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

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

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 51, normalized size of antiderivative = 1.00, 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 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])

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 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]

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*}

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Mathematica [A]  time = 0.01, size = 67, normalized size = 1.31 \[ \frac {\log (1-a x)}{12 a^6}-\frac {\log (a x+1)}{12 a^6}+\frac {x}{6 a^5}+\frac {x^3}{18 a^3}+\frac {1}{6} x^6 \coth ^{-1}(a x)+\frac {x^5}{30 a} \]

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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fricas [A]  time = 0.70, size = 51, normalized size = 1.00 \[ \frac {6 \, a^{5} x^{5} + 10 \, a^{3} x^{3} + 30 \, a x + 15 \, {\left (a^{6} x^{6} - 1\right )} \log \left (\frac {a x + 1}{a x - 1}\right )}{180 \, a^{6}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{5} \operatorname {arcoth}\left (a x\right )\,{d x} \]

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)

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maple [A]  time = 0.03, size = 55, normalized size = 1.08 \[ \frac {x^{6} \mathrm {arccoth}\left (a x \right )}{6}+\frac {x^{5}}{30 a}+\frac {x^{3}}{18 a^{3}}+\frac {x}{6 a^{5}}+\frac {\ln \left (a x -1\right )}{12 a^{6}}-\frac {\ln \left (a x +1\right )}{12 a^{6}} \]

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*ln(a*x+1)/a^6

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maxima [A]  time = 0.30, size = 61, normalized size = 1.20 \[ \frac {1}{6} \, x^{6} \operatorname {arcoth}\left (a x\right ) + \frac {1}{180} \, a {\left (\frac {2 \, {\left (3 \, a^{4} x^{5} + 5 \, a^{2} x^{3} + 15 \, x\right )}}{a^{6}} - \frac {15 \, \log \left (a x + 1\right )}{a^{7}} + \frac {15 \, \log \left (a x - 1\right )}{a^{7}}\right )} \]

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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mupad [B]  time = 1.31, size = 41, normalized size = 0.80 \[ \frac {\frac {a\,x}{6}-\frac {\mathrm {acoth}\left (a\,x\right )}{6}+\frac {a^3\,x^3}{18}+\frac {a^5\,x^5}{30}}{a^6}+\frac {x^6\,\mathrm {acoth}\left (a\,x\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((a*x)/6 - acoth(a*x)/6 + (a^3*x^3)/18 + (a^5*x^5)/30)/a^6 + (x^6*acoth(a*x))/6

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sympy [A]  time = 1.53, size = 49, normalized size = 0.96 \[ \begin {cases} \frac {x^{6} \operatorname {acoth}{\left (a x \right )}}{6} + \frac {x^{5}}{30 a} + \frac {x^{3}}{18 a^{3}} + \frac {x}{6 a^{5}} - \frac {\operatorname {acoth}{\left (a x \right )}}{6 a^{6}} & \text {for}\: a \neq 0 \\\frac {i \pi x^{6}}{12} & \text {otherwise} \end {cases} \]

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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