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

Optimal. Leaf size=76 \[ -\frac{x}{a^5}-\frac{1}{a^6 (1-a x)}+\frac{1}{8 a^6 (a x+1)}+\frac{1}{8 a^6 (1-a x)^2}-\frac{23 \log (1-a x)}{16 a^6}+\frac{7 \log (a x+1)}{16 a^6} \]

[Out]

-(x/a^5) + 1/(8*a^6*(1 - a*x)^2) - 1/(a^6*(1 - a*x)) + 1/(8*a^6*(1 + a*x)) - (23*Log[1 - a*x])/(16*a^6) + (7*L
og[1 + a*x])/(16*a^6)

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Rubi [A]  time = 0.126824, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {6150, 88} \[ -\frac{x}{a^5}-\frac{1}{a^6 (1-a x)}+\frac{1}{8 a^6 (a x+1)}+\frac{1}{8 a^6 (1-a x)^2}-\frac{23 \log (1-a x)}{16 a^6}+\frac{7 \log (a x+1)}{16 a^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(x/a^5) + 1/(8*a^6*(1 - a*x)^2) - 1/(a^6*(1 - a*x)) + 1/(8*a^6*(1 + a*x)) - (23*Log[1 - a*x])/(16*a^6) + (7*L
og[1 + a*x])/(16*a^6)

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 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)} x^5}{\left (1-a^2 x^2\right )^{5/2}} \, dx &=\int \frac{x^5}{(1-a x)^3 (1+a x)^2} \, dx\\ &=\int \left (-\frac{1}{a^5}-\frac{1}{4 a^5 (-1+a x)^3}-\frac{1}{a^5 (-1+a x)^2}-\frac{23}{16 a^5 (-1+a x)}-\frac{1}{8 a^5 (1+a x)^2}+\frac{7}{16 a^5 (1+a x)}\right ) \, dx\\ &=-\frac{x}{a^5}+\frac{1}{8 a^6 (1-a x)^2}-\frac{1}{a^6 (1-a x)}+\frac{1}{8 a^6 (1+a x)}-\frac{23 \log (1-a x)}{16 a^6}+\frac{7 \log (1+a x)}{16 a^6}\\ \end{align*}

Mathematica [A]  time = 0.0721872, size = 55, normalized size = 0.72 \[ \frac{2 \left (-8 a x+\frac{8}{a x-1}+\frac{1}{a x+1}+\frac{1}{(a x-1)^2}\right )-23 \log (1-a x)+7 \log (a x+1)}{16 a^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-8*a*x + (-1 + a*x)^(-2) + 8/(-1 + a*x) + (1 + a*x)^(-1)) - 23*Log[1 - a*x] + 7*Log[1 + a*x])/(16*a^6)

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Maple [A]  time = 0.036, size = 65, normalized size = 0.9 \begin{align*} -{\frac{x}{{a}^{5}}}+{\frac{1}{8\,{a}^{6} \left ( ax+1 \right ) }}+{\frac{7\,\ln \left ( ax+1 \right ) }{16\,{a}^{6}}}-{\frac{23\,\ln \left ( ax-1 \right ) }{16\,{a}^{6}}}+{\frac{1}{8\,{a}^{6} \left ( ax-1 \right ) ^{2}}}+{\frac{1}{{a}^{6} \left ( ax-1 \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^5,x)

[Out]

-x/a^5+1/8/a^6/(a*x+1)+7/16*ln(a*x+1)/a^6-23/16/a^6*ln(a*x-1)+1/8/a^6/(a*x-1)^2+1/a^6/(a*x-1)

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Maxima [A]  time = 0.952949, size = 97, normalized size = 1.28 \begin{align*} \frac{9 \, a^{2} x^{2} - a x - 6}{8 \,{\left (a^{9} x^{3} - a^{8} x^{2} - a^{7} x + a^{6}\right )}} - \frac{x}{a^{5}} + \frac{7 \, \log \left (a x + 1\right )}{16 \, a^{6}} - \frac{23 \, \log \left (a x - 1\right )}{16 \, a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^3*x^5,x, algorithm="maxima")

[Out]

1/8*(9*a^2*x^2 - a*x - 6)/(a^9*x^3 - a^8*x^2 - a^7*x + a^6) - x/a^5 + 7/16*log(a*x + 1)/a^6 - 23/16*log(a*x -
1)/a^6

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Fricas [A]  time = 1.78357, size = 255, normalized size = 3.36 \begin{align*} -\frac{16 \, a^{4} x^{4} - 16 \, a^{3} x^{3} - 34 \, a^{2} x^{2} + 18 \, a x - 7 \,{\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x + 1\right ) + 23 \,{\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x - 1\right ) + 12}{16 \,{\left (a^{9} x^{3} - a^{8} x^{2} - a^{7} x + a^{6}\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^5,x, algorithm="fricas")

[Out]

-1/16*(16*a^4*x^4 - 16*a^3*x^3 - 34*a^2*x^2 + 18*a*x - 7*(a^3*x^3 - a^2*x^2 - a*x + 1)*log(a*x + 1) + 23*(a^3*
x^3 - a^2*x^2 - a*x + 1)*log(a*x - 1) + 12)/(a^9*x^3 - a^8*x^2 - a^7*x + a^6)

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Sympy [A]  time = 0.615176, size = 70, normalized size = 0.92 \begin{align*} \frac{9 a^{2} x^{2} - a x - 6}{8 a^{9} x^{3} - 8 a^{8} x^{2} - 8 a^{7} x + 8 a^{6}} - \frac{x}{a^{5}} - \frac{\frac{23 \log{\left (x - \frac{1}{a} \right )}}{16} - \frac{7 \log{\left (x + \frac{1}{a} \right )}}{16}}{a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a**2*x**2+1)**3*x**5,x)

[Out]

(9*a**2*x**2 - a*x - 6)/(8*a**9*x**3 - 8*a**8*x**2 - 8*a**7*x + 8*a**6) - x/a**5 - (23*log(x - 1/a)/16 - 7*log
(x + 1/a)/16)/a**6

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Giac [A]  time = 1.2163, size = 86, normalized size = 1.13 \begin{align*} -\frac{x}{a^{5}} + \frac{7 \, \log \left ({\left | a x + 1 \right |}\right )}{16 \, a^{6}} - \frac{23 \, \log \left ({\left | a x - 1 \right |}\right )}{16 \, a^{6}} + \frac{9 \, a^{2} x^{2} - a x - 6}{8 \,{\left (a x + 1\right )}{\left (a x - 1\right )}^{2} a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^3*x^5,x, algorithm="giac")

[Out]

-x/a^5 + 7/16*log(abs(a*x + 1))/a^6 - 23/16*log(abs(a*x - 1))/a^6 + 1/8*(9*a^2*x^2 - a*x - 6)/((a*x + 1)*(a*x
- 1)^2*a^6)