3.3 \(\int x^3 \coth ^{-1}(a x) \, dx\)

Optimal. Leaf size=41 \[ \frac{x}{4 a^3}-\frac{\tanh ^{-1}(a x)}{4 a^4}+\frac{x^3}{12 a}+\frac{1}{4} x^4 \coth ^{-1}(a x) \]

[Out]

x/(4*a^3) + x^3/(12*a) + (x^4*ArcCoth[a*x])/4 - ArcTanh[a*x]/(4*a^4)

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Rubi [A]  time = 0.0237262, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {5917, 302, 206} \[ \frac{x}{4 a^3}-\frac{\tanh ^{-1}(a x)}{4 a^4}+\frac{x^3}{12 a}+\frac{1}{4} x^4 \coth ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[x^3*ArcCoth[a*x],x]

[Out]

x/(4*a^3) + x^3/(12*a) + (x^4*ArcCoth[a*x])/4 - ArcTanh[a*x]/(4*a^4)

Rule 5917

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcC
oth[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCoth[c*x])^(p - 1))/(1 -
c^2*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0082089, size = 57, normalized size = 1.39 \[ \frac{x}{4 a^3}+\frac{\log (1-a x)}{8 a^4}-\frac{\log (a x+1)}{8 a^4}+\frac{x^3}{12 a}+\frac{1}{4} x^4 \coth ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*ArcCoth[a*x],x]

[Out]

x/(4*a^3) + x^3/(12*a) + (x^4*ArcCoth[a*x])/4 + Log[1 - a*x]/(8*a^4) - Log[1 + a*x]/(8*a^4)

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Maple [A]  time = 0.031, size = 47, normalized size = 1.2 \begin{align*}{\frac{{x}^{4}{\rm arccoth} \left (ax\right )}{4}}+{\frac{{x}^{3}}{12\,a}}+{\frac{x}{4\,{a}^{3}}}+{\frac{\ln \left ( ax-1 \right ) }{8\,{a}^{4}}}-{\frac{\ln \left ( ax+1 \right ) }{8\,{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*arccoth(a*x),x)

[Out]

1/4*x^4*arccoth(a*x)+1/12*x^3/a+1/4*x/a^3+1/8/a^4*ln(a*x-1)-1/8/a^4*ln(a*x+1)

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Maxima [A]  time = 0.966109, size = 70, normalized size = 1.71 \begin{align*} \frac{1}{4} \, x^{4} \operatorname{arcoth}\left (a x\right ) + \frac{1}{24} \, a{\left (\frac{2 \,{\left (a^{2} x^{3} + 3 \, x\right )}}{a^{4}} - \frac{3 \, \log \left (a x + 1\right )}{a^{5}} + \frac{3 \, \log \left (a x - 1\right )}{a^{5}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arccoth(a*x),x, algorithm="maxima")

[Out]

1/4*x^4*arccoth(a*x) + 1/24*a*(2*(a^2*x^3 + 3*x)/a^4 - 3*log(a*x + 1)/a^5 + 3*log(a*x - 1)/a^5)

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Fricas [A]  time = 1.59866, size = 99, normalized size = 2.41 \begin{align*} \frac{2 \, a^{3} x^{3} + 6 \, a x + 3 \,{\left (a^{4} x^{4} - 1\right )} \log \left (\frac{a x + 1}{a x - 1}\right )}{24 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arccoth(a*x),x, algorithm="fricas")

[Out]

1/24*(2*a^3*x^3 + 6*a*x + 3*(a^4*x^4 - 1)*log((a*x + 1)/(a*x - 1)))/a^4

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Sympy [A]  time = 1.28445, size = 41, normalized size = 1. \begin{align*} \begin{cases} \frac{x^{4} \operatorname{acoth}{\left (a x \right )}}{4} + \frac{x^{3}}{12 a} + \frac{x}{4 a^{3}} - \frac{\operatorname{acoth}{\left (a x \right )}}{4 a^{4}} & \text{for}\: a \neq 0 \\\frac{i \pi x^{4}}{8} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*acoth(a*x),x)

[Out]

Piecewise((x**4*acoth(a*x)/4 + x**3/(12*a) + x/(4*a**3) - acoth(a*x)/(4*a**4), Ne(a, 0)), (I*pi*x**4/8, True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \operatorname{arcoth}\left (a x\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arccoth(a*x),x, algorithm="giac")

[Out]

integrate(x^3*arccoth(a*x), x)