3.1 \(\int x^5 \coth ^{-1}(a x) \, dx\)

Optimal. Leaf size=51 \[ \frac{x^3}{18 a^3}+\frac{x}{6 a^5}-\frac{\tanh ^{-1}(a x)}{6 a^6}+\frac{x^5}{30 a}+\frac{1}{6} x^6 \coth ^{-1}(a x) \]

[Out]

x/(6*a^5) + x^3/(18*a^3) + x^5/(30*a) + (x^6*ArcCoth[a*x])/6 - ArcTanh[a*x]/(6*a^6)

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Rubi [A]  time = 0.0279481, antiderivative size = 51, 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} \[ \frac{x^3}{18 a^3}+\frac{x}{6 a^5}-\frac{\tanh ^{-1}(a x)}{6 a^6}+\frac{x^5}{30 a}+\frac{1}{6} x^6 \coth ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[x^5*ArcCoth[a*x],x]

[Out]

x/(6*a^5) + x^3/(18*a^3) + x^5/(30*a) + (x^6*ArcCoth[a*x])/6 - ArcTanh[a*x]/(6*a^6)

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^5 \coth ^{-1}(a x) \, dx &=\frac{1}{6} x^6 \coth ^{-1}(a x)-\frac{1}{6} a \int \frac{x^6}{1-a^2 x^2} \, dx\\ &=\frac{1}{6} x^6 \coth ^{-1}(a x)-\frac{1}{6} a \int \left (-\frac{1}{a^6}-\frac{x^2}{a^4}-\frac{x^4}{a^2}+\frac{1}{a^6 \left (1-a^2 x^2\right )}\right ) \, dx\\ &=\frac{x}{6 a^5}+\frac{x^3}{18 a^3}+\frac{x^5}{30 a}+\frac{1}{6} x^6 \coth ^{-1}(a x)-\frac{\int \frac{1}{1-a^2 x^2} \, dx}{6 a^5}\\ &=\frac{x}{6 a^5}+\frac{x^3}{18 a^3}+\frac{x^5}{30 a}+\frac{1}{6} x^6 \coth ^{-1}(a x)-\frac{\tanh ^{-1}(a x)}{6 a^6}\\ \end{align*}

Mathematica [A]  time = 0.0089957, size = 67, normalized size = 1.31 \[ \frac{x^3}{18 a^3}+\frac{x}{6 a^5}+\frac{\log (1-a x)}{12 a^6}-\frac{\log (a x+1)}{12 a^6}+\frac{x^5}{30 a}+\frac{1}{6} x^6 \coth ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*ArcCoth[a*x],x]

[Out]

x/(6*a^5) + x^3/(18*a^3) + x^5/(30*a) + (x^6*ArcCoth[a*x])/6 + Log[1 - a*x]/(12*a^6) - Log[1 + a*x]/(12*a^6)

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Maple [A]  time = 0.035, size = 55, normalized size = 1.1 \begin{align*}{\frac{{x}^{6}{\rm arccoth} \left (ax\right )}{6}}+{\frac{{x}^{5}}{30\,a}}+{\frac{{x}^{3}}{18\,{a}^{3}}}+{\frac{x}{6\,{a}^{5}}}+{\frac{\ln \left ( ax-1 \right ) }{12\,{a}^{6}}}-{\frac{\ln \left ( ax+1 \right ) }{12\,{a}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*arccoth(a*x),x)

[Out]

1/6*x^6*arccoth(a*x)+1/30*x^5/a+1/18*x^3/a^3+1/6*x/a^5+1/12/a^6*ln(a*x-1)-1/12/a^6*ln(a*x+1)

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Maxima [A]  time = 0.964772, size = 82, normalized size = 1.61 \begin{align*} \frac{1}{6} \, x^{6} \operatorname{arcoth}\left (a x\right ) + \frac{1}{180} \, a{\left (\frac{2 \,{\left (3 \, a^{4} x^{5} + 5 \, a^{2} x^{3} + 15 \, x\right )}}{a^{6}} - \frac{15 \, \log \left (a x + 1\right )}{a^{7}} + \frac{15 \, \log \left (a x - 1\right )}{a^{7}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*arccoth(a*x),x, algorithm="maxima")

[Out]

1/6*x^6*arccoth(a*x) + 1/180*a*(2*(3*a^4*x^5 + 5*a^2*x^3 + 15*x)/a^6 - 15*log(a*x + 1)/a^7 + 15*log(a*x - 1)/a
^7)

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Fricas [A]  time = 1.54015, size = 120, normalized size = 2.35 \begin{align*} \frac{6 \, a^{5} x^{5} + 10 \, a^{3} x^{3} + 30 \, a x + 15 \,{\left (a^{6} x^{6} - 1\right )} \log \left (\frac{a x + 1}{a x - 1}\right )}{180 \, a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*arccoth(a*x),x, algorithm="fricas")

[Out]

1/180*(6*a^5*x^5 + 10*a^3*x^3 + 30*a*x + 15*(a^6*x^6 - 1)*log((a*x + 1)/(a*x - 1)))/a^6

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Sympy [A]  time = 2.07, size = 49, normalized size = 0.96 \begin{align*} \begin{cases} \frac{x^{6} \operatorname{acoth}{\left (a x \right )}}{6} + \frac{x^{5}}{30 a} + \frac{x^{3}}{18 a^{3}} + \frac{x}{6 a^{5}} - \frac{\operatorname{acoth}{\left (a x \right )}}{6 a^{6}} & \text{for}\: a \neq 0 \\\frac{i \pi x^{6}}{12} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*acoth(a*x),x)

[Out]

Piecewise((x**6*acoth(a*x)/6 + x**5/(30*a) + x**3/(18*a**3) + x/(6*a**5) - acoth(a*x)/(6*a**6), Ne(a, 0)), (I*
pi*x**6/12, True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{5} \operatorname{arcoth}\left (a x\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*arccoth(a*x),x, algorithm="giac")

[Out]

integrate(x^5*arccoth(a*x), x)