3.947 \(\int \frac{e^{\tanh ^{-1}(a x)}}{x^3 (1-a^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=89 \[ \frac{a^2}{1-a x}+\frac{a^2}{8 (a x+1)}+\frac{a^2}{8 (1-a x)^2}+3 a^2 \log (x)-\frac{39}{16} a^2 \log (1-a x)-\frac{9}{16} a^2 \log (a x+1)-\frac{a}{x}-\frac{1}{2 x^2} \]

[Out]

-1/(2*x^2) - a/x + a^2/(8*(1 - a*x)^2) + a^2/(1 - a*x) + a^2/(8*(1 + a*x)) + 3*a^2*Log[x] - (39*a^2*Log[1 - a*
x])/16 - (9*a^2*Log[1 + a*x])/16

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Rubi [A]  time = 0.13547, antiderivative size = 89, 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} \[ \frac{a^2}{1-a x}+\frac{a^2}{8 (a x+1)}+\frac{a^2}{8 (1-a x)^2}+3 a^2 \log (x)-\frac{39}{16} a^2 \log (1-a x)-\frac{9}{16} a^2 \log (a x+1)-\frac{a}{x}-\frac{1}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]/(x^3*(1 - a^2*x^2)^(5/2)),x]

[Out]

-1/(2*x^2) - a/x + a^2/(8*(1 - a*x)^2) + a^2/(1 - a*x) + a^2/(8*(1 + a*x)) + 3*a^2*Log[x] - (39*a^2*Log[1 - a*
x])/16 - (9*a^2*Log[1 + a*x])/16

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

Mathematica [A]  time = 0.071539, size = 83, normalized size = 0.93 \[ \frac{1}{16} \left (\frac{16 a^2}{1-a x}+\frac{2 a^2}{a x+1}+\frac{2 a^2}{(a x-1)^2}+48 a^2 \log (x)-39 a^2 \log (1-a x)-9 a^2 \log (a x+1)-\frac{16 a}{x}-\frac{8}{x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcTanh[a*x]/(x^3*(1 - a^2*x^2)^(5/2)),x]

[Out]

(-8/x^2 - (16*a)/x + (16*a^2)/(1 - a*x) + (2*a^2)/(-1 + a*x)^2 + (2*a^2)/(1 + a*x) + 48*a^2*Log[x] - 39*a^2*Lo
g[1 - a*x] - 9*a^2*Log[1 + a*x])/16

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Maple [A]  time = 0.041, size = 78, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,{x}^{2}}}-{\frac{a}{x}}+3\,{a}^{2}\ln \left ( x \right ) +{\frac{{a}^{2}}{8\,ax+8}}-{\frac{9\,{a}^{2}\ln \left ( ax+1 \right ) }{16}}-{\frac{{a}^{2}}{ax-1}}+{\frac{{a}^{2}}{8\, \left ( ax-1 \right ) ^{2}}}-{\frac{39\,{a}^{2}\ln \left ( ax-1 \right ) }{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/x^2-a/x+3*a^2*ln(x)+1/8*a^2/(a*x+1)-9/16*a^2*ln(a*x+1)-a^2/(a*x-1)+1/8*a^2/(a*x-1)^2-39/16*a^2*ln(a*x-1)

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Maxima [A]  time = 0.961134, size = 120, normalized size = 1.35 \begin{align*} -\frac{9}{16} \, a^{2} \log \left (a x + 1\right ) - \frac{39}{16} \, a^{2} \log \left (a x - 1\right ) + 3 \, a^{2} \log \left (x\right ) - \frac{15 \, a^{4} x^{4} - 3 \, a^{3} x^{3} - 22 \, a^{2} x^{2} + 4 \, a x + 4}{8 \,{\left (a^{3} x^{5} - a^{2} x^{4} - a x^{3} + x^{2}\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^3,x, algorithm="maxima")

[Out]

-9/16*a^2*log(a*x + 1) - 39/16*a^2*log(a*x - 1) + 3*a^2*log(x) - 1/8*(15*a^4*x^4 - 3*a^3*x^3 - 22*a^2*x^2 + 4*
a*x + 4)/(a^3*x^5 - a^2*x^4 - a*x^3 + x^2)

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Fricas [B]  time = 2.14004, size = 348, normalized size = 3.91 \begin{align*} -\frac{30 \, a^{4} x^{4} - 6 \, a^{3} x^{3} - 44 \, a^{2} x^{2} + 8 \, a x + 9 \,{\left (a^{5} x^{5} - a^{4} x^{4} - a^{3} x^{3} + a^{2} x^{2}\right )} \log \left (a x + 1\right ) + 39 \,{\left (a^{5} x^{5} - a^{4} x^{4} - a^{3} x^{3} + a^{2} x^{2}\right )} \log \left (a x - 1\right ) - 48 \,{\left (a^{5} x^{5} - a^{4} x^{4} - a^{3} x^{3} + a^{2} x^{2}\right )} \log \left (x\right ) + 8}{16 \,{\left (a^{3} x^{5} - a^{2} x^{4} - a x^{3} + x^{2}\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^3,x, algorithm="fricas")

[Out]

-1/16*(30*a^4*x^4 - 6*a^3*x^3 - 44*a^2*x^2 + 8*a*x + 9*(a^5*x^5 - a^4*x^4 - a^3*x^3 + a^2*x^2)*log(a*x + 1) +
39*(a^5*x^5 - a^4*x^4 - a^3*x^3 + a^2*x^2)*log(a*x - 1) - 48*(a^5*x^5 - a^4*x^4 - a^3*x^3 + a^2*x^2)*log(x) +
8)/(a^3*x^5 - a^2*x^4 - a*x^3 + x^2)

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Sympy [A]  time = 0.917268, size = 95, normalized size = 1.07 \begin{align*} 3 a^{2} \log{\left (x \right )} - \frac{39 a^{2} \log{\left (x - \frac{1}{a} \right )}}{16} - \frac{9 a^{2} \log{\left (x + \frac{1}{a} \right )}}{16} - \frac{15 a^{4} x^{4} - 3 a^{3} x^{3} - 22 a^{2} x^{2} + 4 a x + 4}{8 a^{3} x^{5} - 8 a^{2} x^{4} - 8 a x^{3} + 8 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*a**2*log(x) - 39*a**2*log(x - 1/a)/16 - 9*a**2*log(x + 1/a)/16 - (15*a**4*x**4 - 3*a**3*x**3 - 22*a**2*x**2
+ 4*a*x + 4)/(8*a**3*x**5 - 8*a**2*x**4 - 8*a*x**3 + 8*x**2)

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Giac [A]  time = 1.23689, size = 111, normalized size = 1.25 \begin{align*} -\frac{9}{16} \, a^{2} \log \left ({\left | a x + 1 \right |}\right ) - \frac{39}{16} \, a^{2} \log \left ({\left | a x - 1 \right |}\right ) + 3 \, a^{2} \log \left ({\left | x \right |}\right ) - \frac{15 \, a^{4} x^{4} - 3 \, a^{3} x^{3} - 22 \, a^{2} x^{2} + 4 \, a x + 4}{8 \,{\left (a x + 1\right )}{\left (a x - 1\right )}^{2} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/16*a^2*log(abs(a*x + 1)) - 39/16*a^2*log(abs(a*x - 1)) + 3*a^2*log(abs(x)) - 1/8*(15*a^4*x^4 - 3*a^3*x^3 -
22*a^2*x^2 + 4*a*x + 4)/((a*x + 1)*(a*x - 1)^2*x^2)