Integrand size = 10, antiderivative size = 127 \[ \int x^4 \coth ^{-1}(a x)^2 \, dx=\frac {3 x}{10 a^4}+\frac {x^3}{30 a^2}+\frac {x^2 \coth ^{-1}(a x)}{5 a^3}+\frac {x^4 \coth ^{-1}(a x)}{10 a}+\frac {\coth ^{-1}(a x)^2}{5 a^5}+\frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {3 \text {arctanh}(a x)}{10 a^5}-\frac {2 \coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{5 a^5}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{5 a^5} \] Output:
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*arccoth(a*x)^2-3/10*arctanh(a*x)/a^5-2/5*arc coth(a*x)*ln(2/(-a*x+1))/a^5-1/5*polylog(2,1-2/(-a*x+1))/a^5
Time = 0.30 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.69 \[ \int x^4 \coth ^{-1}(a x)^2 \, dx=\frac {a x \left (9+a^2 x^2\right )+6 \left (-1+a^5 x^5\right ) \coth ^{-1}(a x)^2+3 \coth ^{-1}(a x) \left (-3+2 a^2 x^2+a^4 x^4-4 \log \left (1-e^{-2 \coth ^{-1}(a x)}\right )\right )+6 \operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(a x)}\right )}{30 a^5} \] Input:
Integrate[x^4*ArcCoth[a*x]^2,x]
Output:
(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)
Time = 1.13 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.35, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.300, Rules used = {6453, 6543, 6453, 254, 2009, 6543, 6453, 262, 219, 6547, 6471, 2849, 2752}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^4 \coth ^{-1}(a x)^2 \, dx\) |
| \(\Big \downarrow \) 6453 |
| \(\displaystyle \frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {2}{5} a \int \frac {x^5 \coth ^{-1}(a x)}{1-a^2 x^2}dx\) |
| \(\Big \downarrow \) 6543 |
| \(\displaystyle \frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {2}{5} a \left (\frac {\int \frac {x^3 \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\int x^3 \coth ^{-1}(a x)dx}{a^2}\right )\) |
| \(\Big \downarrow \) 6453 |
| \(\displaystyle \frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {2}{5} a \left (\frac {\int \frac {x^3 \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{4} x^4 \coth ^{-1}(a x)-\frac {1}{4} a \int \frac {x^4}{1-a^2 x^2}dx}{a^2}\right )\) |
| \(\Big \downarrow \) 254 |
| \(\displaystyle \frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {2}{5} a \left (\frac {\int \frac {x^3 \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{4} x^4 \coth ^{-1}(a x)-\frac {1}{4} a \int \left (-\frac {x^2}{a^2}+\frac {1}{a^4 \left (1-a^2 x^2\right )}-\frac {1}{a^4}\right )dx}{a^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {2}{5} a \left (\frac {\int \frac {x^3 \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{4} x^4 \coth ^{-1}(a x)-\frac {1}{4} a \left (\frac {\text {arctanh}(a x)}{a^5}-\frac {x}{a^4}-\frac {x^3}{3 a^2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 6543 |
| \(\displaystyle \frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {2}{5} a \left (\frac {\frac {\int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\int x \coth ^{-1}(a x)dx}{a^2}}{a^2}-\frac {\frac {1}{4} x^4 \coth ^{-1}(a x)-\frac {1}{4} a \left (\frac {\text {arctanh}(a x)}{a^5}-\frac {x}{a^4}-\frac {x^3}{3 a^2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 6453 |
| \(\displaystyle \frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {2}{5} a \left (\frac {\frac {\int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)-\frac {1}{2} a \int \frac {x^2}{1-a^2 x^2}dx}{a^2}}{a^2}-\frac {\frac {1}{4} x^4 \coth ^{-1}(a x)-\frac {1}{4} a \left (\frac {\text {arctanh}(a x)}{a^5}-\frac {x}{a^4}-\frac {x^3}{3 a^2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {2}{5} a \left (\frac {\frac {\int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)-\frac {1}{2} a \left (\frac {\int \frac {1}{1-a^2 x^2}dx}{a^2}-\frac {x}{a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{4} x^4 \coth ^{-1}(a x)-\frac {1}{4} a \left (\frac {\text {arctanh}(a x)}{a^5}-\frac {x}{a^4}-\frac {x^3}{3 a^2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {2}{5} a \left (\frac {\frac {\int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)-\frac {1}{2} a \left (\frac {\text {arctanh}(a x)}{a^3}-\frac {x}{a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{4} x^4 \coth ^{-1}(a x)-\frac {1}{4} a \left (\frac {\text {arctanh}(a x)}{a^5}-\frac {x}{a^4}-\frac {x^3}{3 a^2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 6547 |
| \(\displaystyle \frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {2}{5} a \left (\frac {\frac {\frac {\int \frac {\coth ^{-1}(a x)}{1-a x}dx}{a}-\frac {\coth ^{-1}(a x)^2}{2 a^2}}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)-\frac {1}{2} a \left (\frac {\text {arctanh}(a x)}{a^3}-\frac {x}{a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{4} x^4 \coth ^{-1}(a x)-\frac {1}{4} a \left (\frac {\text {arctanh}(a x)}{a^5}-\frac {x}{a^4}-\frac {x^3}{3 a^2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 6471 |
| \(\displaystyle \frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {2}{5} a \left (\frac {\frac {\frac {\frac {\log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{a}-\int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2}dx}{a}-\frac {\coth ^{-1}(a x)^2}{2 a^2}}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)-\frac {1}{2} a \left (\frac {\text {arctanh}(a x)}{a^3}-\frac {x}{a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{4} x^4 \coth ^{-1}(a x)-\frac {1}{4} a \left (\frac {\text {arctanh}(a x)}{a^5}-\frac {x}{a^4}-\frac {x^3}{3 a^2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle \frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {2}{5} a \left (\frac {\frac {\frac {\frac {\int \frac {\log \left (\frac {2}{1-a x}\right )}{1-\frac {2}{1-a x}}d\frac {1}{1-a x}}{a}+\frac {\log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{a}}{a}-\frac {\coth ^{-1}(a x)^2}{2 a^2}}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)-\frac {1}{2} a \left (\frac {\text {arctanh}(a x)}{a^3}-\frac {x}{a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{4} x^4 \coth ^{-1}(a x)-\frac {1}{4} a \left (\frac {\text {arctanh}(a x)}{a^5}-\frac {x}{a^4}-\frac {x^3}{3 a^2}\right )}{a^2}\right )\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle \frac {1}{5} x^5 \coth ^{-1}(a x)^2-\frac {2}{5} a \left (\frac {\frac {\frac {\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a}+\frac {\log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{a}}{a}-\frac {\coth ^{-1}(a x)^2}{2 a^2}}{a^2}-\frac {\frac {1}{2} x^2 \coth ^{-1}(a x)-\frac {1}{2} a \left (\frac {\text {arctanh}(a x)}{a^3}-\frac {x}{a^2}\right )}{a^2}}{a^2}-\frac {\frac {1}{4} x^4 \coth ^{-1}(a x)-\frac {1}{4} a \left (\frac {\text {arctanh}(a x)}{a^5}-\frac {x}{a^4}-\frac {x^3}{3 a^2}\right )}{a^2}\right )\) |
Input:
Int[x^4*ArcCoth[a*x]^2,x]
Output:
(x^5*ArcCoth[a*x]^2)/5 - (2*a*(-(((x^4*ArcCoth[a*x])/4 - (a*(-(x/a^4) - x^ 3/(3*a^2) + ArcTanh[a*x]/a^5))/4)/a^2) + (-(((x^2*ArcCoth[a*x])/2 - (a*(-( x/a^2) + ArcTanh[a*x]/a^3))/2)/a^2) + (-1/2*ArcCoth[a*x]^2/a^2 + ((ArcCoth [a*x]*Log[2/(1 - a*x)])/a + PolyLog[2, 1 - 2/(1 - a*x)]/(2*a))/a)/a^2)/a^2 ))/5
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^2, x], x] /; FreeQ[{a, b}, x] && IGtQ[m, 3]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp [-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]
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] - Simp[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]
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] + Simp[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]
Int[(((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + ( e_.)*(x_)^2), x_Symbol] :> Simp[f^2/e Int[(f*x)^(m - 2)*(a + b*ArcCoth[c* x])^p, x], x] - Simp[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]
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] + Simp[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]
Time = 0.32 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.29
| method | result | size |
| parts | \(\frac {x^{5} \operatorname {arccoth}\left (x a \right )^{2}}{5}+\frac {\frac {x^{4} a^{4} \operatorname {arccoth}\left (x a \right )}{10}+\frac {\operatorname {arccoth}\left (x a \right ) a^{2} x^{2}}{5}+\frac {\operatorname {arccoth}\left (x a \right ) \ln \left (x a -1\right )}{5}+\frac {\operatorname {arccoth}\left (x a \right ) \ln \left (x a +1\right )}{5}+\frac {x^{3} a^{3}}{30}+\frac {3 x a}{10}+\frac {3 \ln \left (x a -1\right )}{20}-\frac {3 \ln \left (x a +1\right )}{20}-\frac {\operatorname {dilog}\left (\frac {x a}{2}+\frac {1}{2}\right )}{5}-\frac {\ln \left (x a -1\right ) \ln \left (\frac {x a}{2}+\frac {1}{2}\right )}{10}+\frac {\ln \left (x a -1\right )^{2}}{20}-\frac {\ln \left (x a +1\right )^{2}}{20}+\frac {\left (\ln \left (x a +1\right )-\ln \left (\frac {x a}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {x a}{2}+\frac {1}{2}\right )}{10}}{a^{5}}\) | \(164\) |
| derivativedivides | \(\frac {\frac {x^{5} a^{5} \operatorname {arccoth}\left (x a \right )^{2}}{5}+\frac {x^{4} a^{4} \operatorname {arccoth}\left (x a \right )}{10}+\frac {\operatorname {arccoth}\left (x a \right ) a^{2} x^{2}}{5}+\frac {\operatorname {arccoth}\left (x a \right ) \ln \left (x a -1\right )}{5}+\frac {\operatorname {arccoth}\left (x a \right ) \ln \left (x a +1\right )}{5}+\frac {x^{3} a^{3}}{30}+\frac {3 x a}{10}+\frac {3 \ln \left (x a -1\right )}{20}-\frac {3 \ln \left (x a +1\right )}{20}-\frac {\operatorname {dilog}\left (\frac {x a}{2}+\frac {1}{2}\right )}{5}-\frac {\ln \left (x a -1\right ) \ln \left (\frac {x a}{2}+\frac {1}{2}\right )}{10}+\frac {\ln \left (x a -1\right )^{2}}{20}-\frac {\ln \left (x a +1\right )^{2}}{20}+\frac {\left (\ln \left (x a +1\right )-\ln \left (\frac {x a}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {x a}{2}+\frac {1}{2}\right )}{10}}{a^{5}}\) | \(165\) |
| default | \(\frac {\frac {x^{5} a^{5} \operatorname {arccoth}\left (x a \right )^{2}}{5}+\frac {x^{4} a^{4} \operatorname {arccoth}\left (x a \right )}{10}+\frac {\operatorname {arccoth}\left (x a \right ) a^{2} x^{2}}{5}+\frac {\operatorname {arccoth}\left (x a \right ) \ln \left (x a -1\right )}{5}+\frac {\operatorname {arccoth}\left (x a \right ) \ln \left (x a +1\right )}{5}+\frac {x^{3} a^{3}}{30}+\frac {3 x a}{10}+\frac {3 \ln \left (x a -1\right )}{20}-\frac {3 \ln \left (x a +1\right )}{20}-\frac {\operatorname {dilog}\left (\frac {x a}{2}+\frac {1}{2}\right )}{5}-\frac {\ln \left (x a -1\right ) \ln \left (\frac {x a}{2}+\frac {1}{2}\right )}{10}+\frac {\ln \left (x a -1\right )^{2}}{20}-\frac {\ln \left (x a +1\right )^{2}}{20}+\frac {\left (\ln \left (x a +1\right )-\ln \left (\frac {x a}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {x a}{2}+\frac {1}{2}\right )}{10}}{a^{5}}\) | \(165\) |
| risch | \(\frac {413}{2250 a^{5}}-\frac {x^{5} \ln \left (x a +1\right )}{50}+\frac {\ln \left (x a +1\right ) x^{4}}{40 a}-\frac {\ln \left (x a +1\right ) x^{3}}{30 a^{2}}+\frac {\ln \left (x a +1\right ) x^{2}}{20 a^{3}}-\frac {\ln \left (x a +1\right ) x}{10 a^{4}}-\frac {\ln \left (x a -1\right ) x^{4}}{40 a}-\frac {\ln \left (x a -1\right ) x^{3}}{30 a^{2}}-\frac {\ln \left (x a -1\right ) x^{2}}{20 a^{3}}-\frac {\ln \left (x a -1\right ) x}{10 a^{4}}+\frac {\ln \left (x a +1\right )^{2} x^{5}}{20}+\frac {\ln \left (x a +1\right )^{2}}{20 a^{5}}+\frac {\ln \left (x a -1\right )^{2} x^{5}}{20}-\frac {\ln \left (x a -1\right ) x^{5}}{50}-\frac {\ln \left (x a -1\right )^{2}}{20 a^{5}}+\frac {47 \ln \left (x a +1\right )}{600 a^{5}}+\frac {137 \ln \left (x a -1\right )}{600 a^{5}}+\frac {\ln \left (x a -1\right ) \left (x a -1\right )}{10 a^{5}}-\frac {\ln \left (x a -1\right ) \ln \left (\frac {x a}{2}+\frac {1}{2}\right )}{5 a^{5}}+\frac {3 \left (x a -1\right )^{4} \ln \left (x a -1\right )}{40 a^{5}}+\frac {\ln \left (x a -1\right ) \left (x a -1\right )^{2}}{10 a^{5}}+\frac {\left (x a -1\right )^{5} \ln \left (x a -1\right )}{50 a^{5}}+\frac {2 \left (x a -1\right )^{3} \ln \left (x a -1\right )}{15 a^{5}}-\frac {\left (\left (-\frac {1}{25}+\frac {\ln \left (x a -1\right )}{5}\right ) \left (x a -1\right )^{5}+\left (-\frac {1}{4}+\ln \left (x a -1\right )\right ) \left (x a -1\right )^{4}+\left (-\frac {2}{3}+2 \ln \left (x a -1\right )\right ) \left (x a -1\right )^{3}+\left (-1+2 \ln \left (x a -1\right )\right ) \left (x a -1\right )^{2}+\left (-1+\ln \left (x a -1\right )\right ) \left (x a -1\right )\right ) \ln \left (x a +1\right )}{2 a^{5}}-\frac {\operatorname {dilog}\left (\frac {x a}{2}+\frac {1}{2}\right )}{5 a^{5}}+\frac {3 x}{10 a^{4}}+\frac {x^{3}}{30 a^{2}}\) | \(439\) |
Input:
int(x^4*arccoth(x*a)^2,x,method=_RETURNVERBOSE)
Output:
1/5*x^5*arccoth(x*a)^2+2/5/a^5*(1/4*x^4*a^4*arccoth(x*a)+1/2*arccoth(x*a)* a^2*x^2+1/2*arccoth(x*a)*ln(a*x-1)+1/2*arccoth(x*a)*ln(a*x+1)+1/12*x^3*a^3 +3/4*x*a+3/8*ln(a*x-1)-3/8*ln(a*x+1)-1/2*dilog(1/2*x*a+1/2)-1/4*ln(a*x-1)* ln(1/2*x*a+1/2)+1/8*ln(a*x-1)^2-1/8*ln(a*x+1)^2+1/4*(ln(a*x+1)-ln(1/2*x*a+ 1/2))*ln(-1/2*x*a+1/2))
\[ \int x^4 \coth ^{-1}(a x)^2 \, dx=\int { x^{4} \operatorname {arcoth}\left (a x\right )^{2} \,d x } \] Input:
integrate(x^4*arccoth(a*x)^2,x, algorithm="fricas")
Output:
integral(x^4*arccoth(a*x)^2, x)
\[ \int x^4 \coth ^{-1}(a x)^2 \, dx=\int x^{4} \operatorname {acoth}^{2}{\left (a x \right )}\, dx \] Input:
integrate(x**4*acoth(a*x)**2,x)
Output:
Integral(x**4*acoth(a*x)**2, x)
Time = 0.03 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.22 \[ \int x^4 \coth ^{-1}(a x)^2 \, dx=\frac {1}{5} \, x^{5} \operatorname {arcoth}\left (a x\right )^{2} + \frac {1}{60} \, a^{2} {\left (\frac {2 \, a^{3} x^{3} + 18 \, a x - 3 \, \log \left (a x + 1\right )^{2} + 6 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) + 3 \, \log \left (a x - 1\right )^{2} + 9 \, \log \left (a x - 1\right )}{a^{7}} - \frac {12 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a^{7}} - \frac {9 \, \log \left (a x + 1\right )}{a^{7}}\right )} + \frac {1}{10} \, a {\left (\frac {a^{2} x^{4} + 2 \, x^{2}}{a^{4}} + \frac {2 \, \log \left (a^{2} x^{2} - 1\right )}{a^{6}}\right )} \operatorname {arcoth}\left (a x\right ) \] Input:
integrate(x^4*arccoth(a*x)^2,x, algorithm="maxima")
Output:
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*log(a*x - 1)^2 + 9*log(a*x - 1))/a^7 - 1 2*(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)*arcc oth(a*x)
\[ \int x^4 \coth ^{-1}(a x)^2 \, dx=\int { x^{4} \operatorname {arcoth}\left (a x\right )^{2} \,d x } \] Input:
integrate(x^4*arccoth(a*x)^2,x, algorithm="giac")
Output:
integrate(x^4*arccoth(a*x)^2, x)
Timed out. \[ \int x^4 \coth ^{-1}(a x)^2 \, dx=\int x^4\,{\mathrm {acoth}\left (a\,x\right )}^2 \,d x \] Input:
int(x^4*acoth(a*x)^2,x)
Output:
int(x^4*acoth(a*x)^2, x)
\[ \int x^4 \coth ^{-1}(a x)^2 \, dx=\frac {6 \mathit {acoth} \left (a x \right )^{2} a^{5} x^{5}-6 \mathit {acoth} \left (a x \right )^{2} a x -3 \mathit {acoth} \left (a x \right ) a^{4} x^{4}-6 \mathit {acoth} \left (a x \right ) a^{2} x^{2}+9 \mathit {acoth} \left (a x \right )+6 \left (\int \mathit {acoth} \left (a x \right )^{2}d x \right ) a +a^{3} x^{3}+9 a x}{30 a^{5}} \] Input:
int(x^4*acoth(a*x)^2,x)
Output:
(6*acoth(a*x)**2*a**5*x**5 - 6*acoth(a*x)**2*a*x - 3*acoth(a*x)*a**4*x**4 - 6*acoth(a*x)*a**2*x**2 + 9*acoth(a*x) + 6*int(acoth(a*x)**2,x)*a + a**3* x**3 + 9*a*x)/(30*a**5)