3.851 \(\int e^{-2 \tanh ^{-1}(a+b x)} x^4 \, dx\)

Optimal. Leaf size=71 \[ -\frac{2 (a+1) x^3}{3 b^2}+\frac{(a+1)^2 x^2}{b^3}-\frac{2 (a+1)^3 x}{b^4}+\frac{2 (a+1)^4 \log (a+b x+1)}{b^5}+\frac{x^4}{2 b}-\frac{x^5}{5} \]

[Out]

(-2*(1 + a)^3*x)/b^4 + ((1 + a)^2*x^2)/b^3 - (2*(1 + a)*x^3)/(3*b^2) + x^4/(2*b) - x^5/5 + (2*(1 + a)^4*Log[1
+ a + b*x])/b^5

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Rubi [A]  time = 0.0785129, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {6163, 77} \[ -\frac{2 (a+1) x^3}{3 b^2}+\frac{(a+1)^2 x^2}{b^3}-\frac{2 (a+1)^3 x}{b^4}+\frac{2 (a+1)^4 \log (a+b x+1)}{b^5}+\frac{x^4}{2 b}-\frac{x^5}{5} \]

Antiderivative was successfully verified.

[In]

Int[x^4/E^(2*ArcTanh[a + b*x]),x]

[Out]

(-2*(1 + a)^3*x)/b^4 + ((1 + a)^2*x^2)/b^3 - (2*(1 + a)*x^3)/(3*b^2) + x^4/(2*b) - x^5/5 + (2*(1 + a)^4*Log[1
+ a + b*x])/b^5

Rule 6163

Int[E^(ArcTanh[(c_.)*((a_) + (b_.)*(x_))]*(n_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[((d + e*x)^m*(1
+ a*c + b*c*x)^(n/2))/(1 - a*c - b*c*x)^(n/2), x] /; FreeQ[{a, b, c, d, e, m, n}, 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]))))

Rubi steps

\begin{align*} \int e^{-2 \tanh ^{-1}(a+b x)} x^4 \, dx &=\int \frac{x^4 (1-a-b x)}{1+a+b x} \, dx\\ &=\int \left (-\frac{2 (1+a)^3}{b^4}+\frac{2 (1+a)^2 x}{b^3}-\frac{2 (1+a) x^2}{b^2}+\frac{2 x^3}{b}-x^4+\frac{2 (1+a)^4}{b^4 (1+a+b x)}\right ) \, dx\\ &=-\frac{2 (1+a)^3 x}{b^4}+\frac{(1+a)^2 x^2}{b^3}-\frac{2 (1+a) x^3}{3 b^2}+\frac{x^4}{2 b}-\frac{x^5}{5}+\frac{2 (1+a)^4 \log (1+a+b x)}{b^5}\\ \end{align*}

Mathematica [A]  time = 0.0570517, size = 71, normalized size = 1. \[ -\frac{2 (a+1) x^3}{3 b^2}+\frac{(a+1)^2 x^2}{b^3}-\frac{2 (a+1)^3 x}{b^4}+\frac{2 (a+1)^4 \log (a+b x+1)}{b^5}+\frac{x^4}{2 b}-\frac{x^5}{5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 + a)^3*x)/b^4 + ((1 + a)^2*x^2)/b^3 - (2*(1 + a)*x^3)/(3*b^2) + x^4/(2*b) - x^5/5 + (2*(1 + a)^4*Log[1
+ a + b*x])/b^5

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Maple [B]  time = 0.029, size = 159, normalized size = 2.2 \begin{align*} -{\frac{{x}^{5}}{5}}+{\frac{{x}^{4}}{2\,b}}-{\frac{2\,{x}^{3}a}{3\,{b}^{2}}}-{\frac{2\,{x}^{3}}{3\,{b}^{2}}}+{\frac{{a}^{2}{x}^{2}}{{b}^{3}}}+2\,{\frac{a{x}^{2}}{{b}^{3}}}-2\,{\frac{x{a}^{3}}{{b}^{4}}}+{\frac{{x}^{2}}{{b}^{3}}}-6\,{\frac{{a}^{2}x}{{b}^{4}}}-6\,{\frac{ax}{{b}^{4}}}-2\,{\frac{x}{{b}^{4}}}+2\,{\frac{\ln \left ( bx+a+1 \right ){a}^{4}}{{b}^{5}}}+8\,{\frac{\ln \left ( bx+a+1 \right ){a}^{3}}{{b}^{5}}}+12\,{\frac{\ln \left ( bx+a+1 \right ){a}^{2}}{{b}^{5}}}+8\,{\frac{\ln \left ( bx+a+1 \right ) a}{{b}^{5}}}+2\,{\frac{\ln \left ( bx+a+1 \right ) }{{b}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*x^5+1/2*x^4/b-2/3/b^2*x^3*a-2/3/b^2*x^3+1/b^3*x^2*a^2+2/b^3*x^2*a-2/b^4*x*a^3+1/b^3*x^2-6/b^4*x*a^2-6/b^4
*a*x-2/b^4*x+2/b^5*ln(b*x+a+1)*a^4+8/b^5*ln(b*x+a+1)*a^3+12/b^5*ln(b*x+a+1)*a^2+8/b^5*ln(b*x+a+1)*a+2/b^5*ln(b
*x+a+1)

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Maxima [A]  time = 0.955928, size = 127, normalized size = 1.79 \begin{align*} -\frac{6 \, b^{4} x^{5} - 15 \, b^{3} x^{4} + 20 \,{\left (a + 1\right )} b^{2} x^{3} - 30 \,{\left (a^{2} + 2 \, a + 1\right )} b x^{2} + 60 \,{\left (a^{3} + 3 \, a^{2} + 3 \, a + 1\right )} x}{30 \, b^{4}} + \frac{2 \,{\left (a^{4} + 4 \, a^{3} + 6 \, a^{2} + 4 \, a + 1\right )} \log \left (b x + a + 1\right )}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x+a+1)^2*(1-(b*x+a)^2),x, algorithm="maxima")

[Out]

-1/30*(6*b^4*x^5 - 15*b^3*x^4 + 20*(a + 1)*b^2*x^3 - 30*(a^2 + 2*a + 1)*b*x^2 + 60*(a^3 + 3*a^2 + 3*a + 1)*x)/
b^4 + 2*(a^4 + 4*a^3 + 6*a^2 + 4*a + 1)*log(b*x + a + 1)/b^5

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Fricas [A]  time = 1.41104, size = 234, normalized size = 3.3 \begin{align*} -\frac{6 \, b^{5} x^{5} - 15 \, b^{4} x^{4} + 20 \,{\left (a + 1\right )} b^{3} x^{3} - 30 \,{\left (a^{2} + 2 \, a + 1\right )} b^{2} x^{2} + 60 \,{\left (a^{3} + 3 \, a^{2} + 3 \, a + 1\right )} b x - 60 \,{\left (a^{4} + 4 \, a^{3} + 6 \, a^{2} + 4 \, a + 1\right )} \log \left (b x + a + 1\right )}{30 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x+a+1)^2*(1-(b*x+a)^2),x, algorithm="fricas")

[Out]

-1/30*(6*b^5*x^5 - 15*b^4*x^4 + 20*(a + 1)*b^3*x^3 - 30*(a^2 + 2*a + 1)*b^2*x^2 + 60*(a^3 + 3*a^2 + 3*a + 1)*b
*x - 60*(a^4 + 4*a^3 + 6*a^2 + 4*a + 1)*log(b*x + a + 1))/b^5

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Sympy [A]  time = 0.447723, size = 78, normalized size = 1.1 \begin{align*} - \frac{x^{5}}{5} + \frac{x^{4}}{2 b} - \frac{x^{3} \left (2 a + 2\right )}{3 b^{2}} + \frac{x^{2} \left (a^{2} + 2 a + 1\right )}{b^{3}} - \frac{x \left (2 a^{3} + 6 a^{2} + 6 a + 2\right )}{b^{4}} + \frac{2 \left (a + 1\right )^{4} \log{\left (a + b x + 1 \right )}}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(b*x+a+1)**2*(1-(b*x+a)**2),x)

[Out]

-x**5/5 + x**4/(2*b) - x**3*(2*a + 2)/(3*b**2) + x**2*(a**2 + 2*a + 1)/b**3 - x*(2*a**3 + 6*a**2 + 6*a + 2)/b*
*4 + 2*(a + 1)**4*log(a + b*x + 1)/b**5

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Giac [B]  time = 1.20884, size = 273, normalized size = 3.85 \begin{align*} \frac{{\left (b x + a + 1\right )}^{5}{\left (\frac{15 \,{\left (2 \, a b + 3 \, b\right )}}{{\left (b x + a + 1\right )} b} - \frac{20 \,{\left (3 \, a^{2} b^{2} + 10 \, a b^{2} + 7 \, b^{2}\right )}}{{\left (b x + a + 1\right )}^{2} b^{2}} + \frac{60 \,{\left (a^{3} b^{3} + 6 \, a^{2} b^{3} + 9 \, a b^{3} + 4 \, b^{3}\right )}}{{\left (b x + a + 1\right )}^{3} b^{3}} - \frac{30 \,{\left (a^{4} b^{4} + 12 \, a^{3} b^{4} + 30 \, a^{2} b^{4} + 28 \, a b^{4} + 9 \, b^{4}\right )}}{{\left (b x + a + 1\right )}^{4} b^{4}} - 6\right )}}{30 \, b^{5}} - \frac{2 \,{\left (a^{4} + 4 \, a^{3} + 6 \, a^{2} + 4 \, a + 1\right )} \log \left (\frac{{\left | b x + a + 1 \right |}}{{\left (b x + a + 1\right )}^{2}{\left | b \right |}}\right )}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x+a+1)^2*(1-(b*x+a)^2),x, algorithm="giac")

[Out]

1/30*(b*x + a + 1)^5*(15*(2*a*b + 3*b)/((b*x + a + 1)*b) - 20*(3*a^2*b^2 + 10*a*b^2 + 7*b^2)/((b*x + a + 1)^2*
b^2) + 60*(a^3*b^3 + 6*a^2*b^3 + 9*a*b^3 + 4*b^3)/((b*x + a + 1)^3*b^3) - 30*(a^4*b^4 + 12*a^3*b^4 + 30*a^2*b^
4 + 28*a*b^4 + 9*b^4)/((b*x + a + 1)^4*b^4) - 6)/b^5 - 2*(a^4 + 4*a^3 + 6*a^2 + 4*a + 1)*log(abs(b*x + a + 1)/
((b*x + a + 1)^2*abs(b)))/b^5