### 3.764 $$\int e^{-2 p \coth ^{-1}(a x)} (c-a^2 c x^2)^p \, dx$$

Optimal. Leaf size=52 $\frac{x \left (1-\frac{1}{a^2 x^2}\right )^{-p} \left (1-\frac{1}{a x}\right )^{2 p+1} \left (c-a^2 c x^2\right )^p}{2 p+1}$

[Out]

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

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Rubi [A]  time = 0.121951, antiderivative size = 52, 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 = {6192, 6196, 37} $\frac{x \left (1-\frac{1}{a^2 x^2}\right )^{-p} \left (1-\frac{1}{a x}\right )^{2 p+1} \left (c-a^2 c x^2\right )^p}{2 p+1}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((1 - 1/(a*x))^(1 + 2*p)*x*(c - a^2*c*x^2)^p)/((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 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0621937, size = 36, normalized size = 0.69 $\frac{(a x-1) \left (c-a^2 c x^2\right )^p e^{-2 p \coth ^{-1}(a x)}}{2 a p+a}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.037, size = 40, normalized size = 0.8 \begin{align*}{\frac{ \left ( ax-1 \right ) \left ( -{a}^{2}c{x}^{2}+c \right ) ^{p}}{a \left ( 1+2\,p \right ){{\rm e}^{2\,p{\rm arccoth} \left (ax\right )}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*x-1)/a/(1+2*p)*(-a^2*c*x^2+c)^p/exp(2*p*arccoth(a*x))

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Maxima [A]  time = 1.07306, size = 46, normalized size = 0.88 \begin{align*} \frac{{\left (a \left (-c\right )^{p} x + \left (-c\right )^{p}\right )}{\left (a x + 1\right )}^{2 \, p}}{a{\left (2 \, p + 1\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.31117, size = 92, normalized size = 1.77 \begin{align*} \frac{{\left (a x + 1\right )}{\left (-a^{2} c x^{2} + c\right )}^{p}}{{\left (2 \, a p + a\right )} \left (\frac{a x - 1}{a x + 1}\right )^{p}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-a^{2} c x^{2} + c\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((-a^2*c*x^2+c)^p/exp(2*p*arccoth(a*x)),x, algorithm="giac")

[Out]

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