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

Optimal. Leaf size=75 \[ -\frac {\left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (\frac {1}{a x}+1\right )^{2 p+1} \left (c-\frac {c}{a^2 x^2}\right )^p \, _2F_1\left (2,2 p+1;2 (p+1);1+\frac {1}{a x}\right )}{a (2 p+1)} \]

[Out]

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

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Rubi [A]  time = 0.08, antiderivative size = 75, 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 = {6197, 6194, 65} \[ -\frac {\left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (\frac {1}{a x}+1\right )^{2 p+1} \left (c-\frac {c}{a^2 x^2}\right )^p \, _2F_1\left (2,2 p+1;2 (p+1);1+\frac {1}{a x}\right )}{a (2 p+1)} \]

Antiderivative was successfully verified.

[In]

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

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 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])

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*}

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Mathematica [F]  time = 0.60, size = 0, normalized size = 0.00 \[ \int e^{2 p \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*p*arccoth(a*x))*(c-c/a^2/x^2)^p,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 = 0.10, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{2 p \,\mathrm {arccoth}\left (a x \right )} \left (c -\frac {c}{a^{2} x^{2}}\right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*p*arccoth(a*x))*(c-c/a^2/x^2)^p,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 = -0.01 \[ \int {\mathrm {e}}^{2\,p\,\mathrm {acoth}\left (a\,x\right )}\,{\left (c-\frac {c}{a^2\,x^2}\right )}^p \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(exp(2*p*acoth(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 \[ \int \left (- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )\right )^{p} e^{2 p \operatorname {acoth}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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