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

Optimal. Leaf size=82 \[ -\frac{2^{\frac{n}{2}+1} (1-a x)^{-n/2} (c-a c x)^{p+1} \text{Hypergeometric2F1}\left (-\frac{n}{2},-\frac{n}{2}+p+1,-\frac{n}{2}+p+2,\frac{1}{2} (1-a x)\right )}{a c (-n+2 p+2)} \]

[Out]

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

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Rubi [A]  time = 0.0623685, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6130, 23, 69} \[ -\frac{2^{\frac{n}{2}+1} (1-a x)^{-n/2} (c-a c x)^{p+1} \, _2F_1\left (-\frac{n}{2},-\frac{n}{2}+p+1;-\frac{n}{2}+p+2;\frac{1}{2} (1-a x)\right )}{a c (-n+2 p+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0240423, size = 77, normalized size = 0.94 \[ \frac{2^{\frac{n}{2}+1} (1-a x)^{1-\frac{n}{2}} (c-a c x)^p \text{Hypergeometric2F1}\left (-\frac{n}{2},-\frac{n}{2}+p+1,-\frac{n}{2}+p+2,\frac{1}{2}-\frac{a x}{2}\right )}{a (n-2 (p+1))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.261, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n{\it Artanh} \left ( ax \right ) }} \left ( -acx+c \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))*(-a*c*x+c)^p,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (-a c x + c\right )}^{p} \left (\frac{a x + 1}{a x - 1}\right )^{\frac{1}{2} \, n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((-c*(a*x - 1))**p*exp(n*atanh(a*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctanh(a*x))*(-a*c*x+c)^p,x, algorithm="giac")

[Out]

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