### 3.31 $$\int \frac{e^{4 \coth ^{-1}(a x)}}{x^3} \, dx$$

Optimal. Leaf size=46 $\frac{4 a^2}{1-a x}+8 a^2 \log (x)-8 a^2 \log (1-a x)-\frac{4 a}{x}-\frac{1}{2 x^2}$

[Out]

-1/(2*x^2) - (4*a)/x + (4*a^2)/(1 - a*x) + 8*a^2*Log[x] - 8*a^2*Log[1 - a*x]

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Rubi [A]  time = 0.054295, antiderivative size = 46, 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, 88} $\frac{4 a^2}{1-a x}+8 a^2 \log (x)-8 a^2 \log (1-a x)-\frac{4 a}{x}-\frac{1}{2 x^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^(4*ArcCoth[a*x])/x^3,x]

[Out]

-1/(2*x^2) - (4*a)/x + (4*a^2)/(1 - a*x) + 8*a^2*Log[x] - 8*a^2*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 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^{4 \coth ^{-1}(a x)}}{x^3} \, dx &=\int \frac{e^{4 \tanh ^{-1}(a x)}}{x^3} \, dx\\ &=\int \frac{(1+a x)^2}{x^3 (1-a x)^2} \, dx\\ &=\int \left (\frac{1}{x^3}+\frac{4 a}{x^2}+\frac{8 a^2}{x}+\frac{4 a^3}{(-1+a x)^2}-\frac{8 a^3}{-1+a x}\right ) \, dx\\ &=-\frac{1}{2 x^2}-\frac{4 a}{x}+\frac{4 a^2}{1-a x}+8 a^2 \log (x)-8 a^2 \log (1-a x)\\ \end{align*}

Mathematica [A]  time = 0.0300503, size = 46, normalized size = 1. $\frac{4 a^2}{1-a x}+8 a^2 \log (x)-8 a^2 \log (1-a x)-\frac{4 a}{x}-\frac{1}{2 x^2}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[E^(4*ArcCoth[a*x])/x^3,x]

[Out]

-1/(2*x^2) - (4*a)/x + (4*a^2)/(1 - a*x) + 8*a^2*Log[x] - 8*a^2*Log[1 - a*x]

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Maple [A]  time = 0.053, size = 43, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,{x}^{2}}}-4\,{\frac{a}{x}}+8\,{a}^{2}\ln \left ( x \right ) -4\,{\frac{{a}^{2}}{ax-1}}-8\,{a}^{2}\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*x+1)^2/x^3,x)

[Out]

-1/2/x^2-4*a/x+8*a^2*ln(x)-4*a^2/(a*x-1)-8*a^2*ln(a*x-1)

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Maxima [A]  time = 1.01859, size = 65, normalized size = 1.41 \begin{align*} -8 \, a^{2} \log \left (a x - 1\right ) + 8 \, a^{2} \log \left (x\right ) - \frac{16 \, a^{2} x^{2} - 7 \, a x - 1}{2 \,{\left (a x^{3} - x^{2}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8*a^2*log(a*x - 1) + 8*a^2*log(x) - 1/2*(16*a^2*x^2 - 7*a*x - 1)/(a*x^3 - x^2)

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Fricas [A]  time = 1.81316, size = 155, normalized size = 3.37 \begin{align*} -\frac{16 \, a^{2} x^{2} - 7 \, a x + 16 \,{\left (a^{3} x^{3} - a^{2} x^{2}\right )} \log \left (a x - 1\right ) - 16 \,{\left (a^{3} x^{3} - a^{2} x^{2}\right )} \log \left (x\right ) - 1}{2 \,{\left (a x^{3} - x^{2}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(16*a^2*x^2 - 7*a*x + 16*(a^3*x^3 - a^2*x^2)*log(a*x - 1) - 16*(a^3*x^3 - a^2*x^2)*log(x) - 1)/(a*x^3 - x
^2)

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Sympy [A]  time = 0.459218, size = 41, normalized size = 0.89 \begin{align*} 8 a^{2} \left (\log{\left (x \right )} - \log{\left (x - \frac{1}{a} \right )}\right ) - \frac{16 a^{2} x^{2} - 7 a x - 1}{2 a x^{3} - 2 x^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8*a**2*(log(x) - log(x - 1/a)) - (16*a**2*x**2 - 7*a*x - 1)/(2*a*x**3 - 2*x**2)

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Giac [A]  time = 1.15712, size = 84, normalized size = 1.83 \begin{align*} 8 \, a^{2} \log \left ({\left | -\frac{1}{a x - 1} - 1 \right |}\right ) - \frac{4 \, a^{2}}{a x - 1} + \frac{9 \, a^{2} + \frac{10 \, a^{2}}{a x - 1}}{2 \,{\left (\frac{1}{a x - 1} + 1\right )}^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8*a^2*log(abs(-1/(a*x - 1) - 1)) - 4*a^2/(a*x - 1) + 1/2*(9*a^2 + 10*a^2/(a*x - 1))/(1/(a*x - 1) + 1)^2