\(\int e^{n \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx\) [745]

Optimal result

Integrand size = 24, antiderivative size = 116 \[ \int e^{n \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\frac {8 \left (1-\frac {1}{a x}\right )^{\frac {3-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} \sqrt {c-a^2 c x^2} \operatorname {Hypergeometric2F1}\left (3,\frac {3-n}{2},\frac {5-n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a^2 (3-n) \sqrt {1-\frac {1}{a^2 x^2}} x} \]

[Out]

8*(1-1/a/x)^(3/2-1/2*n)*(1+1/a/x)^(-3/2+1/2*n)*hypergeom([3, 3/2-1/2*n],[5/2-1/2*n],(a-1/x)/(a+1/x))*(-a^2*c*x
^2+c)^(1/2)/a^2/(3-n)/x/(1-1/a^2/x^2)^(1/2)

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6327, 6330, 133} \[ \int e^{n \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\frac {8 \sqrt {c-a^2 c x^2} \left (1-\frac {1}{a x}\right )^{\frac {3-n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \operatorname {Hypergeometric2F1}\left (3,\frac {3-n}{2},\frac {5-n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a^2 (3-n) x \sqrt {1-\frac {1}{a^2 x^2}}} \]

[In]

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

[Out]

(8*(1 - 1/(a*x))^((3 - n)/2)*(1 + 1/(a*x))^((-3 + n)/2)*Sqrt[c - a^2*c*x^2]*Hypergeometric2F1[3, (3 - n)/2, (5
- n)/2, (a - x^(-1))/(a + x^(-1))])/(a^2*(3 - n)*Sqrt[1 - 1/(a^2*x^2)]*x)

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 6327

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(c + d*x^2)^p/(x^(2*p)*(
1 - 1/(a^2*x^2))^p), Int[u*x^(2*p)*(1 - 1/(a^2*x^2))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x]
&& EqQ[a^2*c + d, 0] &&  !IntegerQ[n/2] &&  !IntegerQ[p]

Rule 6330

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_)^2)^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[-c^p, Subst[Int[(1
- x/a)^(p - n/2)*((1 + x/a)^(p + n/2)/x^(m + 2)), x], x, 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2
*d, 0] &&  !IntegerQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) &&  !IntegersQ[2*p, p + n/2] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c-a^2 c x^2} \int e^{n \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}} x \, dx}{\sqrt {1-\frac {1}{a^2 x^2}} x} \\ & = -\frac {\sqrt {c-a^2 c x^2} \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{\frac {1}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{\frac {1}{2}+\frac {n}{2}}}{x^3} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}} x} \\ & = \frac {8 \left (1-\frac {1}{a x}\right )^{\frac {3-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} \sqrt {c-a^2 c x^2} \operatorname {Hypergeometric2F1}\left (3,\frac {3-n}{2},\frac {5-n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a^2 (3-n) \sqrt {1-\frac {1}{a^2 x^2}} x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.85 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.87 \[ \int e^{n \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=-\frac {c e^{n \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}} x \left (a \sqrt {1-\frac {1}{a^2 x^2}} x (n+a x)+2 e^{\coth ^{-1}(a x)} (-1+n) \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},e^{2 \coth ^{-1}(a x)}\right )\right )}{2 \sqrt {c-a^2 c x^2}} \]

[In]

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

[Out]

-1/2*(c*E^(n*ArcCoth[a*x])*Sqrt[1 - 1/(a^2*x^2)]*x*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(n + a*x) + 2*E^ArcCoth[a*x]*(-1
+ n)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, E^(2*ArcCoth[a*x])]))/Sqrt[c - a^2*c*x^2]

Maple [F]

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

[In]

int(exp(n*arccoth(a*x))*(-a^2*c*x^2+c)^(1/2),x)

[Out]

int(exp(n*arccoth(a*x))*(-a^2*c*x^2+c)^(1/2),x)

Fricas [F]

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

[In]

integrate(exp(n*arccoth(a*x))*(-a^2*c*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

integrate(exp(n*arccoth(a*x))*(-a^2*c*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int e^{n \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(exp(n*arccoth(a*x))*(-a^2*c*x^2+c)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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