\(\int x^4 \coth ^{-1}(a x)^2 \, dx\) [13]

Optimal result

Integrand size = 10, antiderivative size = 127 \[ \int x^4 \coth ^{-1}(a x)^2 \, dx=\frac {3 x}{10 a^4}+\frac {x^3}{30 a^2}+\frac {x^2 \coth ^{-1}(a x)}{5 a^3}+\frac {x^4 \coth ^{-1}(a x)}{10 a}+\frac {\coth ^{-1}(a x)^2}{5 a^5}+\frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {3 \text {arctanh}(a x)}{10 a^5}-\frac {2 \coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{5 a^5}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{5 a^5} \]

[Out]

3/10*x/a^4+1/30*x^3/a^2+1/5*x^2*arccoth(a*x)/a^3+1/10*x^4*arccoth(a*x)/a+1/5*arccoth(a*x)^2/a^5+1/5*x^5*arccot
h(a*x)^2-3/10*arctanh(a*x)/a^5-2/5*arccoth(a*x)*ln(2/(-a*x+1))/a^5-1/5*polylog(2,1-2/(-a*x+1))/a^5

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {6038, 6128, 308, 212, 327, 6132, 6056, 2449, 2352} \[ \int x^4 \coth ^{-1}(a x)^2 \, dx=-\frac {3 \text {arctanh}(a x)}{10 a^5}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{5 a^5}+\frac {\coth ^{-1}(a x)^2}{5 a^5}-\frac {2 \log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{5 a^5}+\frac {3 x}{10 a^4}+\frac {x^2 \coth ^{-1}(a x)}{5 a^3}+\frac {x^3}{30 a^2}+\frac {1}{5} x^5 \coth ^{-1}(a x)^2+\frac {x^4 \coth ^{-1}(a x)}{10 a} \]

[In]

Int[x^4*ArcCoth[a*x]^2,x]

[Out]

(3*x)/(10*a^4) + x^3/(30*a^2) + (x^2*ArcCoth[a*x])/(5*a^3) + (x^4*ArcCoth[a*x])/(10*a) + ArcCoth[a*x]^2/(5*a^5
) + (x^5*ArcCoth[a*x]^2)/5 - (3*ArcTanh[a*x])/(10*a^5) - (2*ArcCoth[a*x]*Log[2/(1 - a*x)])/(5*a^5) - PolyLog[2
, 1 - 2/(1 - a*x)]/(5*a^5)

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 327

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

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

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
]

Rule 6056

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

Rule 6128

Int[(((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[f^2
/e, Int[(f*x)^(m - 2)*(a + b*ArcCoth[c*x])^p, x], x] - Dist[d*(f^2/e), Int[(f*x)^(m - 2)*((a + b*ArcCoth[c*x])
^p/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]

Rule 6132

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {1}{5} (2 a) \int \frac {x^5 \coth ^{-1}(a x)}{1-a^2 x^2} \, dx \\ & = \frac {1}{5} x^5 \coth ^{-1}(a x)^2+\frac {2 \int x^3 \coth ^{-1}(a x) \, dx}{5 a}-\frac {2 \int \frac {x^3 \coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{5 a} \\ & = \frac {x^4 \coth ^{-1}(a x)}{10 a}+\frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {1}{10} \int \frac {x^4}{1-a^2 x^2} \, dx+\frac {2 \int x \coth ^{-1}(a x) \, dx}{5 a^3}-\frac {2 \int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{5 a^3} \\ & = \frac {x^2 \coth ^{-1}(a x)}{5 a^3}+\frac {x^4 \coth ^{-1}(a x)}{10 a}+\frac {\coth ^{-1}(a x)^2}{5 a^5}+\frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {1}{10} \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 {2 \int \frac {\coth ^{-1}(a x)}{1-a x} \, dx}{5 a^4}-\frac {\int \frac {x^2}{1-a^2 x^2} \, dx}{5 a^2} \\ & = \frac {3 x}{10 a^4}+\frac {x^3}{30 a^2}+\frac {x^2 \coth ^{-1}(a x)}{5 a^3}+\frac {x^4 \coth ^{-1}(a x)}{10 a}+\frac {\coth ^{-1}(a x)^2}{5 a^5}+\frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {2 \coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{5 a^5}-\frac {\int \frac {1}{1-a^2 x^2} \, dx}{10 a^4}-\frac {\int \frac {1}{1-a^2 x^2} \, dx}{5 a^4}+\frac {2 \int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{5 a^4} \\ & = \frac {3 x}{10 a^4}+\frac {x^3}{30 a^2}+\frac {x^2 \coth ^{-1}(a x)}{5 a^3}+\frac {x^4 \coth ^{-1}(a x)}{10 a}+\frac {\coth ^{-1}(a x)^2}{5 a^5}+\frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {3 \text {arctanh}(a x)}{10 a^5}-\frac {2 \coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{5 a^5}-\frac {2 \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a x}\right )}{5 a^5} \\ & = \frac {3 x}{10 a^4}+\frac {x^3}{30 a^2}+\frac {x^2 \coth ^{-1}(a x)}{5 a^3}+\frac {x^4 \coth ^{-1}(a x)}{10 a}+\frac {\coth ^{-1}(a x)^2}{5 a^5}+\frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {3 \text {arctanh}(a x)}{10 a^5}-\frac {2 \coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{5 a^5}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{5 a^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.69 \[ \int x^4 \coth ^{-1}(a x)^2 \, dx=\frac {a x \left (9+a^2 x^2\right )+6 \left (-1+a^5 x^5\right ) \coth ^{-1}(a x)^2+3 \coth ^{-1}(a x) \left (-3+2 a^2 x^2+a^4 x^4-4 \log \left (1-e^{-2 \coth ^{-1}(a x)}\right )\right )+6 \operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(a x)}\right )}{30 a^5} \]

[In]

Integrate[x^4*ArcCoth[a*x]^2,x]

[Out]

(a*x*(9 + a^2*x^2) + 6*(-1 + a^5*x^5)*ArcCoth[a*x]^2 + 3*ArcCoth[a*x]*(-3 + 2*a^2*x^2 + a^4*x^4 - 4*Log[1 - E^
(-2*ArcCoth[a*x])]) + 6*PolyLog[2, E^(-2*ArcCoth[a*x])])/(30*a^5)

Maple [A] (verified)

Time = 0.47 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.29

method	result	size
parts	\(\frac {x^{5} \operatorname {arccoth}\left (a x \right )^{2}}{5}+\frac {\frac {a^{4} x^{4} \operatorname {arccoth}\left (a x \right )}{10}+\frac {\operatorname {arccoth}\left (a x \right ) a^{2} x^{2}}{5}+\frac {\operatorname {arccoth}\left (a x \right ) \ln \left (a x -1\right )}{5}+\frac {\operatorname {arccoth}\left (a x \right ) \ln \left (a x +1\right )}{5}+\frac {a^{3} x^{3}}{30}+\frac {3 a x}{10}+\frac {3 \ln \left (a x -1\right )}{20}-\frac {3 \ln \left (a x +1\right )}{20}-\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{5}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{10}+\frac {\ln \left (a x -1\right )^{2}}{20}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{10}-\frac {\ln \left (a x +1\right )^{2}}{20}}{a^{5}}\)	\(164\)
derivativedivides	\(\frac {\frac {a^{5} x^{5} \operatorname {arccoth}\left (a x \right )^{2}}{5}+\frac {a^{4} x^{4} \operatorname {arccoth}\left (a x \right )}{10}+\frac {\operatorname {arccoth}\left (a x \right ) a^{2} x^{2}}{5}+\frac {\operatorname {arccoth}\left (a x \right ) \ln \left (a x -1\right )}{5}+\frac {\operatorname {arccoth}\left (a x \right ) \ln \left (a x +1\right )}{5}+\frac {a^{3} x^{3}}{30}+\frac {3 a x}{10}+\frac {3 \ln \left (a x -1\right )}{20}-\frac {3 \ln \left (a x +1\right )}{20}-\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{5}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{10}+\frac {\ln \left (a x -1\right )^{2}}{20}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{10}-\frac {\ln \left (a x +1\right )^{2}}{20}}{a^{5}}\)	\(165\)
default	\(\frac {\frac {a^{5} x^{5} \operatorname {arccoth}\left (a x \right )^{2}}{5}+\frac {a^{4} x^{4} \operatorname {arccoth}\left (a x \right )}{10}+\frac {\operatorname {arccoth}\left (a x \right ) a^{2} x^{2}}{5}+\frac {\operatorname {arccoth}\left (a x \right ) \ln \left (a x -1\right )}{5}+\frac {\operatorname {arccoth}\left (a x \right ) \ln \left (a x +1\right )}{5}+\frac {a^{3} x^{3}}{30}+\frac {3 a x}{10}+\frac {3 \ln \left (a x -1\right )}{20}-\frac {3 \ln \left (a x +1\right )}{20}-\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{5}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{10}+\frac {\ln \left (a x -1\right )^{2}}{20}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{10}-\frac {\ln \left (a x +1\right )^{2}}{20}}{a^{5}}\)	\(165\)
risch	\(\frac {413}{2250 a^{5}}+\frac {x^{3}}{30 a^{2}}+\frac {47 \ln \left (a x +1\right )}{600 a^{5}}+\frac {\ln \left (a x +1\right ) x^{4}}{40 a}-\frac {\ln \left (a x +1\right ) x^{3}}{30 a^{2}}+\frac {\ln \left (a x +1\right ) x^{2}}{20 a^{3}}-\frac {\ln \left (a x +1\right ) x}{10 a^{4}}-\frac {\ln \left (a x -1\right ) x^{4}}{40 a}-\frac {\ln \left (a x -1\right ) x^{3}}{30 a^{2}}-\frac {\ln \left (a x -1\right ) x^{2}}{20 a^{3}}-\frac {\ln \left (a x -1\right ) x}{10 a^{4}}+\frac {3 x}{10 a^{4}}+\frac {137 \ln \left (a x -1\right )}{600 a^{5}}+\frac {\ln \left (a x +1\right )^{2} x^{5}}{20}+\frac {\ln \left (a x +1\right )^{2}}{20 a^{5}}+\frac {\ln \left (a x -1\right )^{2} x^{5}}{20}-\frac {\ln \left (a x -1\right ) x^{5}}{50}-\frac {\ln \left (a x -1\right )^{2}}{20 a^{5}}-\frac {x^{5} \ln \left (a x +1\right )}{50}-\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{5 a^{5}}+\frac {2 \left (a x -1\right )^{3} \ln \left (a x -1\right )}{15 a^{5}}-\frac {\left (\left (-\frac {1}{25}+\frac {\ln \left (a x -1\right )}{5}\right ) \left (a x -1\right )^{5}+\left (-\frac {1}{4}+\ln \left (a x -1\right )\right ) \left (a x -1\right )^{4}+\left (-\frac {2}{3}+2 \ln \left (a x -1\right )\right ) \left (a x -1\right )^{3}+\left (-1+2 \ln \left (a x -1\right )\right ) \left (a x -1\right )^{2}+\left (-1+\ln \left (a x -1\right )\right ) \left (a x -1\right )\right ) \ln \left (a x +1\right )}{2 a^{5}}+\frac {\ln \left (a x -1\right ) \left (a x -1\right )}{10 a^{5}}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{5 a^{5}}+\frac {3 \left (a x -1\right )^{4} \ln \left (a x -1\right )}{40 a^{5}}+\frac {\ln \left (a x -1\right ) \left (a x -1\right )^{2}}{10 a^{5}}+\frac {\left (a x -1\right )^{5} \ln \left (a x -1\right )}{50 a^{5}}\)	\(439\)

[In]

int(x^4*arccoth(a*x)^2,x,method=_RETURNVERBOSE)

[Out]

1/5*x^5*arccoth(a*x)^2+2/5/a^5*(1/4*a^4*x^4*arccoth(a*x)+1/2*arccoth(a*x)*a^2*x^2+1/2*arccoth(a*x)*ln(a*x-1)+1
/2*arccoth(a*x)*ln(a*x+1)+1/12*a^3*x^3+3/4*a*x+3/8*ln(a*x-1)-3/8*ln(a*x+1)-1/2*dilog(1/2*a*x+1/2)-1/4*ln(a*x-1
)*ln(1/2*a*x+1/2)+1/8*ln(a*x-1)^2+1/4*(ln(a*x+1)-ln(1/2*a*x+1/2))*ln(-1/2*a*x+1/2)-1/8*ln(a*x+1)^2)

Fricas [F]

\[ \int x^4 \coth ^{-1}(a x)^2 \, dx=\int { x^{4} \operatorname {arcoth}\left (a x\right )^{2} \,d x } \]

[In]

integrate(x^4*arccoth(a*x)^2,x, algorithm="fricas")

[Out]

integral(x^4*arccoth(a*x)^2, x)

Sympy [F]

\[ \int x^4 \coth ^{-1}(a x)^2 \, dx=\int x^{4} \operatorname {acoth}^{2}{\left (a x \right )}\, dx \]

[In]

integrate(x**4*acoth(a*x)**2,x)

[Out]

Integral(x**4*acoth(a*x)**2, x)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.22 \[ \int x^4 \coth ^{-1}(a x)^2 \, dx=\frac {1}{5} \, x^{5} \operatorname {arcoth}\left (a x\right )^{2} + \frac {1}{60} \, a^{2} {\left (\frac {2 \, a^{3} x^{3} + 18 \, a x - 3 \, \log \left (a x + 1\right )^{2} + 6 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) + 3 \, \log \left (a x - 1\right )^{2} + 9 \, \log \left (a x - 1\right )}{a^{7}} - \frac {12 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a^{7}} - \frac {9 \, \log \left (a x + 1\right )}{a^{7}}\right )} + \frac {1}{10} \, a {\left (\frac {a^{2} x^{4} + 2 \, x^{2}}{a^{4}} + \frac {2 \, \log \left (a^{2} x^{2} - 1\right )}{a^{6}}\right )} \operatorname {arcoth}\left (a x\right ) \]

[In]

integrate(x^4*arccoth(a*x)^2,x, algorithm="maxima")

[Out]

1/5*x^5*arccoth(a*x)^2 + 1/60*a^2*((2*a^3*x^3 + 18*a*x - 3*log(a*x + 1)^2 + 6*log(a*x + 1)*log(a*x - 1) + 3*lo
g(a*x - 1)^2 + 9*log(a*x - 1))/a^7 - 12*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2))/a^7 - 9*log(
a*x + 1)/a^7) + 1/10*a*((a^2*x^4 + 2*x^2)/a^4 + 2*log(a^2*x^2 - 1)/a^6)*arccoth(a*x)

Giac [F]

\[ \int x^4 \coth ^{-1}(a x)^2 \, dx=\int { x^{4} \operatorname {arcoth}\left (a x\right )^{2} \,d x } \]

[In]

integrate(x^4*arccoth(a*x)^2,x, algorithm="giac")

[Out]

integrate(x^4*arccoth(a*x)^2, x)

Mupad [F(-1)]

Timed out. \[ \int x^4 \coth ^{-1}(a x)^2 \, dx=\int x^4\,{\mathrm {acoth}\left (a\,x\right )}^2 \,d x \]

[In]

int(x^4*acoth(a*x)^2,x)

[Out]

int(x^4*acoth(a*x)^2, x)