3.1008 \(\int e^{\tanh ^{-1}(a x)} (1-a^2 x^2)^p \, dx\)

Optimal. Leaf size=59 \[ -\frac {2^{p+\frac {3}{2}} (1-a x)^{p+\frac {1}{2}} \, _2F_1\left (-p-\frac {1}{2},p+\frac {1}{2};p+\frac {3}{2};\frac {1}{2} (1-a x)\right )}{a (2 p+1)} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {6140, 69} \[ -\frac {2^{p+\frac {3}{2}} (1-a x)^{p+\frac {1}{2}} \, _2F_1\left (-p-\frac {1}{2},p+\frac {1}{2};p+\frac {3}{2};\frac {1}{2} (1-a x)\right )}{a (2 p+1)} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]*(1 - a^2*x^2)^p,x]

[Out]

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

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]))

Rule 6140

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

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 59, normalized size = 1.00 \[ -\frac {2^{p+\frac {1}{2}} (1-a x)^{p+\frac {1}{2}} \, _2F_1\left (-p-\frac {1}{2},p+\frac {1}{2};p+\frac {3}{2};\frac {1}{2} (1-a x)\right )}{a \left (p+\frac {1}{2}\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcTanh[a*x]*(1 - a^2*x^2)^p,x]

[Out]

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

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fricas [F]  time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1} {\left (-a^{2} x^{2} + 1\right )}^{p}}{a x - 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.34, size = 44, normalized size = 0.75 \[ \frac {a \,x^{2} \hypergeom \left (\left [1, \frac {1}{2}-p \right ], \relax [2], a^{2} x^{2}\right )}{2}+x \hypergeom \left (\left [\frac {1}{2}, \frac {1}{2}-p \right ], \left [\frac {3}{2}\right ], a^{2} x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a x + 1\right )} {\left (-a^{2} x^{2} + 1\right )}^{p - \frac {1}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [C]  time = 11.47, size = 292, normalized size = 4.95 \[ - \frac {a a^{2 p} x^{2} x^{2 p} e^{i \pi p} \Gamma \left (- p - 1\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, 1 \\ p + 2 \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \sqrt {\pi } \Gamma \left (- p\right ) \Gamma \left (p + 1\right )} - \frac {a a^{2 p} x^{2} x^{2 p} e^{i \pi p} \Gamma \left (- p - 1\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} 1, - p - 1 \\ \frac {1}{2} \end {matrix}\middle | {\frac {1}{a^{2} x^{2}}} \right )}}{2 \sqrt {\pi } \Gamma \left (- p\right ) \Gamma \left (p + 1\right )} - \frac {a^{2 p} x x^{2 p} e^{i \pi p} \Gamma \left (- p - \frac {1}{2}\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} \frac {1}{2}, 1, p + \frac {1}{2} \\ p + 1, p + \frac {3}{2} \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \sqrt {\pi } \Gamma \left (\frac {1}{2} - p\right ) \Gamma \left (p + 1\right )} - \frac {a^{2 p} x x^{2 p} e^{i \pi p} \Gamma \left (- p - \frac {1}{2}\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} 1, - p, - p - \frac {1}{2} \\ \frac {1}{2}, \frac {1}{2} - p \end {matrix}\middle | {\frac {1}{a^{2} x^{2}}} \right )}}{2 \sqrt {\pi } \Gamma \left (\frac {1}{2} - p\right ) \Gamma \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*a**(2*p)*x**2*x**(2*p)*exp(I*pi*p)*gamma(-p - 1)*gamma(p + 1/2)*hyper((1/2, 1), (p + 2,), a**2*x**2*exp_pol
ar(2*I*pi))/(2*sqrt(pi)*gamma(-p)*gamma(p + 1)) - a*a**(2*p)*x**2*x**(2*p)*exp(I*pi*p)*gamma(-p - 1)*gamma(p +
1/2)*hyper((1, -p - 1), (1/2,), 1/(a**2*x**2))/(2*sqrt(pi)*gamma(-p)*gamma(p + 1)) - a**(2*p)*x*x**(2*p)*exp(
I*pi*p)*gamma(-p - 1/2)*gamma(p + 1/2)*hyper((1/2, 1, p + 1/2), (p + 1, p + 3/2), a**2*x**2*exp_polar(2*I*pi))
/(2*sqrt(pi)*gamma(1/2 - p)*gamma(p + 1)) - a**(2*p)*x*x**(2*p)*exp(I*pi*p)*gamma(-p - 1/2)*gamma(p + 1/2)*hyp
er((1, -p, -p - 1/2), (1/2, 1/2 - p), 1/(a**2*x**2))/(2*sqrt(pi)*gamma(1/2 - p)*gamma(p + 1))

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