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

Optimal. Leaf size=61 \[ -\frac{2^{-p} (1-a x)^p (c-a c x)^{p+1} \text{Hypergeometric2F1}\left (p,2 p+1,2 (p+1),\frac{1}{2} (1-a x)\right )}{a c (2 p+1)} \]

[Out]

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

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Rubi [A]  time = 0.0587585, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {6130, 23, 69} \[ -\frac{2^{-p} (1-a x)^p (c-a c x)^{p+1} \, _2F_1\left (p,2 p+1;2 (p+1);\frac{1}{2} (1-a x)\right )}{a c (2 p+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6130

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

Rule 23

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((c_) + (d_.)*(v_))^(n_), x_Symbol] :> Dist[(a + b*v)^m/(c + d*v)^m, Int[u*
(c + d*v)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] &&  !(IntegerQ[m] || IntegerQ[n
] || GtQ[b/d, 0])

Rule 69

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

Rubi steps

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

Mathematica [A]  time = 0.0258814, size = 56, normalized size = 0.92 \[ -\frac{2^{-p} (1-a x)^{p+1} (c-a c x)^p \text{Hypergeometric2F1}\left (p,2 p+1,2 p+2,\frac{1}{2}-\frac{a x}{2}\right )}{2 a p+a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.343, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( -acx+c \right ) ^{p}}{{{\rm e}^{2\,p{\it Artanh} \left ( ax \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

[Out]

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