3.937 \(\int \frac{e^{\tanh ^{-1}(a x)}}{x^4 (1-a^2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=73 \[ \frac{a^3}{2 (1-a x)}-\frac{2 a^2}{x}+2 a^3 \log (x)-\frac{9}{4} a^3 \log (1-a x)+\frac{1}{4} a^3 \log (a x+1)-\frac{a}{2 x^2}-\frac{1}{3 x^3} \]

[Out]

-1/(3*x^3) - a/(2*x^2) - (2*a^2)/x + a^3/(2*(1 - a*x)) + 2*a^3*Log[x] - (9*a^3*Log[1 - a*x])/4 + (a^3*Log[1 +
a*x])/4

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Rubi [A]  time = 0.121673, antiderivative size = 73, 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} \[ \frac{a^3}{2 (1-a x)}-\frac{2 a^2}{x}+2 a^3 \log (x)-\frac{9}{4} a^3 \log (1-a x)+\frac{1}{4} a^3 \log (a x+1)-\frac{a}{2 x^2}-\frac{1}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]/(x^4*(1 - a^2*x^2)^(3/2)),x]

[Out]

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

Mathematica [A]  time = 0.0514541, size = 67, normalized size = 0.92 \[ \frac{1}{12} \left (\frac{6 a^3}{1-a x}-\frac{24 a^2}{x}+24 a^3 \log (x)-27 a^3 \log (1-a x)+3 a^3 \log (a x+1)-\frac{6 a}{x^2}-\frac{4}{x^3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcTanh[a*x]/(x^4*(1 - a^2*x^2)^(3/2)),x]

[Out]

(-4/x^3 - (6*a)/x^2 - (24*a^2)/x + (6*a^3)/(1 - a*x) + 24*a^3*Log[x] - 27*a^3*Log[1 - a*x] + 3*a^3*Log[1 + a*x
])/12

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Maple [A]  time = 0.036, size = 62, normalized size = 0.9 \begin{align*} -{\frac{1}{3\,{x}^{3}}}-{\frac{a}{2\,{x}^{2}}}-2\,{\frac{{a}^{2}}{x}}+2\,{a}^{3}\ln \left ( x \right ) +{\frac{{a}^{3}\ln \left ( ax+1 \right ) }{4}}-{\frac{{a}^{3}}{2\,ax-2}}-{\frac{9\,{a}^{3}\ln \left ( ax-1 \right ) }{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^4,x)

[Out]

-1/3/x^3-1/2*a/x^2-2*a^2/x+2*a^3*ln(x)+1/4*a^3*ln(a*x+1)-1/2*a^3/(a*x-1)-9/4*a^3*ln(a*x-1)

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Maxima [A]  time = 0.964068, size = 90, normalized size = 1.23 \begin{align*} \frac{1}{4} \, a^{3} \log \left (a x + 1\right ) - \frac{9}{4} \, a^{3} \log \left (a x - 1\right ) + 2 \, a^{3} \log \left (x\right ) - \frac{15 \, a^{3} x^{3} - 9 \, a^{2} x^{2} - a x - 2}{6 \,{\left (a x^{4} - x^{3}\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^4,x, algorithm="maxima")

[Out]

1/4*a^3*log(a*x + 1) - 9/4*a^3*log(a*x - 1) + 2*a^3*log(x) - 1/6*(15*a^3*x^3 - 9*a^2*x^2 - a*x - 2)/(a*x^4 - x
^3)

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Fricas [A]  time = 1.71931, size = 224, normalized size = 3.07 \begin{align*} -\frac{30 \, a^{3} x^{3} - 18 \, a^{2} x^{2} - 2 \, a x - 3 \,{\left (a^{4} x^{4} - a^{3} x^{3}\right )} \log \left (a x + 1\right ) + 27 \,{\left (a^{4} x^{4} - a^{3} x^{3}\right )} \log \left (a x - 1\right ) - 24 \,{\left (a^{4} x^{4} - a^{3} x^{3}\right )} \log \left (x\right ) - 4}{12 \,{\left (a x^{4} - x^{3}\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^4,x, algorithm="fricas")

[Out]

-1/12*(30*a^3*x^3 - 18*a^2*x^2 - 2*a*x - 3*(a^4*x^4 - a^3*x^3)*log(a*x + 1) + 27*(a^4*x^4 - a^3*x^3)*log(a*x -
1) - 24*(a^4*x^4 - a^3*x^3)*log(x) - 4)/(a*x^4 - x^3)

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Sympy [A]  time = 0.631542, size = 66, normalized size = 0.9 \begin{align*} 2 a^{3} \log{\left (x \right )} - \frac{9 a^{3} \log{\left (x - \frac{1}{a} \right )}}{4} + \frac{a^{3} \log{\left (x + \frac{1}{a} \right )}}{4} - \frac{15 a^{3} x^{3} - 9 a^{2} x^{2} - a x - 2}{6 a x^{4} - 6 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a**2*x**2+1)**2/x**4,x)

[Out]

2*a**3*log(x) - 9*a**3*log(x - 1/a)/4 + a**3*log(x + 1/a)/4 - (15*a**3*x**3 - 9*a**2*x**2 - a*x - 2)/(6*a*x**4
- 6*x**3)

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Giac [A]  time = 1.14705, size = 90, normalized size = 1.23 \begin{align*} \frac{1}{4} \, a^{3} \log \left ({\left | a x + 1 \right |}\right ) - \frac{9}{4} \, a^{3} \log \left ({\left | a x - 1 \right |}\right ) + 2 \, a^{3} \log \left ({\left | x \right |}\right ) - \frac{15 \, a^{3} x^{3} - 9 \, a^{2} x^{2} - a x - 2}{6 \,{\left (a x - 1\right )} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^2/x^4,x, algorithm="giac")

[Out]

1/4*a^3*log(abs(a*x + 1)) - 9/4*a^3*log(abs(a*x - 1)) + 2*a^3*log(abs(x)) - 1/6*(15*a^3*x^3 - 9*a^2*x^2 - a*x
- 2)/((a*x - 1)*x^3)