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

Optimal. Leaf size=40 \[ \frac{2 a^2}{x}+2 a^3 \log (x)-2 a^3 \log (a x+1)-\frac{a}{x^2}+\frac{1}{3 x^3} \]

[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 = 0.0496812, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6167, 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*ArcCoth[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 6167

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

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 \coth ^{-1}(a x)}}{x^4} \, dx &=-\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 = 0.0132844, size = 40, 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*ArcCoth[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 = 0.046, size = 39, 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)*(a*x-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 = 0.985085, size = 51, normalized size = 1.27 \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((a*x-1)/(a*x+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 = 1.47225, size = 105, normalized size = 2.62 \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((a*x-1)/(a*x+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 = 0.360706, size = 34, normalized size = 0.85 \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((a*x-1)/(a*x+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 = 1.15059, size = 54, normalized size = 1.35 \begin{align*} -2 \, a^{3} \log \left ({\left | a x + 1 \right |}\right ) + 2 \, a^{3} \log \left ({\left | x \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((a*x-1)/(a*x+1)/x^4,x, algorithm="giac")

[Out]

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