3.938 \(\int \frac{e^{\tanh ^{-1}(a x)} x^6}{(1-a^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=88 \[ -\frac{x^2}{2 a^5}-\frac{x}{a^6}-\frac{5}{4 a^7 (1-a x)}-\frac{1}{8 a^7 (a x+1)}+\frac{1}{8 a^7 (1-a x)^2}-\frac{39 \log (1-a x)}{16 a^7}-\frac{9 \log (a x+1)}{16 a^7} \]

[Out]

-(x/a^6) - x^2/(2*a^5) + 1/(8*a^7*(1 - a*x)^2) - 5/(4*a^7*(1 - a*x)) - 1/(8*a^7*(1 + a*x)) - (39*Log[1 - a*x])
/(16*a^7) - (9*Log[1 + a*x])/(16*a^7)

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Rubi [A]  time = 0.140105, antiderivative size = 88, 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} \[ -\frac{x^2}{2 a^5}-\frac{x}{a^6}-\frac{5}{4 a^7 (1-a x)}-\frac{1}{8 a^7 (a x+1)}+\frac{1}{8 a^7 (1-a x)^2}-\frac{39 \log (1-a x)}{16 a^7}-\frac{9 \log (a x+1)}{16 a^7} \]

Antiderivative was successfully verified.

[In]

Int[(E^ArcTanh[a*x]*x^6)/(1 - a^2*x^2)^(5/2),x]

[Out]

-(x/a^6) - x^2/(2*a^5) + 1/(8*a^7*(1 - a*x)^2) - 5/(4*a^7*(1 - a*x)) - 1/(8*a^7*(1 + a*x)) - (39*Log[1 - a*x])
/(16*a^7) - (9*Log[1 + a*x])/(16*a^7)

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^6}{\left (1-a^2 x^2\right )^{5/2}} \, dx &=\int \frac{x^6}{(1-a x)^3 (1+a x)^2} \, dx\\ &=\int \left (-\frac{1}{a^6}-\frac{x}{a^5}-\frac{1}{4 a^6 (-1+a x)^3}-\frac{5}{4 a^6 (-1+a x)^2}-\frac{39}{16 a^6 (-1+a x)}+\frac{1}{8 a^6 (1+a x)^2}-\frac{9}{16 a^6 (1+a x)}\right ) \, dx\\ &=-\frac{x}{a^6}-\frac{x^2}{2 a^5}+\frac{1}{8 a^7 (1-a x)^2}-\frac{5}{4 a^7 (1-a x)}-\frac{1}{8 a^7 (1+a x)}-\frac{39 \log (1-a x)}{16 a^7}-\frac{9 \log (1+a x)}{16 a^7}\\ \end{align*}

Mathematica [A]  time = 0.0901663, size = 65, normalized size = 0.74 \[ \frac{2 \left (-4 a^2 x^2-8 a x+\frac{10}{a x-1}-\frac{1}{a x+1}+\frac{1}{(a x-1)^2}\right )-39 \log (1-a x)-9 \log (a x+1)}{16 a^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^ArcTanh[a*x]*x^6)/(1 - a^2*x^2)^(5/2),x]

[Out]

(2*(-8*a*x - 4*a^2*x^2 + (-1 + a*x)^(-2) + 10/(-1 + a*x) - (1 + a*x)^(-1)) - 39*Log[1 - a*x] - 9*Log[1 + a*x])
/(16*a^7)

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Maple [A]  time = 0.035, size = 74, normalized size = 0.8 \begin{align*} -{\frac{{x}^{2}}{2\,{a}^{5}}}-{\frac{x}{{a}^{6}}}-{\frac{1}{8\,{a}^{7} \left ( ax+1 \right ) }}-{\frac{9\,\ln \left ( ax+1 \right ) }{16\,{a}^{7}}}+{\frac{1}{8\,{a}^{7} \left ( ax-1 \right ) ^{2}}}+{\frac{5}{4\,{a}^{7} \left ( ax-1 \right ) }}-{\frac{39\,\ln \left ( ax-1 \right ) }{16\,{a}^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)/(-a^2*x^2+1)^3*x^6,x)

[Out]

-1/2*x^2/a^5-x/a^6-1/8/a^7/(a*x+1)-9/16*ln(a*x+1)/a^7+1/8/a^7/(a*x-1)^2+5/4/a^7/(a*x-1)-39/16/a^7*ln(a*x-1)

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Maxima [A]  time = 0.955406, size = 108, normalized size = 1.23 \begin{align*} \frac{9 \, a^{2} x^{2} + 3 \, a x - 10}{8 \,{\left (a^{10} x^{3} - a^{9} x^{2} - a^{8} x + a^{7}\right )}} - \frac{a x^{2} + 2 \, x}{2 \, a^{6}} - \frac{9 \, \log \left (a x + 1\right )}{16 \, a^{7}} - \frac{39 \, \log \left (a x - 1\right )}{16 \, a^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^3*x^6,x, algorithm="maxima")

[Out]

1/8*(9*a^2*x^2 + 3*a*x - 10)/(a^10*x^3 - a^9*x^2 - a^8*x + a^7) - 1/2*(a*x^2 + 2*x)/a^6 - 9/16*log(a*x + 1)/a^
7 - 39/16*log(a*x - 1)/a^7

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Fricas [A]  time = 1.77636, size = 271, normalized size = 3.08 \begin{align*} -\frac{8 \, a^{5} x^{5} + 8 \, a^{4} x^{4} - 24 \, a^{3} x^{3} - 26 \, a^{2} x^{2} + 10 \, a x + 9 \,{\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x + 1\right ) + 39 \,{\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x - 1\right ) + 20}{16 \,{\left (a^{10} x^{3} - a^{9} x^{2} - a^{8} x + a^{7}\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^6,x, algorithm="fricas")

[Out]

-1/16*(8*a^5*x^5 + 8*a^4*x^4 - 24*a^3*x^3 - 26*a^2*x^2 + 10*a*x + 9*(a^3*x^3 - a^2*x^2 - a*x + 1)*log(a*x + 1)
+ 39*(a^3*x^3 - a^2*x^2 - a*x + 1)*log(a*x - 1) + 20)/(a^10*x^3 - a^9*x^2 - a^8*x + a^7)

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Sympy [A]  time = 0.62623, size = 80, normalized size = 0.91 \begin{align*} \frac{9 a^{2} x^{2} + 3 a x - 10}{8 a^{10} x^{3} - 8 a^{9} x^{2} - 8 a^{8} x + 8 a^{7}} - \frac{x^{2}}{2 a^{5}} - \frac{x}{a^{6}} - \frac{3 \left (\frac{13 \log{\left (x - \frac{1}{a} \right )}}{16} + \frac{3 \log{\left (x + \frac{1}{a} \right )}}{16}\right )}{a^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a**2*x**2+1)**3*x**6,x)

[Out]

(9*a**2*x**2 + 3*a*x - 10)/(8*a**10*x**3 - 8*a**9*x**2 - 8*a**8*x + 8*a**7) - x**2/(2*a**5) - x/a**6 - 3*(13*l
og(x - 1/a)/16 + 3*log(x + 1/a)/16)/a**7

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Giac [A]  time = 1.17428, size = 104, normalized size = 1.18 \begin{align*} -\frac{9 \, \log \left ({\left | a x + 1 \right |}\right )}{16 \, a^{7}} - \frac{39 \, \log \left ({\left | a x - 1 \right |}\right )}{16 \, a^{7}} - \frac{a^{5} x^{2} + 2 \, a^{4} x}{2 \, a^{10}} + \frac{9 \, a^{2} x^{2} + 3 \, a x - 10}{8 \,{\left (a x + 1\right )}{\left (a x - 1\right )}^{2} a^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^3*x^6,x, algorithm="giac")

[Out]

-9/16*log(abs(a*x + 1))/a^7 - 39/16*log(abs(a*x - 1))/a^7 - 1/2*(a^5*x^2 + 2*a^4*x)/a^10 + 1/8*(9*a^2*x^2 + 3*
a*x - 10)/((a*x + 1)*(a*x - 1)^2*a^7)