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

Optimal. Leaf size=48 \[ \frac{x}{a^3}+\frac{1}{2 a^4 (1-a x)}+\frac{5 \log (1-a x)}{4 a^4}-\frac{\log (a x+1)}{4 a^4} \]

[Out]

x/a^3 + 1/(2*a^4*(1 - a*x)) + (5*Log[1 - a*x])/(4*a^4) - Log[1 + a*x]/(4*a^4)

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Rubi [A]  time = 0.113325, antiderivative size = 48, 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} \[ \frac{x}{a^3}+\frac{1}{2 a^4 (1-a x)}+\frac{5 \log (1-a x)}{4 a^4}-\frac{\log (a x+1)}{4 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/a^3 + 1/(2*a^4*(1 - a*x)) + (5*Log[1 - a*x])/(4*a^4) - Log[1 + a*x]/(4*a^4)

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

Mathematica [A]  time = 0.0306449, size = 39, normalized size = 0.81 \[ \frac{4 a x+\frac{2}{1-a x}+5 \log (1-a x)-\log (a x+1)}{4 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*a*x + 2/(1 - a*x) + 5*Log[1 - a*x] - Log[1 + a*x])/(4*a^4)

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Maple [A]  time = 0.033, size = 41, normalized size = 0.9 \begin{align*}{\frac{x}{{a}^{3}}}-{\frac{\ln \left ( ax+1 \right ) }{4\,{a}^{4}}}-{\frac{1}{2\,{a}^{4} \left ( ax-1 \right ) }}+{\frac{5\,\ln \left ( ax-1 \right ) }{4\,{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/a^3-1/4/a^4*ln(a*x+1)-1/2/a^4/(a*x-1)+5/4/a^4*ln(a*x-1)

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Maxima [A]  time = 0.948588, size = 58, normalized size = 1.21 \begin{align*} -\frac{1}{2 \,{\left (a^{5} x - a^{4}\right )}} + \frac{x}{a^{3}} - \frac{\log \left (a x + 1\right )}{4 \, a^{4}} + \frac{5 \, \log \left (a x - 1\right )}{4 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(a^5*x - a^4) + x/a^3 - 1/4*log(a*x + 1)/a^4 + 5/4*log(a*x - 1)/a^4

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Fricas [A]  time = 1.77264, size = 128, normalized size = 2.67 \begin{align*} \frac{4 \, a^{2} x^{2} - 4 \, a x -{\left (a x - 1\right )} \log \left (a x + 1\right ) + 5 \,{\left (a x - 1\right )} \log \left (a x - 1\right ) - 2}{4 \,{\left (a^{5} x - a^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(4*a^2*x^2 - 4*a*x - (a*x - 1)*log(a*x + 1) + 5*(a*x - 1)*log(a*x - 1) - 2)/(a^5*x - a^4)

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Sympy [A]  time = 0.412554, size = 39, normalized size = 0.81 \begin{align*} - \frac{1}{2 a^{5} x - 2 a^{4}} + \frac{x}{a^{3}} + \frac{\frac{5 \log{\left (x - \frac{1}{a} \right )}}{4} - \frac{\log{\left (x + \frac{1}{a} \right )}}{4}}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(2*a**5*x - 2*a**4) + x/a**3 + (5*log(x - 1/a)/4 - log(x + 1/a)/4)/a**4

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Giac [A]  time = 1.15815, size = 57, normalized size = 1.19 \begin{align*} \frac{x}{a^{3}} - \frac{\log \left ({\left | a x + 1 \right |}\right )}{4 \, a^{4}} + \frac{5 \, \log \left ({\left | a x - 1 \right |}\right )}{4 \, a^{4}} - \frac{1}{2 \,{\left (a x - 1\right )} a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/a^3 - 1/4*log(abs(a*x + 1))/a^4 + 5/4*log(abs(a*x - 1))/a^4 - 1/2/((a*x - 1)*a^4)