3.2.57 \(\int e^{\tanh ^{-1}(a x)} (c-a c x)^p \, dx\) [157]

Optimal. Leaf size=65 \[ -\frac {2 \sqrt {2} (c-a c x)^{1+p} \, _2F_1\left (-\frac {1}{2},\frac {1}{2}+p;\frac {3}{2}+p;\frac {1}{2} (1-a x)\right )}{a c (1+2 p) \sqrt {1-a x}} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 65, normalized size of antiderivative = 1.00, 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 = {6265, 23, 71} \begin {gather*} -\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}} \end {gather*}

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 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 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], 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]))

Rule 6265

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]
)

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*}

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Mathematica [A]
time = 0.02, size = 53, normalized size = 0.82 \begin {gather*} -\frac {2 \sqrt {2-2 a x} (c-a c x)^p \, _2F_1\left (-\frac {1}{2},\frac {1}{2}+p;\frac {3}{2}+p;\frac {1}{2}-\frac {a x}{2}\right )}{a+2 a p} \end {gather*}

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)

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Maple [F]
time = 0.81, size = 0, normalized size = 0.00 \[\int \frac {\left (a x +1\right ) \left (-c x a +c \right )^{p}}{\sqrt {-a^{2} x^{2}+1}}\, dx\]

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)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (c-a\,c\,x\right )}^p\,\left (a\,x+1\right )}{\sqrt {1-a^2\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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