### 3.934 $$\int e^{-2 p \coth ^{-1}(a x)} (c-\frac{c}{a^2 x^2})^p \, dx$$

Optimal. Leaf size=76 $\frac{\left (1-\frac{1}{a^2 x^2}\right )^{-p} \left (1-\frac{1}{a x}\right )^{2 p+1} \left (c-\frac{c}{a^2 x^2}\right )^p \text{Hypergeometric2F1}\left (2,2 p+1,2 (p+1),1-\frac{1}{a x}\right )}{a (2 p+1)}$

[Out]

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

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Rubi [A]  time = 0.0850554, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.13, Rules used = {6197, 6194, 65} $\frac{\left (1-\frac{1}{a^2 x^2}\right )^{-p} \left (1-\frac{1}{a x}\right )^{2 p+1} \left (c-\frac{c}{a^2 x^2}\right )^p \, _2F_1\left (2,2 p+1;2 (p+1);1-\frac{1}{a x}\right )}{a (2 p+1)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 6197

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

Rule 6194

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]

Rule 65

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x)^(n + 1)*Hypergeometric2F1[-m, n +
1, n + 2, 1 + (d*x)/c])/(d*(n + 1)*(-(d/(b*c)))^m), x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (Inte
gerQ[m] || GtQ[-(d/(b*c)), 0])

Rubi steps

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

Mathematica [F]  time = 0.454987, size = 0, normalized size = 0. $\int e^{-2 p \coth ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^p \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 0.189, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{{\rm e}^{2\,p{\rm arccoth} \left (ax\right )}}} \left ( c-{\frac{c}{{a}^{2}{x}^{2}}} \right ) ^{p}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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