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

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

[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 = 0.03, antiderivative size = 39, 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} \[ -\frac {2 a^2}{x}-2 a^3 \log (x)+2 a^3 \log (a x+1)+\frac {a}{x^2}-\frac {1}{3 x^3} \]

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 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 = 39, normalized size = 1.00 \[ -2 a^3 \log (x)+2 a^3 \log (a x+1)-\frac {2 a^2}{x}+\frac {a}{x^2}-\frac {1}{3 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*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 = 0.52, size = 43, normalized size = 1.10 \[ \frac {6 \, a^{3} x^{3} \log \left (a x + 1\right ) - 6 \, a^{3} x^{3} \log \relax (x) - 6 \, a^{2} x^{2} + 3 \, a x - 1}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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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giac [A]  time = 0.16, size = 62, normalized size = 1.59 \[ -2 \, a^{3} \log \left ({\left | -\frac {1}{a x + 1} + 1 \right |}\right ) + \frac {10 \, a^{3} - \frac {24 \, a^{3}}{a x + 1} + \frac {15 \, a^{3}}{{\left (a x + 1\right )}^{2}}}{3 \, {\left (\frac {1}{a x + 1} - 1\right )}^{3}} \]

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

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maple [A]  time = 0.03, size = 38, normalized size = 0.97 \[ -\frac {1}{3 x^{3}}+\frac {a}{x^{2}}-\frac {2 a^{2}}{x}-2 a^{3} \ln \relax (x )+2 a^{3} \ln \left (a x +1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(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 = 0.31, size = 38, normalized size = 0.97 \[ 2 \, a^{3} \log \left (a x + 1\right ) - 2 \, a^{3} \log \relax (x) - \frac {6 \, a^{2} x^{2} - 3 \, a x + 1}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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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mupad [B]  time = 0.05, size = 32, normalized size = 0.82 \[ 4\,a^3\,\mathrm {atanh}\left (2\,a\,x+1\right )-\frac {2\,a^2\,x^2-a\,x+\frac {1}{3}}{x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.18, size = 36, normalized size = 0.92 \[ - 2 a^{3} \left (\log {\relax (x )} - \log {\left (x + \frac {1}{a} \right )}\right ) - \frac {6 a^{2} x^{2} - 3 a x + 1}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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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