3.25 \(\int e^{4 \coth ^{-1}(a x)} x^3 \, dx\)

Optimal. Leaf size=57 \[ \frac{4 x^2}{a^2}+\frac{12 x}{a^3}+\frac{4}{a^4 (1-a x)}+\frac{16 \log (1-a x)}{a^4}+\frac{4 x^3}{3 a}+\frac{x^4}{4} \]

[Out]

(12*x)/a^3 + (4*x^2)/a^2 + (4*x^3)/(3*a) + x^4/4 + 4/(a^4*(1 - a*x)) + (16*Log[1 - a*x])/a^4

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Rubi [A]  time = 0.0658147, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6167, 6126, 88} \[ \frac{4 x^2}{a^2}+\frac{12 x}{a^3}+\frac{4}{a^4 (1-a x)}+\frac{16 \log (1-a x)}{a^4}+\frac{4 x^3}{3 a}+\frac{x^4}{4} \]

Antiderivative was successfully verified.

[In]

Int[E^(4*ArcCoth[a*x])*x^3,x]

[Out]

(12*x)/a^3 + (4*x^2)/a^2 + (4*x^3)/(3*a) + x^4/4 + 4/(a^4*(1 - a*x)) + (16*Log[1 - a*x])/a^4

Rule 6167

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

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]

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int e^{4 \coth ^{-1}(a x)} x^3 \, dx &=\int e^{4 \tanh ^{-1}(a x)} x^3 \, dx\\ &=\int \frac{x^3 (1+a x)^2}{(1-a x)^2} \, dx\\ &=\int \left (\frac{12}{a^3}+\frac{8 x}{a^2}+\frac{4 x^2}{a}+x^3+\frac{4}{a^3 (-1+a x)^2}+\frac{16}{a^3 (-1+a x)}\right ) \, dx\\ &=\frac{12 x}{a^3}+\frac{4 x^2}{a^2}+\frac{4 x^3}{3 a}+\frac{x^4}{4}+\frac{4}{a^4 (1-a x)}+\frac{16 \log (1-a x)}{a^4}\\ \end{align*}

Mathematica [A]  time = 0.045886, size = 57, normalized size = 1. \[ \frac{4 x^2}{a^2}+\frac{12 x}{a^3}+\frac{4}{a^4 (1-a x)}+\frac{16 \log (1-a x)}{a^4}+\frac{4 x^3}{3 a}+\frac{x^4}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(4*ArcCoth[a*x])*x^3,x]

[Out]

(12*x)/a^3 + (4*x^2)/a^2 + (4*x^3)/(3*a) + x^4/4 + 4/(a^4*(1 - a*x)) + (16*Log[1 - a*x])/a^4

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Maple [A]  time = 0.047, size = 52, normalized size = 0.9 \begin{align*}{\frac{{x}^{4}}{4}}+{\frac{4\,{x}^{3}}{3\,a}}+4\,{\frac{{x}^{2}}{{a}^{2}}}+12\,{\frac{x}{{a}^{3}}}-4\,{\frac{1}{{a}^{4} \left ( ax-1 \right ) }}+16\,{\frac{\ln \left ( ax-1 \right ) }{{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^4+4/3*x^3/a+4*x^2/a^2+12*x/a^3-4/a^4/(a*x-1)+16/a^4*ln(a*x-1)

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Maxima [A]  time = 0.984836, size = 78, normalized size = 1.37 \begin{align*} -\frac{4}{a^{5} x - a^{4}} + \frac{3 \, a^{3} x^{4} + 16 \, a^{2} x^{3} + 48 \, a x^{2} + 144 \, x}{12 \, a^{3}} + \frac{16 \, \log \left (a x - 1\right )}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/(a^5*x - a^4) + 1/12*(3*a^3*x^4 + 16*a^2*x^3 + 48*a*x^2 + 144*x)/a^3 + 16*log(a*x - 1)/a^4

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Fricas [A]  time = 1.70644, size = 155, normalized size = 2.72 \begin{align*} \frac{3 \, a^{5} x^{5} + 13 \, a^{4} x^{4} + 32 \, a^{3} x^{3} + 96 \, a^{2} x^{2} - 144 \, a x + 192 \,{\left (a x - 1\right )} \log \left (a x - 1\right ) - 48}{12 \,{\left (a^{5} x - a^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*a^5*x^5 + 13*a^4*x^4 + 32*a^3*x^3 + 96*a^2*x^2 - 144*a*x + 192*(a*x - 1)*log(a*x - 1) - 48)/(a^5*x - a
^4)

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Sympy [A]  time = 0.353346, size = 49, normalized size = 0.86 \begin{align*} \frac{x^{4}}{4} - \frac{4}{a^{5} x - a^{4}} + \frac{4 x^{3}}{3 a} + \frac{4 x^{2}}{a^{2}} + \frac{12 x}{a^{3}} + \frac{16 \log{\left (a x - 1 \right )}}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**4/4 - 4/(a**5*x - a**4) + 4*x**3/(3*a) + 4*x**2/a**2 + 12*x/a**3 + 16*log(a*x - 1)/a**4

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Giac [A]  time = 1.21016, size = 105, normalized size = 1.84 \begin{align*} \frac{{\left (a x - 1\right )}^{4}{\left (\frac{28}{a x - 1} + \frac{114}{{\left (a x - 1\right )}^{2}} + \frac{300}{{\left (a x - 1\right )}^{3}} + 3\right )}}{12 \, a^{4}} - \frac{16 \, \log \left (\frac{{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2}{\left | a \right |}}\right )}{a^{4}} - \frac{4}{{\left (a x - 1\right )} a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(a*x - 1)^4*(28/(a*x - 1) + 114/(a*x - 1)^2 + 300/(a*x - 1)^3 + 3)/a^4 - 16*log(abs(a*x - 1)/((a*x - 1)^2
*abs(a)))/a^4 - 4/((a*x - 1)*a^4)