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

Optimal. Leaf size=50 \[ \frac {x (1-a x)^{-p} \left (c-\frac {c}{a x}\right )^p \, _2F_1(1-p,-p;2-p;-a x)}{1-p} \]

[Out]

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

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Rubi [A]  time = 0.10, antiderivative size = 50, 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 = {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 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 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 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]

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*}

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Mathematica [A]  time = 0.02, size = 50, normalized size = 1.00 \[ \frac {x (1-a x)^{-p} \left (c-\frac {c}{a x}\right )^p \, _2F_1(1-p,-p;2-p;-a x)}{1-p} \]

Antiderivative was successfully verified.

[In]

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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fricas [F]  time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (\frac {a x + 1}{a x - 1}\right )^{p} \left (\frac {a c x - c}{a x}\right )^{p}, x\right ) \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*p*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)

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maple [F]  time = 0.12, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{2 p \arctanh \left (a x \right )} \left (c -\frac {c}{a x}\right )^{p}\, dx \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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