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

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

[Out]

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

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Rubi [A]  time = 0.0317947, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6126, 77} \[ -\frac{2 a^2}{x}-2 a^3 \log (x)+2 a^3 \log (a x+1)+\frac{a}{x^2}-\frac{1}{3 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6126

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_)^(m_.), x_Symbol] :> Int[(x^m*(1 + a*x)^(n/2))/(1 - a*x)^(n/2), x] /; Fre
eQ[{a, m, n}, x] &&  !IntegerQ[(n - 1)/2]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

Mathematica [A]  time = 0.0117499, size = 39, normalized size = 1. \[ -\frac{2 a^2}{x}-2 a^3 \log (x)+2 a^3 \log (a x+1)+\frac{a}{x^2}-\frac{1}{3 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.038, size = 38, normalized size = 1. \begin{align*} -{\frac{1}{3\,{x}^{3}}}+{\frac{a}{{x}^{2}}}-2\,{\frac{{a}^{2}}{x}}-2\,{a}^{3}\ln \left ( x \right ) +2\,{a}^{3}\ln \left ( ax+1 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.949299, size = 51, normalized size = 1.31 \begin{align*} 2 \, a^{3} \log \left (a x + 1\right ) - 2 \, a^{3} \log \left (x\right ) - \frac{6 \, a^{2} x^{2} - 3 \, a x + 1}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.82843, size = 104, normalized size = 2.67 \begin{align*} \frac{6 \, a^{3} x^{3} \log \left (a x + 1\right ) - 6 \, a^{3} x^{3} \log \left (x\right ) - 6 \, a^{2} x^{2} + 3 \, a x - 1}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.603, size = 36, normalized size = 0.92 \begin{align*} - 2 a^{3} \left (\log{\left (x \right )} - \log{\left (x + \frac{1}{a} \right )}\right ) - \frac{6 a^{2} x^{2} - 3 a x + 1}{3 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.19352, size = 84, normalized size = 2.15 \begin{align*} -2 \, a^{3} \log \left ({\left | -\frac{1}{a x + 1} + 1 \right |}\right ) + \frac{10 \, a^{3} - \frac{24 \, a^{3}}{a x + 1} + \frac{15 \, a^{3}}{{\left (a x + 1\right )}^{2}}}{3 \,{\left (\frac{1}{a x + 1} - 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a^3*log(abs(-1/(a*x + 1) + 1)) + 1/3*(10*a^3 - 24*a^3/(a*x + 1) + 15*a^3/(a*x + 1)^2)/(1/(a*x + 1) - 1)^3