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

Optimal. Leaf size=67 \[ -\frac{\left (1-\frac{1}{a x}\right )^{-p} \left (\frac{1}{a x}+1\right )^{p+1} \left (c-\frac{c}{a x}\right )^p \text{Hypergeometric2F1}\left (2,p+1,p+2,\frac{1}{a x}+1\right )}{a (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.0769245, antiderivative size = 67, 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 = {6182, 6179, 65} \[ -\frac{\left (1-\frac{1}{a x}\right )^{-p} \left (\frac{1}{a x}+1\right )^{p+1} \left (c-\frac{c}{a x}\right )^p \, _2F_1\left (2,p+1;p+2;1+\frac{1}{a x}\right )}{a (p+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6182

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

Rule 6179

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

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

Mathematica [A]  time = 0.0278952, size = 67, normalized size = 1. \[ -\frac{\left (1-\frac{1}{a x}\right )^{-p} \left (\frac{1}{a x}+1\right )^{p+1} \left (c-\frac{c}{a x}\right )^p \text{Hypergeometric2F1}\left (2,p+1,p+2,\frac{1}{a x}+1\right )}{a (p+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.182, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{2\,p{\rm arccoth} \left (ax\right )}} \left ( c-{\frac{c}{ax}} \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

integrate((c - c/(a*x))^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 (\left (\frac{a x - 1}{a x + 1}\right )^{p} \left (\frac{a c x - c}{a x}\right )^{p}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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