3.789 \(\int e^{-2 p \tanh ^{-1}(a x)} (c-\frac {c}{a^2 x^2})^p \, dx\)

Optimal. Leaf size=53 \[ \frac {x \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \, _2F_1(1-2 p,-2 p;2-2 p;a x)}{1-2 p} \]

[Out]

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

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Rubi [A]  time = 0.12, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6160, 6150, 64} \[ \frac {x \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \, _2F_1(1-2 p,-2 p;2-2 p;a x)}{1-2 p} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 64

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

Rule 6150

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

Rule 6160

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

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 53, normalized size = 1.00 \[ \frac {x \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \, _2F_1(1-2 p,-2 p;2-2 p;a x)}{1-2 p} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}\right )^{p}}{\left (\frac {a x + 1}{a x - 1}\right )^{p}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a^2/x^2)^p/exp(2*p*arctanh(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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p}}{\left (\frac {a x + 1}{a x - 1}\right )^{p}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.21, size = 0, normalized size = 0.00 \[ \int \left (c -\frac {c}{a^{2} x^{2}}\right )^{p} {\mathrm e}^{-2 p \arctanh \left (a x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p}}{\left (\frac {a x + 1}{a x - 1}\right )^{p}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a^2/x^2)^p/exp(2*p*arctanh(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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\mathrm {e}}^{-2\,p\,\mathrm {atanh}\left (a\,x\right )}\,{\left (c-\frac {c}{a^2\,x^2}\right )}^p \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )\right )^{p} e^{- 2 p \operatorname {atanh}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**p*exp(-2*p*atanh(a*x)), x)

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