3.995 \(\int \frac{e^{\tanh ^{-1}(a x)} x^m}{\sqrt{1-a^2 x^2}} \, dx\)

Optimal. Leaf size=22 \[ \frac{x^{m+1} \text{Hypergeometric2F1}(1,m+1,m+2,a x)}{m+1} \]

[Out]

(x^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, a*x])/(1 + m)

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Rubi [A]  time = 0.081731, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {6150, 64} \[ \frac{x^{m+1} \, _2F_1(1,m+1;m+2;a x)}{m+1} \]

Antiderivative was successfully verified.

[In]

Int[(E^ArcTanh[a*x]*x^m)/Sqrt[1 - a^2*x^2],x]

[Out]

(x^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, a*x])/(1 + m)

Rule 6150

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

Rule 64

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c^n*(b*x)^(m + 1)*Hypergeometric2F1[-n, m +
1, m + 2, -((d*x)/c)])/(b*(m + 1)), x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[
c, 0] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-(d/(b*c)), 0])))

Rubi steps

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

Mathematica [A]  time = 0.01085, size = 22, normalized size = 1. \[ \frac{x^{m+1} \text{Hypergeometric2F1}(1,m+1,m+2,a x)}{m+1} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^ArcTanh[a*x]*x^m)/Sqrt[1 - a^2*x^2],x]

[Out]

(x^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, a*x])/(1 + m)

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Maple [C]  time = 0.193, size = 100, normalized size = 4.6 \begin{align*} -{\frac{1}{2\,a} \left ( -{a}^{2} \right ) ^{-{\frac{m}{2}}} \left ( -2\,{\frac{{x}^{m} \left ( -{a}^{2} \right ) ^{m/2} \left ( -m-2 \right ) }{ \left ( 2+m \right ) m}}-{x}^{m} \left ( -{a}^{2} \right ) ^{{\frac{m}{2}}}{\it LerchPhi} \left ({a}^{2}{x}^{2},1,{\frac{m}{2}} \right ) \right ) }+{\frac{{x}^{1+m}}{1+m} \left ({\frac{1}{2}}+{\frac{m}{2}} \right ){\it LerchPhi} \left ({a}^{2}{x}^{2},1,{\frac{1}{2}}+{\frac{m}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/a*(-a^2)^(-1/2*m)*(-2/(2+m)*x^m*(-a^2)^(1/2*m)*(-m-2)/m-x^m*(-a^2)^(1/2*m)*LerchPhi(a^2*x^2,1,1/2*m))+1/(
1+m)*x^(1+m)*(1/2+1/2*m)*LerchPhi(a^2*x^2,1,1/2+1/2*m)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{{\left (a x + 1\right )} x^{m}}{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)*x^m,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{x^{m}}{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)*x^m,x, algorithm="fricas")

[Out]

integral(-x^m/(a*x - 1), x)

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Sympy [B]  time = 2.0484, size = 44, normalized size = 2. \begin{align*} \frac{m x x^{m} \Phi \left (a x, 1, m + 1\right ) \Gamma \left (m + 1\right )}{\Gamma \left (m + 2\right )} + \frac{x x^{m} \Phi \left (a x, 1, m + 1\right ) \Gamma \left (m + 1\right )}{\Gamma \left (m + 2\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

m*x*x**m*lerchphi(a*x, 1, m + 1)*gamma(m + 1)/gamma(m + 2) + x*x**m*lerchphi(a*x, 1, m + 1)*gamma(m + 1)/gamma
(m + 2)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{{\left (a x + 1\right )} x^{m}}{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)*x^m,x, algorithm="giac")

[Out]

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