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

Optimal. Leaf size=50 \[ \frac{x (1-a x)^{-p} \left (c-\frac{c}{a x}\right )^p \text{Hypergeometric2F1}(1-p,-p,2-p,-a x)}{1-p} \]

[Out]

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

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Rubi [A]  time = 0.0970773, antiderivative size = 50, 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 = {6134, 6129, 64} \[ \frac{x (1-a x)^{-p} \left (c-\frac{c}{a x}\right )^p \, _2F_1(1-p,-p;2-p;-a x)}{1-p} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6134

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

Rule 6129

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

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])))

Rubi steps

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

Mathematica [A]  time = 0.0198358, size = 50, normalized size = 1. \[ \frac{x (1-a x)^{-p} \left (c-\frac{c}{a x}\right )^p \text{Hypergeometric2F1}(1-p,-p,2-p,-a x)}{1-p} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.161, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{2\,p{\it Artanh} \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*arctanh(a*x))*(c-c/a/x)^p,x)

[Out]

int(exp(2*p*arctanh(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*arctanh(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*arctanh(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{atanh}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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