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

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

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 41, normalized size of antiderivative = 1.00, 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 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]))))

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]

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*}

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Mathematica [A]  time = 0.01, size = 41, normalized size = 1.00 \[ 2 a^3 \log (x)-2 a^3 \log (1-a x)-\frac {2 a^2}{x}-\frac {a}{x^2}-\frac {1}{3 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*1/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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fricas [A]  time = 0.47, size = 43, normalized size = 1.05 \[ -\frac {6 \, a^{3} x^{3} \log \left (a x - 1\right ) - 6 \, a^{3} x^{3} \log \relax (x) + 6 \, a^{2} x^{2} + 3 \, a x + 1}{3 \, x^{3}} \]

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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giac [A]  time = 0.19, size = 40, normalized size = 0.98 \[ -2 \, a^{3} \log \left ({\left | a x - 1 \right |}\right ) + 2 \, a^{3} \log \left ({\left | x \right |}\right ) - \frac {6 \, a^{2} x^{2} + 3 \, a x + 1}{3 \, x^{3}} \]

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

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maple [A]  time = 0.03, size = 39, normalized size = 0.95 \[ -\frac {1}{3 x^{3}}-\frac {a}{x^{2}}-\frac {2 a^{2}}{x}+2 a^{3} \ln \relax (x )-2 a^{3} \ln \left (a x -1\right ) \]

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 = 0.31, size = 38, normalized size = 0.93 \[ -2 \, a^{3} \log \left (a x - 1\right ) + 2 \, a^{3} \log \relax (x) - \frac {6 \, a^{2} x^{2} + 3 \, a x + 1}{3 \, x^{3}} \]

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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mupad [B]  time = 0.05, size = 31, normalized size = 0.76 \[ 4\,a^3\,\mathrm {atanh}\left (2\,a\,x-1\right )-\frac {2\,a^2\,x^2+a\,x+\frac {1}{3}}{x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.18, size = 36, normalized size = 0.88 \[ - 2 a^{3} \left (- \log {\relax (x )} + \log {\left (x - \frac {1}{a} \right )}\right ) - \frac {6 a^{2} x^{2} + 3 a x + 1}{3 x^{3}} \]

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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