/(c - c/(a*x)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}}{c-\frac {c}{a\,x}} \,d x \]

[In]

int(exp(n*acoth(a*x))/(c - c/(a*x)),x)

[Out]

int(exp(n*acoth(a*x))/(c - c/(a*x)), x)