3.1.44 \(\int e^{-2 \tanh ^{-1}(a x)} x^2 \, dx\) [44]

Optimal. Leaf size=32 \[ -\frac {2 x}{a^2}+\frac {x^2}{a}-\frac {x^3}{3}+\frac {2 \log (1+a x)}{a^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 32, 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 = {6261, 78} \begin {gather*} \frac {2 \log (a x+1)}{a^3}-\frac {2 x}{a^2}+\frac {x^2}{a}-\frac {x^3}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 78

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 6261

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

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 32, normalized size = 1.00 \begin {gather*} -\frac {2 x}{a^2}+\frac {x^2}{a}-\frac {x^3}{3}+\frac {2 \log (1+a x)}{a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.42, size = 36, normalized size = 1.12

method	result	size
risch	\(-\frac {2 x}{a^{2}}+\frac {x^{2}}{a}-\frac {x^{3}}{3}+\frac {2 \ln \left (a x +1\right )}{a^{3}}\)	\(31\)
default	\(-\frac {\frac {1}{3} a^{2} x^{3}-a \,x^{2}+2 x}{a^{2}}+\frac {2 \ln \left (a x +1\right )}{a^{3}}\)	\(36\)
norman	\(\frac {-\frac {2 x}{a^{2}}+\frac {2 x^{3}}{3}-\frac {x^{2}}{a}-\frac {x^{4} a}{3}}{a x +1}+\frac {2 \ln \left (a x +1\right )}{a^{3}}\)	\(47\)
meijerg	\(-\frac {\frac {a x \left (5 a^{3} x^{3}-10 a^{2} x^{2}+30 a x +60\right )}{15 a x +15}-4 \ln \left (a x +1\right )}{a^{3}}+\frac {\frac {a x \left (3 a x +6\right )}{3 a x +3}-2 \ln \left (a x +1\right )}{a^{3}}\)	\(79\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a*x+1)^2*(-a^2*x^2+1),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 34, normalized size = 1.06 \begin {gather*} -\frac {a^{2} x^{3} - 3 \, a x^{2} + 6 \, x}{3 \, a^{2}} + \frac {2 \, \log \left (a x + 1\right )}{a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(a^2*x^3 - 3*a*x^2 + 6*x)/a^2 + 2*log(a*x + 1)/a^3

________________________________________________________________________________________

Fricas [A]
time = 0.36, size = 33, normalized size = 1.03 \begin {gather*} -\frac {a^{3} x^{3} - 3 \, a^{2} x^{2} + 6 \, a x - 6 \, \log \left (a x + 1\right )}{3 \, a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(a^3*x^3 - 3*a^2*x^2 + 6*a*x - 6*log(a*x + 1))/a^3

________________________________________________________________________________________

Sympy [A]
time = 0.04, size = 27, normalized size = 0.84 \begin {gather*} - \frac {x^{3}}{3} + \frac {x^{2}}{a} - \frac {2 x}{a^{2}} + \frac {2 \log {\left (a x + 1 \right )}}{a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 0.41, size = 57, normalized size = 1.78 \begin {gather*} \frac {{\left (a x + 1\right )}^{3} {\left (\frac {6}{a x + 1} - \frac {15}{{\left (a x + 1\right )}^{2}} - 1\right )}}{3 \, a^{3}} - \frac {2 \, \log \left (\frac {{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2} {\left | a \right |}}\right )}{a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(a*x + 1)^3*(6/(a*x + 1) - 15/(a*x + 1)^2 - 1)/a^3 - 2*log(abs(a*x + 1)/((a*x + 1)^2*abs(a)))/a^3

________________________________________________________________________________________

Mupad [B]
time = 0.04, size = 30, normalized size = 0.94 \begin {gather*} \frac {2\,\ln \left (a\,x+1\right )}{a^3}-\frac {2\,x}{a^2}-\frac {x^3}{3}+\frac {x^2}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________