3.2 \(\int x^4 \coth ^{-1}(a x) \, dx\)

Optimal. Leaf size=50 \[ \frac{x^2}{10 a^3}+\frac{\log \left (1-a^2 x^2\right )}{10 a^5}+\frac{x^4}{20 a}+\frac{1}{5} x^5 \coth ^{-1}(a x) \]

[Out]

x^2/(10*a^3) + x^4/(20*a) + (x^5*ArcCoth[a*x])/5 + Log[1 - a^2*x^2]/(10*a^5)

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Rubi [A]  time = 0.0371068, antiderivative size = 50, 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, 266, 43} \[ \frac{x^2}{10 a^3}+\frac{\log \left (1-a^2 x^2\right )}{10 a^5}+\frac{x^4}{20 a}+\frac{1}{5} x^5 \coth ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[x^4*ArcCoth[a*x],x]

[Out]

x^2/(10*a^3) + x^4/(20*a) + (x^5*ArcCoth[a*x])/5 + Log[1 - a^2*x^2]/(10*a^5)

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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x^4 \coth ^{-1}(a x) \, dx &=\frac{1}{5} x^5 \coth ^{-1}(a x)-\frac{1}{5} a \int \frac{x^5}{1-a^2 x^2} \, dx\\ &=\frac{1}{5} x^5 \coth ^{-1}(a x)-\frac{1}{10} a \operatorname{Subst}\left (\int \frac{x^2}{1-a^2 x} \, dx,x,x^2\right )\\ &=\frac{1}{5} x^5 \coth ^{-1}(a x)-\frac{1}{10} a \operatorname{Subst}\left (\int \left (-\frac{1}{a^4}-\frac{x}{a^2}-\frac{1}{a^4 \left (-1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac{x^2}{10 a^3}+\frac{x^4}{20 a}+\frac{1}{5} x^5 \coth ^{-1}(a x)+\frac{\log \left (1-a^2 x^2\right )}{10 a^5}\\ \end{align*}

Mathematica [A]  time = 0.0081977, size = 50, normalized size = 1. \[ \frac{x^2}{10 a^3}+\frac{\log \left (1-a^2 x^2\right )}{10 a^5}+\frac{x^4}{20 a}+\frac{1}{5} x^5 \coth ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*ArcCoth[a*x],x]

[Out]

x^2/(10*a^3) + x^4/(20*a) + (x^5*ArcCoth[a*x])/5 + Log[1 - a^2*x^2]/(10*a^5)

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Maple [A]  time = 0.035, size = 49, normalized size = 1. \begin{align*}{\frac{{x}^{5}{\rm arccoth} \left (ax\right )}{5}}+{\frac{{x}^{4}}{20\,a}}+{\frac{{x}^{2}}{10\,{a}^{3}}}+{\frac{\ln \left ( ax-1 \right ) }{10\,{a}^{5}}}+{\frac{\ln \left ( ax+1 \right ) }{10\,{a}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*arccoth(a*x),x)

[Out]

1/5*x^5*arccoth(a*x)+1/20*x^4/a+1/10*x^2/a^3+1/10/a^5*ln(a*x-1)+1/10/a^5*ln(a*x+1)

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Maxima [A]  time = 0.958241, size = 62, normalized size = 1.24 \begin{align*} \frac{1}{5} \, x^{5} \operatorname{arcoth}\left (a x\right ) + \frac{1}{20} \, a{\left (\frac{a^{2} x^{4} + 2 \, x^{2}}{a^{4}} + \frac{2 \, \log \left (a^{2} x^{2} - 1\right )}{a^{6}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arccoth(a*x),x, algorithm="maxima")

[Out]

1/5*x^5*arccoth(a*x) + 1/20*a*((a^2*x^4 + 2*x^2)/a^4 + 2*log(a^2*x^2 - 1)/a^6)

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Fricas [A]  time = 1.57557, size = 122, normalized size = 2.44 \begin{align*} \frac{2 \, a^{5} x^{5} \log \left (\frac{a x + 1}{a x - 1}\right ) + a^{4} x^{4} + 2 \, a^{2} x^{2} + 2 \, \log \left (a^{2} x^{2} - 1\right )}{20 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arccoth(a*x),x, algorithm="fricas")

[Out]

1/20*(2*a^5*x^5*log((a*x + 1)/(a*x - 1)) + a^4*x^4 + 2*a^2*x^2 + 2*log(a^2*x^2 - 1))/a^5

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Sympy [A]  time = 3.79828, size = 54, normalized size = 1.08 \begin{align*} \begin{cases} \frac{x^{5} \operatorname{acoth}{\left (a x \right )}}{5} + \frac{x^{4}}{20 a} + \frac{x^{2}}{10 a^{3}} + \frac{\log{\left (a x + 1 \right )}}{5 a^{5}} - \frac{\operatorname{acoth}{\left (a x \right )}}{5 a^{5}} & \text{for}\: a \neq 0 \\\frac{i \pi x^{5}}{10} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*acoth(a*x),x)

[Out]

Piecewise((x**5*acoth(a*x)/5 + x**4/(20*a) + x**2/(10*a**3) + log(a*x + 1)/(5*a**5) - acoth(a*x)/(5*a**5), Ne(
a, 0)), (I*pi*x**5/10, True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \operatorname{arcoth}\left (a x\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arccoth(a*x),x, algorithm="giac")

[Out]

integrate(x^4*arccoth(a*x), x)