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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0490697, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {6130, 23, 69} \[ -\frac{2 \sqrt{2} (c-a c x)^{p+1} \, _2F_1\left (-\frac{1}{2},p+\frac{1}{2};p+\frac{3}{2};\frac{1}{2} (1-a x)\right )}{a c (2 p+1) \sqrt{1-a x}} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]*(c - a*c*x)^p,x]

[Out]

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

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

Mathematica [A]  time = 0.0307093, size = 53, normalized size = 0.82 \[ -\frac{2 \sqrt{2-2 a x} (c-a c x)^p \text{Hypergeometric2F1}\left (-\frac{1}{2},p+\frac{1}{2},p+\frac{3}{2},\frac{1}{2}-\frac{a x}{2}\right )}{2 a p+a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.378, size = 0, normalized size = 0. \begin{align*} \int{ \left ( ax+1 \right ) \left ( -acx+c \right ) ^{p}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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