### 3.762 $$\int e^{n \coth ^{-1}(a x)} (c-a^2 c x^2)^p \, dx$$

Optimal. Leaf size=127 $\frac{x \left (1-\frac{1}{a^2 x^2}\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{\frac{1}{2} (n-2 p)} \left (1-\frac{1}{a x}\right )^{p-\frac{n}{2}} \left (\frac{1}{a x}+1\right )^{\frac{n}{2}+p+1} \text{Hypergeometric2F1}\left (-2 p-1,\frac{1}{2} (n-2 p),-2 p,\frac{2}{x \left (a+\frac{1}{x}\right )}\right )}{2 p+1}$

[Out]

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

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Rubi [A]  time = 0.149586, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.136, Rules used = {6192, 6196, 132} $\frac{x \left (1-\frac{1}{a^2 x^2}\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{\frac{1}{2} (n-2 p)} \left (1-\frac{1}{a x}\right )^{p-\frac{n}{2}} \left (\frac{1}{a x}+1\right )^{\frac{n}{2}+p+1} \, _2F_1\left (-2 p-1,\frac{1}{2} (n-2 p);-2 p;\frac{2}{\left (a+\frac{1}{x}\right ) x}\right )}{2 p+1}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 6192

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 6196

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

Rule 132

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((a + b*x
)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1)*Hypergeometric2F1[m + 1, -n, m + 2, -(((d*e - c*f)*(a + b*x))/((b*c -
a*d)*(e + f*x)))])/(((b*e - a*f)*(m + 1))*(((b*e - a*f)*(c + d*x))/((b*c - a*d)*(e + f*x)))^n), x] /; FreeQ[{a
, b, c, d, e, f, m, n, p}, x] && EqQ[m + n + p + 2, 0] &&  !IntegerQ[n]

Rubi steps

\begin{align*} \int e^{n \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx &=\left (\left (1-\frac{1}{a^2 x^2}\right )^{-p} x^{-2 p} \left (c-a^2 c x^2\right )^p\right ) \int e^{n \coth ^{-1}(a x)} \left (1-\frac{1}{a^2 x^2}\right )^p x^{2 p} \, dx\\ &=-\left (\left (\left (1-\frac{1}{a^2 x^2}\right )^{-p} \left (\frac{1}{x}\right )^{2 p} \left (c-a^2 c x^2\right )^p\right ) \operatorname{Subst}\left (\int x^{-2-2 p} \left (1-\frac{x}{a}\right )^{-\frac{n}{2}+p} \left (1+\frac{x}{a}\right )^{\frac{n}{2}+p} \, dx,x,\frac{1}{x}\right )\right )\\ &=\frac{\left (1-\frac{1}{a^2 x^2}\right )^{-p} \left (\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right )^{\frac{1}{2} (n-2 p)} \left (1-\frac{1}{a x}\right )^{-\frac{n}{2}+p} \left (1+\frac{1}{a x}\right )^{1+\frac{n}{2}+p} x \left (c-a^2 c x^2\right )^p \, _2F_1\left (-1-2 p,\frac{1}{2} (n-2 p);-2 p;\frac{2}{\left (a+\frac{1}{x}\right ) x}\right )}{1+2 p}\\ \end{align*}

Mathematica [A]  time = 0.188116, size = 83, normalized size = 0.65 $-\frac{\left (a^2 x^2-1\right ) \left (e^{2 \coth ^{-1}(a x)}-1\right ) \left (c-a^2 c x^2\right )^p e^{(n-2) \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (1,-\frac{n}{2}-p,-\frac{n}{2}+p+2,e^{-2 \coth ^{-1}(a x)}\right )}{a (n-2 (p+1))}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.378, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n{\rm arccoth} \left (ax\right )}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{p}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-a^{2} c x^{2} + c\right )}^{p} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (-a^{2} c x^{2} + c\right )}^{p} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{p} e^{n \operatorname{acoth}{\left (a x \right )}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-a^{2} c x^{2} + c\right )}^{p} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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