\(\int \frac {e^{n \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx\) [929]

Optimal result

Integrand size = 22, antiderivative size = 150 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=-\frac {(1+n) \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2}}{a c n}+\frac {\left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2} x}{c}+\frac {2 \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2} \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},\frac {2+n}{2},\frac {a+\frac {1}{x}}{a-\frac {1}{x}}\right )}{a c} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6329, 105, 160, 12, 133} \[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\frac {2 \left (\frac {1}{a x}+1\right )^{n/2} \left (1-\frac {1}{a x}\right )^{-n/2} \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},\frac {n+2}{2},\frac {a+\frac {1}{x}}{a-\frac {1}{x}}\right )}{a c}-\frac {(n+1) \left (\frac {1}{a x}+1\right )^{n/2} \left (1-\frac {1}{a x}\right )^{-n/2}}{a c n}+\frac {x \left (\frac {1}{a x}+1\right )^{n/2} \left (1-\frac {1}{a x}\right )^{-n/2}}{c} \]

[In]

Int[E^(n*ArcCoth[a*x])/(c - c/(a^2*x^2)),x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 105

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[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*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

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 160

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(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[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*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m + n + p + 2, 0] && NeQ[m, -1] && (Sum
SimplerQ[m, 1] || ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) &&  !(NeQ[p, -1] && SumSimplerQ[p, 1])))

Rule 6329

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_)^2)^(p_.), x_Symbol] :> Dist[-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] &&  !Integ
erQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) &&  !IntegersQ[2*p, p + n/2]

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

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.63 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\frac {e^{n \coth ^{-1}(a x)} \left (e^{2 \coth ^{-1}(a x)} n^2 \operatorname {Hypergeometric2F1}\left (1,1+\frac {n}{2},2+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )+(2+n) \left (-1+a n x+n \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},1+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )\right )\right )}{a c n (2+n)} \]

[In]

Integrate[E^(n*ArcCoth[a*x])/(c - c/(a^2*x^2)),x]

[Out]

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

Maple [F]

\[\int \frac {{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}}{c -\frac {c}{a^{2} x^{2}}}d x\]

[In]

int(exp(n*arccoth(a*x))/(c-c/a^2/x^2),x)

[Out]

int(exp(n*arccoth(a*x))/(c-c/a^2/x^2),x)

Fricas [F]

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

[In]

integrate(exp(n*arccoth(a*x))/(c-c/a^2/x^2),x, algorithm="fricas")

[Out]

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

Sympy [F]

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

[In]

integrate(exp(n*acoth(a*x))/(c-c/a**2/x**2),x)

[Out]

a**2*Integral(x**2*exp(n*acoth(a*x))/(a**2*x**2 - 1), x)/c

Maxima [F]

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

[In]

integrate(exp(n*arccoth(a*x))/(c-c/a^2/x^2),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate(exp(n*arccoth(a*x))/(c-c/a^2/x^2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

int(exp(n*acoth(a*x))/(c - c/(a^2*x^2)),x)

[Out]

int(exp(n*acoth(a*x))/(c - c/(a^2*x^2)), x)