3.997 \(\int \frac{e^{\tanh ^{-1}(a x)} x^m}{(1-a^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=70 \[ \frac{x^{m+1} \text{Hypergeometric2F1}\left (3,\frac{m+1}{2},\frac{m+3}{2},a^2 x^2\right )}{m+1}+\frac{a x^{m+2} \text{Hypergeometric2F1}\left (3,\frac{m+2}{2},\frac{m+4}{2},a^2 x^2\right )}{m+2} \]

[Out]

(x^(1 + m)*Hypergeometric2F1[3, (1 + m)/2, (3 + m)/2, a^2*x^2])/(1 + m) + (a*x^(2 + m)*Hypergeometric2F1[3, (2
+ m)/2, (4 + m)/2, a^2*x^2])/(2 + m)

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Rubi [A]  time = 0.116348, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6150, 82, 73, 364} \[ \frac{x^{m+1} \, _2F_1\left (3,\frac{m+1}{2};\frac{m+3}{2};a^2 x^2\right )}{m+1}+\frac{a x^{m+2} \, _2F_1\left (3,\frac{m+2}{2};\frac{m+4}{2};a^2 x^2\right )}{m+2} \]

Antiderivative was successfully verified.

[In]

Int[(E^ArcTanh[a*x]*x^m)/(1 - a^2*x^2)^(5/2),x]

[Out]

(x^(1 + m)*Hypergeometric2F1[3, (1 + m)/2, (3 + m)/2, a^2*x^2])/(1 + m) + (a*x^(2 + m)*Hypergeometric2F1[3, (2
+ m)/2, (4 + m)/2, a^2*x^2])/(2 + 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 82

Int[((f_.)*(x_))^(p_.)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Dist[a, Int[(a + b*
x)^n*(c + d*x)^n*(f*x)^p, x], x] + Dist[b/f, Int[(a + b*x)^n*(c + d*x)^n*(f*x)^(p + 1), x], x] /; FreeQ[{a, b,
c, d, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[m - n - 1, 0] &&  !RationalQ[p] &&  !IGtQ[m, 0] && NeQ[m +
n + p + 2, 0]

Rule 73

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[(a*c + b*
d*x^2)^m*(e + f*x)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && Integer
Q[m]

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)} x^m}{\left (1-a^2 x^2\right )^{5/2}} \, dx &=\int \frac{x^m}{(1-a x)^3 (1+a x)^2} \, dx\\ &=a \int \frac{x^{1+m}}{(1-a x)^3 (1+a x)^3} \, dx+\int \frac{x^m}{(1-a x)^3 (1+a x)^3} \, dx\\ &=a \int \frac{x^{1+m}}{\left (1-a^2 x^2\right )^3} \, dx+\int \frac{x^m}{\left (1-a^2 x^2\right )^3} \, dx\\ &=\frac{x^{1+m} \, _2F_1\left (3,\frac{1+m}{2};\frac{3+m}{2};a^2 x^2\right )}{1+m}+\frac{a x^{2+m} \, _2F_1\left (3,\frac{2+m}{2};\frac{4+m}{2};a^2 x^2\right )}{2+m}\\ \end{align*}

Mathematica [A]  time = 0.0333006, size = 67, normalized size = 0.96 \[ x^{m+1} \left (\frac{a x \text{Hypergeometric2F1}\left (3,\frac{m}{2}+1,\frac{m}{2}+2,a^2 x^2\right )}{m+2}+\frac{\text{Hypergeometric2F1}\left (3,\frac{m+1}{2},\frac{m+3}{2},a^2 x^2\right )}{m+1}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(E^ArcTanh[a*x]*x^m)/(1 - a^2*x^2)^(5/2),x]

[Out]

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

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Maple [C]  time = 0.277, size = 224, normalized size = 3.2 \begin{align*}{\frac{1}{4} \left ( -{a}^{2} \right ) ^{-{\frac{1}{2}}-{\frac{m}{2}}} \left ({\frac{{x}^{1+m} \left ({a}^{2}{m}^{2}{x}^{2}-2\,{a}^{2}m{x}^{2}-3\,{a}^{2}{x}^{2}-{m}^{2}+4\,m+5 \right ) }{ \left ( 2\,m+2 \right ) \left ( -{a}^{2}{x}^{2}+1 \right ) ^{2}} \left ( -{a}^{2} \right ) ^{{\frac{1}{2}}+{\frac{m}{2}}}}+4\,{\frac{{x}^{1+m} \left ( -{a}^{2} \right ) ^{1/2+m/2} \left ( 1/16\,{m}^{3}-3/16\,{m}^{2}-m/16+3/16 \right ){\it LerchPhi} \left ({a}^{2}{x}^{2},1,1/2+m/2 \right ) }{1+m}} \right ) }-{\frac{1}{4\,a} \left ( -{a}^{2} \right ) ^{-{\frac{m}{2}}} \left ( -{\frac{{x}^{m} \left ({a}^{2}m{x}^{2}-m+2 \right ) }{2\, \left ( -{a}^{2}{x}^{2}+1 \right ) ^{2}} \left ( -{a}^{2} \right ) ^{{\frac{m}{2}}}}-{\frac{{x}^{m} \left ( -2+m \right ) m}{4} \left ( -{a}^{2} \right ) ^{{\frac{m}{2}}}{\it LerchPhi} \left ({a}^{2}{x}^{2},1,{\frac{m}{2}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)/(-a^2*x^2+1)^3*x^m,x)

[Out]

1/4*(-a^2)^(-1/2-1/2*m)*(1/2/(1+m)*x^(1+m)*(-a^2)^(1/2+1/2*m)*(a^2*m^2*x^2-2*a^2*m*x^2-3*a^2*x^2-m^2+4*m+5)/(-
a^2*x^2+1)^2+4/(1+m)*x^(1+m)*(-a^2)^(1/2+1/2*m)*(1/16*m^3-3/16*m^2-1/16*m+3/16)*LerchPhi(a^2*x^2,1,1/2+1/2*m))
-1/4/a*(-a^2)^(-1/2*m)*(-1/2*x^m*(-a^2)^(1/2*m)*(a^2*m*x^2-m+2)/(-a^2*x^2+1)^2-1/4*x^m*(-a^2)^(1/2*m)*(-2+m)*m
*LerchPhi(a^2*x^2,1,1/2*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}}{{\left (a^{2} x^{2} - 1\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^3*x^m,x, algorithm="maxima")

[Out]

-integrate((a*x + 1)*x^m/(a^2*x^2 - 1)^3, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{x^{m}}{a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + 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)^3*x^m,x, algorithm="fricas")

[Out]

integral(-x^m/(a^5*x^5 - a^4*x^4 - 2*a^3*x^3 + 2*a^2*x^2 + a*x - 1), x)

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Sympy [C]  time = 7.94969, size = 2152, normalized size = 30.74 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a**2*x**2+1)**3*x**m,x)

[Out]

a**4*m**3*x**5*x**m*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**4*x**4*gamma(m
/2 + 3/2) - 64*a**2*x**2*gamma(m/2 + 3/2) + 32*gamma(m/2 + 3/2)) - 3*a**4*m**2*x**5*x**m*lerchphi(a**2*x**2*ex
p_polar(2*I*pi), 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**4*x**4*gamma(m/2 + 3/2) - 64*a**2*x**2*gamma(m/2 + 3/2)
+ 32*gamma(m/2 + 3/2)) - a**4*m*x**5*x**m*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1/2)*gamma(m/2 + 1/2
)/(32*a**4*x**4*gamma(m/2 + 3/2) - 64*a**2*x**2*gamma(m/2 + 3/2) + 32*gamma(m/2 + 3/2)) + 3*a**4*x**5*x**m*ler
chphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**4*x**4*gamma(m/2 + 3/2) - 64*a**2*x**
2*gamma(m/2 + 3/2) + 32*gamma(m/2 + 3/2)) - 2*a**2*m**3*x**3*x**m*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2
+ 1/2)*gamma(m/2 + 1/2)/(32*a**4*x**4*gamma(m/2 + 3/2) - 64*a**2*x**2*gamma(m/2 + 3/2) + 32*gamma(m/2 + 3/2))
+ 6*a**2*m**2*x**3*x**m*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**4*x**4*ga
mma(m/2 + 3/2) - 64*a**2*x**2*gamma(m/2 + 3/2) + 32*gamma(m/2 + 3/2)) + 2*a**2*m**2*x**3*x**m*gamma(m/2 + 1/2)
/(32*a**4*x**4*gamma(m/2 + 3/2) - 64*a**2*x**2*gamma(m/2 + 3/2) + 32*gamma(m/2 + 3/2)) + 2*a**2*m*x**3*x**m*le
rchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**4*x**4*gamma(m/2 + 3/2) - 64*a**2*x*
*2*gamma(m/2 + 3/2) + 32*gamma(m/2 + 3/2)) - 4*a**2*m*x**3*x**m*gamma(m/2 + 1/2)/(32*a**4*x**4*gamma(m/2 + 3/2
) - 64*a**2*x**2*gamma(m/2 + 3/2) + 32*gamma(m/2 + 3/2)) - 6*a**2*x**3*x**m*lerchphi(a**2*x**2*exp_polar(2*I*p
i), 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**4*x**4*gamma(m/2 + 3/2) - 64*a**2*x**2*gamma(m/2 + 3/2) + 32*gamma(m
/2 + 3/2)) - 6*a**2*x**3*x**m*gamma(m/2 + 1/2)/(32*a**4*x**4*gamma(m/2 + 3/2) - 64*a**2*x**2*gamma(m/2 + 3/2)
+ 32*gamma(m/2 + 3/2)) + a*(a**4*m**3*x**6*x**m*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1)*gamma(m/2 +
1)/(32*a**4*x**4*gamma(m/2 + 2) - 64*a**2*x**2*gamma(m/2 + 2) + 32*gamma(m/2 + 2)) - 4*a**4*m*x**6*x**m*lerchp
hi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1)*gamma(m/2 + 1)/(32*a**4*x**4*gamma(m/2 + 2) - 64*a**2*x**2*gamma(m
/2 + 2) + 32*gamma(m/2 + 2)) - 2*a**2*m**3*x**4*x**m*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1)*gamma(m
/2 + 1)/(32*a**4*x**4*gamma(m/2 + 2) - 64*a**2*x**2*gamma(m/2 + 2) + 32*gamma(m/2 + 2)) + 2*a**2*m**2*x**4*x**
m*gamma(m/2 + 1)/(32*a**4*x**4*gamma(m/2 + 2) - 64*a**2*x**2*gamma(m/2 + 2) + 32*gamma(m/2 + 2)) + 8*a**2*m*x*
*4*x**m*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1)*gamma(m/2 + 1)/(32*a**4*x**4*gamma(m/2 + 2) - 64*a**
2*x**2*gamma(m/2 + 2) + 32*gamma(m/2 + 2)) - 8*a**2*x**4*x**m*gamma(m/2 + 1)/(32*a**4*x**4*gamma(m/2 + 2) - 64
*a**2*x**2*gamma(m/2 + 2) + 32*gamma(m/2 + 2)) + m**3*x**2*x**m*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 +
1)*gamma(m/2 + 1)/(32*a**4*x**4*gamma(m/2 + 2) - 64*a**2*x**2*gamma(m/2 + 2) + 32*gamma(m/2 + 2)) - 2*m**2*x*
*2*x**m*gamma(m/2 + 1)/(32*a**4*x**4*gamma(m/2 + 2) - 64*a**2*x**2*gamma(m/2 + 2) + 32*gamma(m/2 + 2)) - 4*m*x
**2*x**m*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1)*gamma(m/2 + 1)/(32*a**4*x**4*gamma(m/2 + 2) - 64*a*
*2*x**2*gamma(m/2 + 2) + 32*gamma(m/2 + 2)) + 4*m*x**2*x**m*gamma(m/2 + 1)/(32*a**4*x**4*gamma(m/2 + 2) - 64*a
**2*x**2*gamma(m/2 + 2) + 32*gamma(m/2 + 2)) + 16*x**2*x**m*gamma(m/2 + 1)/(32*a**4*x**4*gamma(m/2 + 2) - 64*a
**2*x**2*gamma(m/2 + 2) + 32*gamma(m/2 + 2))) + m**3*x*x**m*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1/2
)*gamma(m/2 + 1/2)/(32*a**4*x**4*gamma(m/2 + 3/2) - 64*a**2*x**2*gamma(m/2 + 3/2) + 32*gamma(m/2 + 3/2)) - 3*m
**2*x*x**m*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**4*x**4*gamma(m/2 + 3/2)
- 64*a**2*x**2*gamma(m/2 + 3/2) + 32*gamma(m/2 + 3/2)) - 2*m**2*x*x**m*gamma(m/2 + 1/2)/(32*a**4*x**4*gamma(m
/2 + 3/2) - 64*a**2*x**2*gamma(m/2 + 3/2) + 32*gamma(m/2 + 3/2)) - m*x*x**m*lerchphi(a**2*x**2*exp_polar(2*I*p
i), 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**4*x**4*gamma(m/2 + 3/2) - 64*a**2*x**2*gamma(m/2 + 3/2) + 32*gamma(m
/2 + 3/2)) + 8*m*x*x**m*gamma(m/2 + 1/2)/(32*a**4*x**4*gamma(m/2 + 3/2) - 64*a**2*x**2*gamma(m/2 + 3/2) + 32*g
amma(m/2 + 3/2)) + 3*x*x**m*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**4*x**4
*gamma(m/2 + 3/2) - 64*a**2*x**2*gamma(m/2 + 3/2) + 32*gamma(m/2 + 3/2)) + 10*x*x**m*gamma(m/2 + 1/2)/(32*a**4
*x**4*gamma(m/2 + 3/2) - 64*a**2*x**2*gamma(m/2 + 3/2) + 32*gamma(m/2 + 3/2))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{{\left (a x + 1\right )} x^{m}}{{\left (a^{2} x^{2} - 1\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^3*x^m,x, algorithm="giac")

[Out]

integrate(-(a*x + 1)*x^m/(a^2*x^2 - 1)^3, x)