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

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

[Out]

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

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Rubi [A]  time = 0.14, antiderivative size = 88, normalized size of antiderivative = 1.00, 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 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]))

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^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*}

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Mathematica [A]  time = 0.09, 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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fricas [A]  time = 0.66, size = 125, normalized size = 1.42 \[ -\frac {8 \, a^{5} x^{5} + 8 \, a^{4} x^{4} - 24 \, a^{3} x^{3} - 26 \, a^{2} x^{2} + 10 \, a x + 9 \, {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x + 1\right ) + 39 \, {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x - 1\right ) + 20}{16 \, {\left (a^{10} x^{3} - a^{9} x^{2} - a^{8} x + a^{7}\right )}} \]

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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giac [A]  time = 0.19, size = 77, normalized size = 0.88 \[ -\frac {9 \, \log \left ({\left | a x + 1 \right |}\right )}{16 \, a^{7}} - \frac {39 \, \log \left ({\left | a x - 1 \right |}\right )}{16 \, a^{7}} - \frac {a^{5} x^{2} + 2 \, a^{4} x}{2 \, a^{10}} + \frac {9 \, a^{2} x^{2} + 3 \, a x - 10}{8 \, {\left (a x + 1\right )} {\left (a x - 1\right )}^{2} a^{7}} \]

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)

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maple [A]  time = 0.04, size = 74, normalized size = 0.84 \[ -\frac {x^{2}}{2 a^{5}}-\frac {x}{a^{6}}+\frac {1}{8 a^{7} \left (a x -1\right )^{2}}+\frac {5}{4 a^{7} \left (a x -1\right )}-\frac {39 \ln \left (a x -1\right )}{16 a^{7}}-\frac {1}{8 a^{7} \left (a x +1\right )}-\frac {9 \ln \left (a x +1\right )}{16 a^{7}} \]

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

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maxima [A]  time = 0.32, size = 80, normalized size = 0.91 \[ \frac {9 \, a^{2} x^{2} + 3 \, a x - 10}{8 \, {\left (a^{10} x^{3} - a^{9} x^{2} - a^{8} x + a^{7}\right )}} - \frac {a x^{2} + 2 \, x}{2 \, a^{6}} - \frac {9 \, \log \left (a x + 1\right )}{16 \, a^{7}} - \frac {39 \, \log \left (a x - 1\right )}{16 \, a^{7}} \]

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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mupad [B]  time = 0.08, size = 82, normalized size = 0.93 \[ -\frac {39\,\ln \left (a\,x-1\right )}{16\,a^7}-\frac {9\,\ln \left (a\,x+1\right )}{16\,a^7}-\frac {x}{a^6}-\frac {\frac {3\,x}{8}+\frac {9\,a\,x^2}{8}-\frac {5}{4\,a}}{-a^9\,x^3+a^8\,x^2+a^7\,x-a^6}-\frac {x^2}{2\,a^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- (39*log(a*x - 1))/(16*a^7) - (9*log(a*x + 1))/(16*a^7) - x/a^6 - ((3*x)/8 + (9*a*x^2)/8 - 5/(4*a))/(a^7*x -
a^6 + a^8*x^2 - a^9*x^3) - x^2/(2*a^5)

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sympy [A]  time = 0.44, size = 82, normalized size = 0.93 \[ - \frac {- 9 a^{2} x^{2} - 3 a x + 10}{8 a^{10} x^{3} - 8 a^{9} x^{2} - 8 a^{8} x + 8 a^{7}} - \frac {x^{2}}{2 a^{5}} - \frac {x}{a^{6}} - \frac {3 \left (\frac {13 \log {\left (x - \frac {1}{a} \right )}}{16} + \frac {3 \log {\left (x + \frac {1}{a} \right )}}{16}\right )}{a^{7}} \]

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
*log(x - 1/a)/16 + 3*log(x + 1/a)/16)/a**7

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