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

Optimal. Leaf size=65 \[ -\frac{\sqrt{2} \sqrt{1-a x} (c-a c x)^{p+1} \text{Hypergeometric2F1}\left (\frac{1}{2},p+\frac{3}{2},p+\frac{5}{2},\frac{1}{2} (1-a x)\right )}{a c (2 p+3)} \]

[Out]

-((Sqrt[2]*Sqrt[1 - a*x]*(c - a*c*x)^(1 + p)*Hypergeometric2F1[1/2, 3/2 + p, 5/2 + p, (1 - a*x)/2])/(a*c*(3 +
2*p)))

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Rubi [A]  time = 0.0493803, antiderivative size = 65, 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} \[ -\frac{\sqrt{2} \sqrt{1-a x} (c-a c x)^{p+1} \, _2F_1\left (\frac{1}{2},p+\frac{3}{2};p+\frac{5}{2};\frac{1}{2} (1-a x)\right )}{a c (2 p+3)} \]

Antiderivative was successfully verified.

[In]

Int[(c - a*c*x)^p/E^ArcTanh[a*x],x]

[Out]

-((Sqrt[2]*Sqrt[1 - a*x]*(c - a*c*x)^(1 + p)*Hypergeometric2F1[1/2, 3/2 + p, 5/2 + p, (1 - a*x)/2])/(a*c*(3 +
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^{-\tanh ^{-1}(a x)} (c-a c x)^p \, dx &=\int \frac{\sqrt{1-a x} (c-a c x)^p}{\sqrt{1+a x}} \, dx\\ &=\frac{\sqrt{1-a x} \int \frac{(c-a c x)^{\frac{1}{2}+p}}{\sqrt{1+a x}} \, dx}{\sqrt{c-a c x}}\\ &=-\frac{\sqrt{2} \sqrt{1-a x} (c-a c x)^{1+p} \, _2F_1\left (\frac{1}{2},\frac{3}{2}+p;\frac{5}{2}+p;\frac{1}{2} (1-a x)\right )}{a c (3+2 p)}\\ \end{align*}

Mathematica [A]  time = 0.0232447, size = 59, normalized size = 0.91 \[ \frac{\sqrt{2-2 a x} (a x-1) (c-a c x)^p \text{Hypergeometric2F1}\left (\frac{1}{2},p+\frac{3}{2},p+\frac{5}{2},\frac{1}{2}-\frac{a x}{2}\right )}{a (2 p+3)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2 - 2*a*x]*(-1 + a*x)*(c - a*c*x)^p*Hypergeometric2F1[1/2, 3/2 + p, 5/2 + p, 1/2 - (a*x)/2])/(a*(3 + 2*p
))

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Maple [F]  time = 0.451, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( -acx+c \right ) ^{p}}{ax+1}\sqrt{-{a}^{2}{x}^{2}+1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^p/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^p/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((-c*(a*x - 1))**p*sqrt(-(a*x - 1)*(a*x + 1))/(a*x + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^p/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

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