3.997 \(\int \frac {e^{\tanh ^{-1}(a x)} x^m}{(1-a^2 x^2)^{5/2}} \, dx\)
Optimal. Leaf size=70 \[ \frac {x^{m+1} \, _2F_1\left (3,\frac {m+1}{2};\frac {m+3}{2};a^2 x^2\right )}{m+1}+\frac {a x^{m+2} \, _2F_1\left (3,\frac {m+2}{2};\frac {m+4}{2};a^2 x^2\right )}{m+2} \]
[Out]
x^(1+m)*hypergeom([3, 1/2+1/2*m],[3/2+1/2*m],a^2*x^2)/(1+m)+a*x^(2+m)*hypergeom([3, 1+1/2*m],[2+1/2*m],a^2*x^2
)/(2+m)
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Rubi [A] time = 0.12, antiderivative size = 70, normalized size of antiderivative = 1.00,
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 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 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 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])
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])
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*}
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Mathematica [A] time = 0.03, size = 67, normalized size = 0.96 \[ x^{m+1} \left (\frac {a x \, _2F_1\left (3,\frac {m}{2}+1;\frac {m}{2}+2;a^2 x^2\right )}{m+2}+\frac {\, _2F_1\left (3,\frac {m+1}{2};\frac {m+3}{2};a^2 x^2\right )}{m+1}\right ) \]
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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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {x^{m}}{a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1}, x\right ) \]
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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {{\left (a x + 1\right )} x^{m}}{{\left (a^{2} x^{2} - 1\right )}^{3}}\,{d x} \]
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)
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maple [C] time = 0.28, size = 224, normalized size = 3.20 \[ \frac {\left (-a^{2}\right )^{-\frac {1}{2}-\frac {m}{2}} \left (\frac {x^{1+m} \left (-a^{2}\right )^{\frac {1}{2}+\frac {m}{2}} \left (a^{2} m^{2} x^{2}-2 a^{2} m \,x^{2}-3 a^{2} x^{2}-m^{2}+4 m +5\right )}{2 \left (1+m \right ) \left (-a^{2} x^{2}+1\right )^{2}}+\frac {4 x^{1+m} \left (-a^{2}\right )^{\frac {1}{2}+\frac {m}{2}} \left (\frac {1}{16} m^{3}-\frac {3}{16} m^{2}-\frac {1}{16} m +\frac {3}{16}\right ) \Phi \left (a^{2} x^{2}, 1, \frac {1}{2}+\frac {m}{2}\right )}{1+m}\right )}{4}-\frac {\left (-a^{2}\right )^{-\frac {m}{2}} \left (-\frac {x^{m} \left (-a^{2}\right )^{\frac {m}{2}} \left (a^{2} m \,x^{2}-m +2\right )}{2 \left (-a^{2} x^{2}+1\right )^{2}}-\frac {x^{m} \left (-a^{2}\right )^{\frac {m}{2}} \left (-2+m \right ) m \Phi \left (a^{2} x^{2}, 1, \frac {m}{2}\right )}{4}\right )}{4 a} \]
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 = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {{\left (a x + 1\right )} x^{m}}{{\left (a^{2} x^{2} - 1\right )}^{3}}\,{d x} \]
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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int -\frac {x^m\,\left (a\,x+1\right )}{{\left (a^2\,x^2-1\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-(x^m*(a*x + 1))/(a^2*x^2 - 1)^3,x)
[Out]
int(-(x^m*(a*x + 1))/(a^2*x^2 - 1)^3, x)
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sympy [C] time = 50.31, size = 2152, normalized size = 30.74 \[ \text {result too large to display} \]
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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