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

Optimal result

Integrand size = 8, antiderivative size = 51 \[ \int x^5 \coth ^{-1}(a x) \, dx=\frac {x}{6 a^5}+\frac {x^3}{18 a^3}+\frac {x^5}{30 a}+\frac {1}{6} x^6 \coth ^{-1}(a x)-\frac {\text {arctanh}(a x)}{6 a^6} \]

[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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6038, 308, 212} \[ \int x^5 \coth ^{-1}(a x) \, dx=-\frac {\text {arctanh}(a x)}{6 a^6}+\frac {x}{6 a^5}+\frac {x^3}{18 a^3}+\frac {1}{6} x^6 \coth ^{-1}(a x)+\frac {x^5}{30 a} \]

[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 212

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

Rule 308

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 6038

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

Rubi steps \begin{align*} \text {integral}& = \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 {\text {arctanh}(a x)}{6 a^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.31 \[ \int x^5 \coth ^{-1}(a x) \, dx=\frac {x}{6 a^5}+\frac {x^3}{18 a^3}+\frac {x^5}{30 a}+\frac {1}{6} x^6 \coth ^{-1}(a x)+\frac {\log (1-a x)}{12 a^6}-\frac {\log (1+a x)}{12 a^6} \]

[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)

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.88

method	result	size
parallelrisch	\(-\frac {-15 a^{6} x^{6} \operatorname {arccoth}\left (a x \right )-3 a^{5} x^{5}-5 a^{3} x^{3}-15 a x +15 \,\operatorname {arccoth}\left (a x \right )}{90 a^{6}}\)	\(45\)
derivativedivides	\(\frac {\frac {a^{6} x^{6} \operatorname {arccoth}\left (a x \right )}{6}+\frac {a^{5} x^{5}}{30}+\frac {a^{3} x^{3}}{18}+\frac {a x}{6}+\frac {\ln \left (a x -1\right )}{12}-\frac {\ln \left (a x +1\right )}{12}}{a^{6}}\)	\(54\)
default	\(\frac {\frac {a^{6} x^{6} \operatorname {arccoth}\left (a x \right )}{6}+\frac {a^{5} x^{5}}{30}+\frac {a^{3} x^{3}}{18}+\frac {a x}{6}+\frac {\ln \left (a x -1\right )}{12}-\frac {\ln \left (a x +1\right )}{12}}{a^{6}}\)	\(54\)
parts	\(\frac {x^{6} \operatorname {arccoth}\left (a x \right )}{6}+\frac {a \left (-\frac {\ln \left (a x +1\right )}{2 a^{7}}+\frac {\frac {1}{5} a^{4} x^{5}+\frac {1}{3} a^{2} x^{3}+x}{a^{6}}+\frac {\ln \left (a x -1\right )}{2 a^{7}}\right )}{6}\)	\(59\)
risch	\(\frac {x^{6} \ln \left (a x +1\right )}{12}-\frac {\ln \left (a x -1\right ) x^{6}}{12}+\frac {x^{5}}{30 a}+\frac {x^{3}}{18 a^{3}}+\frac {x}{6 a^{5}}-\frac {\ln \left (a x +1\right )}{12 a^{6}}+\frac {\ln \left (-a x +1\right )}{12 a^{6}}\)	\(69\)

[In]

int(x^5*arccoth(a*x),x,method=_RETURNVERBOSE)

[Out]

-1/90*(-15*a^6*x^6*arccoth(a*x)-3*a^5*x^5-5*a^3*x^3-15*a*x+15*arccoth(a*x))/a^6

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00 \[ \int x^5 \coth ^{-1}(a x) \, dx=\frac {6 \, a^{5} x^{5} + 10 \, a^{3} x^{3} + 30 \, a x + 15 \, {\left (a^{6} x^{6} - 1\right )} \log \left (\frac {a x + 1}{a x - 1}\right )}{180 \, a^{6}} \]

[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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.33 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.96 \[ \int x^5 \coth ^{-1}(a x) \, dx=\begin {cases} \frac {x^{6} \operatorname {acoth}{\left (a x \right )}}{6} + \frac {x^{5}}{30 a} + \frac {x^{3}}{18 a^{3}} + \frac {x}{6 a^{5}} - \frac {\operatorname {acoth}{\left (a x \right )}}{6 a^{6}} & \text {for}\: a \neq 0 \\\frac {i \pi x^{6}}{12} & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.20 \[ \int x^5 \coth ^{-1}(a x) \, dx=\frac {1}{6} \, x^{6} \operatorname {arcoth}\left (a x\right ) + \frac {1}{180} \, a {\left (\frac {2 \, {\left (3 \, a^{4} x^{5} + 5 \, a^{2} x^{3} + 15 \, x\right )}}{a^{6}} - \frac {15 \, \log \left (a x + 1\right )}{a^{7}} + \frac {15 \, \log \left (a x - 1\right )}{a^{7}}\right )} \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (41) = 82\).

Time = 0.27 (sec) , antiderivative size = 245, normalized size of antiderivative = 4.80 \[ \int x^5 \coth ^{-1}(a x) \, dx=\frac {1}{45} \, a {\left (\frac {\frac {45 \, {\left (a x + 1\right )}^{4}}{{\left (a x - 1\right )}^{4}} - \frac {90 \, {\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} + \frac {140 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} - \frac {70 \, {\left (a x + 1\right )}}{a x - 1} + 23}{a^{7} {\left (\frac {a x + 1}{a x - 1} - 1\right )}^{5}} + \frac {15 \, {\left (\frac {3 \, {\left (a x + 1\right )}^{5}}{{\left (a x - 1\right )}^{5}} + \frac {10 \, {\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} + \frac {3 \, {\left (a x + 1\right )}}{a x - 1}\right )} \log \left (-\frac {\frac {\frac {{\left (a x + 1\right )} a}{a x - 1} - a}{a {\left (\frac {a x + 1}{a x - 1} + 1\right )}} + 1}{\frac {\frac {{\left (a x + 1\right )} a}{a x - 1} - a}{a {\left (\frac {a x + 1}{a x - 1} + 1\right )}} - 1}\right )}{a^{7} {\left (\frac {a x + 1}{a x - 1} - 1\right )}^{6}}\right )} \]

[In]

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

[Out]

1/45*a*((45*(a*x + 1)^4/(a*x - 1)^4 - 90*(a*x + 1)^3/(a*x - 1)^3 + 140*(a*x + 1)^2/(a*x - 1)^2 - 70*(a*x + 1)/
(a*x - 1) + 23)/(a^7*((a*x + 1)/(a*x - 1) - 1)^5) + 15*(3*(a*x + 1)^5/(a*x - 1)^5 + 10*(a*x + 1)^3/(a*x - 1)^3
+ 3*(a*x + 1)/(a*x - 1))*log(-(((a*x + 1)*a/(a*x - 1) - a)/(a*((a*x + 1)/(a*x - 1) + 1)) + 1)/(((a*x + 1)*a/(
a*x - 1) - a)/(a*((a*x + 1)/(a*x - 1) + 1)) - 1))/(a^7*((a*x + 1)/(a*x - 1) - 1)^6))

Mupad [B] (verification not implemented)

Time = 4.49 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.80 \[ \int x^5 \coth ^{-1}(a x) \, dx=\frac {\frac {a\,x}{6}-\frac {\mathrm {acoth}\left (a\,x\right )}{6}+\frac {a^3\,x^3}{18}+\frac {a^5\,x^5}{30}}{a^6}+\frac {x^6\,\mathrm {acoth}\left (a\,x\right )}{6} \]

[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