3.43 \(\int e^{-2 \tanh ^{-1}(a x)} x^3 \, dx\)

Optimal. Leaf size=43 \[ -\frac {2 \log (a x+1)}{a^4}+\frac {2 x}{a^3}-\frac {x^2}{a^2}+\frac {2 x^3}{3 a}-\frac {x^4}{4} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 43, 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} \[ -\frac {x^2}{a^2}+\frac {2 x}{a^3}-\frac {2 \log (a x+1)}{a^4}+\frac {2 x^3}{3 a}-\frac {x^4}{4} \]

Antiderivative was successfully verified.

[In]

Int[x^3/E^(2*ArcTanh[a*x]),x]

[Out]

(2*x)/a^3 - x^2/a^2 + (2*x^3)/(3*a) - x^4/4 - (2*Log[1 + a*x])/a^4

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

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Mathematica [A]  time = 0.02, size = 43, normalized size = 1.00 \[ -\frac {2 \log (a x+1)}{a^4}+\frac {2 x}{a^3}-\frac {x^2}{a^2}+\frac {2 x^3}{3 a}-\frac {x^4}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/E^(2*ArcTanh[a*x]),x]

[Out]

(2*x)/a^3 - x^2/a^2 + (2*x^3)/(3*a) - x^4/4 - (2*Log[1 + a*x])/a^4

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fricas [A]  time = 0.72, size = 42, normalized size = 0.98 \[ -\frac {3 \, a^{4} x^{4} - 8 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 24 \, a x + 24 \, \log \left (a x + 1\right )}{12 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(3*a^4*x^4 - 8*a^3*x^3 + 12*a^2*x^2 - 24*a*x + 24*log(a*x + 1))/a^4

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giac [A]  time = 0.39, size = 66, normalized size = 1.53 \[ \frac {{\left (a x + 1\right )}^{4} {\left (\frac {20}{a x + 1} - \frac {54}{{\left (a x + 1\right )}^{2}} + \frac {84}{{\left (a x + 1\right )}^{3}} - 3\right )}}{12 \, a^{4}} + \frac {2 \, \log \left (\frac {{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2} {\left | a \right |}}\right )}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(a*x + 1)^4*(20/(a*x + 1) - 54/(a*x + 1)^2 + 84/(a*x + 1)^3 - 3)/a^4 + 2*log(abs(a*x + 1)/((a*x + 1)^2*ab
s(a)))/a^4

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maple [A]  time = 0.03, size = 40, normalized size = 0.93 \[ \frac {2 x}{a^{3}}-\frac {x^{2}}{a^{2}}+\frac {2 x^{3}}{3 a}-\frac {x^{4}}{4}-\frac {2 \ln \left (a x +1\right )}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.31, size = 43, normalized size = 1.00 \[ -\frac {3 \, a^{3} x^{4} - 8 \, a^{2} x^{3} + 12 \, a x^{2} - 24 \, x}{12 \, a^{3}} - \frac {2 \, \log \left (a x + 1\right )}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(3*a^3*x^4 - 8*a^2*x^3 + 12*a*x^2 - 24*x)/a^3 - 2*log(a*x + 1)/a^4

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mupad [B]  time = 0.04, size = 39, normalized size = 0.91 \[ \frac {2\,x}{a^3}-\frac {2\,\ln \left (a\,x+1\right )}{a^4}-\frac {x^4}{4}+\frac {2\,x^3}{3\,a}-\frac {x^2}{a^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.11, size = 37, normalized size = 0.86 \[ - \frac {x^{4}}{4} + \frac {2 x^{3}}{3 a} - \frac {x^{2}}{a^{2}} + \frac {2 x}{a^{3}} - \frac {2 \log {\left (a x + 1 \right )}}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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