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

Optimal. Leaf size=41 \[ -\frac{2 a^2}{x}+2 a^3 \log (x)-2 a^3 \log (1-a x)-\frac{a}{x^2}-\frac{1}{3 x^3} \]

[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 = 0.0356681, antiderivative size = 41, 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} \[ -\frac{2 a^2}{x}+2 a^3 \log (x)-2 a^3 \log (1-a x)-\frac{a}{x^2}-\frac{1}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[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 = 0.0131947, size = 41, normalized size = 1. \[ -\frac{2 a^2}{x}+2 a^3 \log (x)-2 a^3 \log (1-a x)-\frac{a}{x^2}-\frac{1}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[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 = 0.037, 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((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 = 0.951531, size = 51, normalized size = 1.24 \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)^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 = 1.69405, size = 105, normalized size = 2.56 \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)^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 = 0.485669, size = 36, normalized size = 0.88 \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)**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 = 1.14824, size = 54, normalized size = 1.32 \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)^2/(-a^2*x^2+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