\(\int e^{n \coth ^{-1}(a x)} (c-a^2 c x^2) \, dx\) [738]

Optimal result

Integrand size = 20, antiderivative size = 79 \[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx=-\frac {16 c \left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-4+n)} \operatorname {Hypergeometric2F1}\left (4,2-\frac {n}{2},3-\frac {n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (4-n)} \]

[Out]

-16*c*(1-1/a/x)^(2-1/2*n)*(1+1/a/x)^(-2+1/2*n)*hypergeom([4, 2-1/2*n],[3-1/2*n],(a-1/x)/(a+1/x))/a/(4-n)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6326, 6330, 133} \[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx=-\frac {16 c \left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-4}{2}} \operatorname {Hypergeometric2F1}\left (4,2-\frac {n}{2},3-\frac {n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (4-n)} \]

[In]

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

[Out]

(-16*c*(1 - 1/(a*x))^(2 - n/2)*(1 + 1/(a*x))^((-4 + n)/2)*Hypergeometric2F1[4, 2 - n/2, 3 - n/2, (a - x^(-1))/
(a + x^(-1))])/(a*(4 - n))

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 6326

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[d^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}, 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}& = -\left (\left (a^2 c\right ) \int e^{n \coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right ) x^2 \, dx\right ) \\ & = \left (a^2 c\right ) \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^4} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {16 c \left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-4+n)} \operatorname {Hypergeometric2F1}\left (4,2-\frac {n}{2},3-\frac {n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (4-n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.80 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.41 \[ \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx=-\frac {c e^{n \coth ^{-1}(a x)} \left (-n-6 a x+a n^2 x+a^2 n x^2+2 a^3 x^3+e^{2 \coth ^{-1}(a x)} (-2+n) n \operatorname {Hypergeometric2F1}\left (1,1+\frac {n}{2},2+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )+\left (-4+n^2\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},1+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )\right )}{6 a} \]

[In]

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

[Out]

-1/6*(c*E^(n*ArcCoth[a*x])*(-n - 6*a*x + a*n^2*x + a^2*n*x^2 + 2*a^3*x^3 + E^(2*ArcCoth[a*x])*(-2 + n)*n*Hyper
geometric2F1[1, 1 + n/2, 2 + n/2, E^(2*ArcCoth[a*x])] + (-4 + n^2)*Hypergeometric2F1[1, n/2, 1 + n/2, E^(2*Arc
Coth[a*x])]))/a

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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