Integrand size = 10, antiderivative size = 154 \[ \int \frac {\coth ^{-1}(a x)^3}{x^4} \, dx=-\frac {a^2 \coth ^{-1}(a x)}{x}+\frac {1}{2} a^3 \coth ^{-1}(a x)^2-\frac {a \coth ^{-1}(a x)^2}{2 x^2}+\frac {1}{3} a^3 \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^3}{3 x^3}+a^3 \log (x)-\frac {1}{2} a^3 \log \left (1-a^2 x^2\right )+a^3 \coth ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-a^3 \coth ^{-1}(a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )-\frac {1}{2} a^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+a x}\right ) \]
-a^2*arccoth(a*x)/x+1/2*a^3*arccoth(a*x)^2-1/2*a*arccoth(a*x)^2/x^2+1/3*a^ 3*arccoth(a*x)^3-1/3*arccoth(a*x)^3/x^3+a^3*ln(x)-1/2*a^3*ln(-a^2*x^2+1)+a ^3*arccoth(a*x)^2*ln(2-2/(a*x+1))-a^3*arccoth(a*x)*polylog(2,-1+2/(a*x+1)) -1/2*a^3*polylog(3,-1+2/(a*x+1))
Time = 0.16 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.92 \[ \int \frac {\coth ^{-1}(a x)^3}{x^4} \, dx=\frac {1}{6} \left (-\frac {6 a^2 \coth ^{-1}(a x)}{x}+3 a^3 \coth ^{-1}(a x)^2-\frac {3 a \coth ^{-1}(a x)^2}{x^2}+2 a^3 \coth ^{-1}(a x)^3-\frac {2 \coth ^{-1}(a x)^3}{x^3}+6 a^3 \coth ^{-1}(a x)^2 \log \left (1+e^{-2 \coth ^{-1}(a x)}\right )+6 a^3 \log \left (\frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )-6 a^3 \coth ^{-1}(a x) \operatorname {PolyLog}\left (2,-e^{-2 \coth ^{-1}(a x)}\right )-3 a^3 \operatorname {PolyLog}\left (3,-e^{-2 \coth ^{-1}(a x)}\right )\right ) \]
((-6*a^2*ArcCoth[a*x])/x + 3*a^3*ArcCoth[a*x]^2 - (3*a*ArcCoth[a*x]^2)/x^2 + 2*a^3*ArcCoth[a*x]^3 - (2*ArcCoth[a*x]^3)/x^3 + 6*a^3*ArcCoth[a*x]^2*Lo g[1 + E^(-2*ArcCoth[a*x])] + 6*a^3*Log[1/Sqrt[1 - 1/(a^2*x^2)]] - 6*a^3*Ar cCoth[a*x]*PolyLog[2, -E^(-2*ArcCoth[a*x])] - 3*a^3*PolyLog[3, -E^(-2*ArcC oth[a*x])])/6
Time = 1.51 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.03, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.400, Rules used = {6453, 6545, 6453, 6545, 6453, 243, 47, 14, 16, 6511, 6551, 6495, 6619, 7164}
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.
\(\displaystyle \int \frac {\coth ^{-1}(a x)^3}{x^4} \, dx\) |
\(\Big \downarrow \) 6453 |
\(\displaystyle a \int \frac {\coth ^{-1}(a x)^2}{x^3 \left (1-a^2 x^2\right )}dx-\frac {\coth ^{-1}(a x)^3}{3 x^3}\) |
\(\Big \downarrow \) 6545 |
\(\displaystyle a \left (a^2 \int \frac {\coth ^{-1}(a x)^2}{x \left (1-a^2 x^2\right )}dx+\int \frac {\coth ^{-1}(a x)^2}{x^3}dx\right )-\frac {\coth ^{-1}(a x)^3}{3 x^3}\) |
\(\Big \downarrow \) 6453 |
\(\displaystyle a \left (a^2 \int \frac {\coth ^{-1}(a x)^2}{x \left (1-a^2 x^2\right )}dx+a \int \frac {\coth ^{-1}(a x)}{x^2 \left (1-a^2 x^2\right )}dx-\frac {\coth ^{-1}(a x)^2}{2 x^2}\right )-\frac {\coth ^{-1}(a x)^3}{3 x^3}\) |
\(\Big \downarrow \) 6545 |
\(\displaystyle a \left (a^2 \int \frac {\coth ^{-1}(a x)^2}{x \left (1-a^2 x^2\right )}dx+a \left (a^2 \int \frac {\coth ^{-1}(a x)}{1-a^2 x^2}dx+\int \frac {\coth ^{-1}(a x)}{x^2}dx\right )-\frac {\coth ^{-1}(a x)^2}{2 x^2}\right )-\frac {\coth ^{-1}(a x)^3}{3 x^3}\) |
\(\Big \downarrow \) 6453 |
\(\displaystyle a \left (a^2 \int \frac {\coth ^{-1}(a x)^2}{x \left (1-a^2 x^2\right )}dx+a \left (a \int \frac {1}{x \left (1-a^2 x^2\right )}dx+a^2 \int \frac {\coth ^{-1}(a x)}{1-a^2 x^2}dx-\frac {\coth ^{-1}(a x)}{x}\right )-\frac {\coth ^{-1}(a x)^2}{2 x^2}\right )-\frac {\coth ^{-1}(a x)^3}{3 x^3}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle a \left (a^2 \int \frac {\coth ^{-1}(a x)^2}{x \left (1-a^2 x^2\right )}dx+a \left (\frac {1}{2} a \int \frac {1}{x^2 \left (1-a^2 x^2\right )}dx^2+a^2 \int \frac {\coth ^{-1}(a x)}{1-a^2 x^2}dx-\frac {\coth ^{-1}(a x)}{x}\right )-\frac {\coth ^{-1}(a x)^2}{2 x^2}\right )-\frac {\coth ^{-1}(a x)^3}{3 x^3}\) |
\(\Big \downarrow \) 47 |
\(\displaystyle a \left (a^2 \int \frac {\coth ^{-1}(a x)^2}{x \left (1-a^2 x^2\right )}dx+a \left (\frac {1}{2} a \left (a^2 \int \frac {1}{1-a^2 x^2}dx^2+\int \frac {1}{x^2}dx^2\right )+a^2 \int \frac {\coth ^{-1}(a x)}{1-a^2 x^2}dx-\frac {\coth ^{-1}(a x)}{x}\right )-\frac {\coth ^{-1}(a x)^2}{2 x^2}\right )-\frac {\coth ^{-1}(a x)^3}{3 x^3}\) |
\(\Big \downarrow \) 14 |
\(\displaystyle a \left (a^2 \int \frac {\coth ^{-1}(a x)^2}{x \left (1-a^2 x^2\right )}dx+a \left (\frac {1}{2} a \left (a^2 \int \frac {1}{1-a^2 x^2}dx^2+\log \left (x^2\right )\right )+a^2 \int \frac {\coth ^{-1}(a x)}{1-a^2 x^2}dx-\frac {\coth ^{-1}(a x)}{x}\right )-\frac {\coth ^{-1}(a x)^2}{2 x^2}\right )-\frac {\coth ^{-1}(a x)^3}{3 x^3}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle a \left (a^2 \int \frac {\coth ^{-1}(a x)^2}{x \left (1-a^2 x^2\right )}dx+a \left (a^2 \int \frac {\coth ^{-1}(a x)}{1-a^2 x^2}dx+\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (1-a^2 x^2\right )\right )-\frac {\coth ^{-1}(a x)}{x}\right )-\frac {\coth ^{-1}(a x)^2}{2 x^2}\right )-\frac {\coth ^{-1}(a x)^3}{3 x^3}\) |
\(\Big \downarrow \) 6511 |
\(\displaystyle a \left (a^2 \int \frac {\coth ^{-1}(a x)^2}{x \left (1-a^2 x^2\right )}dx+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (1-a^2 x^2\right )\right )+\frac {1}{2} a \coth ^{-1}(a x)^2-\frac {\coth ^{-1}(a x)}{x}\right )-\frac {\coth ^{-1}(a x)^2}{2 x^2}\right )-\frac {\coth ^{-1}(a x)^3}{3 x^3}\) |
\(\Big \downarrow \) 6551 |
\(\displaystyle a \left (a^2 \left (\int \frac {\coth ^{-1}(a x)^2}{x (a x+1)}dx+\frac {1}{3} \coth ^{-1}(a x)^3\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (1-a^2 x^2\right )\right )+\frac {1}{2} a \coth ^{-1}(a x)^2-\frac {\coth ^{-1}(a x)}{x}\right )-\frac {\coth ^{-1}(a x)^2}{2 x^2}\right )-\frac {\coth ^{-1}(a x)^3}{3 x^3}\) |
\(\Big \downarrow \) 6495 |
\(\displaystyle a \left (a^2 \left (-2 a \int \frac {\coth ^{-1}(a x) \log \left (2-\frac {2}{a x+1}\right )}{1-a^2 x^2}dx+\frac {1}{3} \coth ^{-1}(a x)^3+\log \left (2-\frac {2}{a x+1}\right ) \coth ^{-1}(a x)^2\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (1-a^2 x^2\right )\right )+\frac {1}{2} a \coth ^{-1}(a x)^2-\frac {\coth ^{-1}(a x)}{x}\right )-\frac {\coth ^{-1}(a x)^2}{2 x^2}\right )-\frac {\coth ^{-1}(a x)^3}{3 x^3}\) |
\(\Big \downarrow \) 6619 |
\(\displaystyle a \left (a^2 \left (-2 a \left (\frac {\operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right ) \coth ^{-1}(a x)}{2 a}-\frac {1}{2} \int \frac {\operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right )}{1-a^2 x^2}dx\right )+\frac {1}{3} \coth ^{-1}(a x)^3+\log \left (2-\frac {2}{a x+1}\right ) \coth ^{-1}(a x)^2\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (1-a^2 x^2\right )\right )+\frac {1}{2} a \coth ^{-1}(a x)^2-\frac {\coth ^{-1}(a x)}{x}\right )-\frac {\coth ^{-1}(a x)^2}{2 x^2}\right )-\frac {\coth ^{-1}(a x)^3}{3 x^3}\) |
\(\Big \downarrow \) 7164 |
\(\displaystyle a \left (a^2 \left (-2 a \left (\frac {\operatorname {PolyLog}\left (3,\frac {2}{a x+1}-1\right )}{4 a}+\frac {\operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right ) \coth ^{-1}(a x)}{2 a}\right )+\frac {1}{3} \coth ^{-1}(a x)^3+\log \left (2-\frac {2}{a x+1}\right ) \coth ^{-1}(a x)^2\right )+a \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (1-a^2 x^2\right )\right )+\frac {1}{2} a \coth ^{-1}(a x)^2-\frac {\coth ^{-1}(a x)}{x}\right )-\frac {\coth ^{-1}(a x)^2}{2 x^2}\right )-\frac {\coth ^{-1}(a x)^3}{3 x^3}\) |
-1/3*ArcCoth[a*x]^3/x^3 + a*(-1/2*ArcCoth[a*x]^2/x^2 + a*(-(ArcCoth[a*x]/x ) + (a*ArcCoth[a*x]^2)/2 + (a*(Log[x^2] - Log[1 - a^2*x^2]))/2) + a^2*(Arc Coth[a*x]^3/3 + ArcCoth[a*x]^2*Log[2 - 2/(1 + a*x)] - 2*a*((ArcCoth[a*x]*P olyLog[2, -1 + 2/(1 + a*x)])/(2*a) + PolyLog[3, -1 + 2/(1 + a*x)]/(4*a))))
3.1.32.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c - a*d) Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d) Int[1/(c + d*x), x ], x] /; FreeQ[{a, b, c, d}, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
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_.)/((x_)*((d_) + (e_.)*(x_))), x _Symbol] :> Simp[(a + b*ArcCoth[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Simp[b*c*(p/d) Int[(a + b*ArcCoth[c*x])^(p - 1)*(Log[2 - 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_.)/((d_) + (e_.)*(x_)^2), x_Symb ol] :> Simp[(a + b*ArcCoth[c*x])^(p + 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b , c, d, e, p}, x] && EqQ[c^2*d + e, 0] && NeQ[p, -1]
Int[(((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + ( e_.)*(x_)^2), x_Symbol] :> Simp[1/d Int[(f*x)^m*(a + b*ArcCoth[c*x])^p, x ], x] - Simp[e/(d*f^2) 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] && LtQ[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*d*(p + 1)), x] + Simp[1/ d Int[(a + b*ArcCoth[c*x])^p/(x*(1 + c*x)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
Int[(Log[u_]*((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^ 2), x_Symbol] :> Simp[(a + b*ArcCoth[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)), x ] - Simp[b*(p/2) Int[(a + b*ArcCoth[c*x])^(p - 1)*(PolyLog[2, 1 - u]/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[(1 - u)^2 - (1 - 2/(1 + c*x))^2, 0]
Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /; !FalseQ[w]] /; FreeQ[n, x]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 6.86 (sec) , antiderivative size = 840, normalized size of antiderivative = 5.45
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(840\) |
default | \(\text {Expression too large to display}\) | \(840\) |
parts | \(\text {Expression too large to display}\) | \(843\) |
a^3*(-1/3/a^3/x^3*arccoth(a*x)^3-1/2*arccoth(a*x)^2*ln(a*x-1)-1/2*arccoth( a*x)^2*ln(a*x+1)-1/2*arccoth(a*x)^2/a^2/x^2+ln(a*x)*arccoth(a*x)^2-1/2*arc coth(a*x)^2*ln((a*x-1)/(a*x+1))+arccoth(a*x)*polylog(2,-(a*x+1)/(a*x-1))-1 /2*polylog(3,-(a*x+1)/(a*x-1))+1/12*arccoth(a*x)*(-3*I*csgn(I/((a*x+1)/(a* x-1)-1))*arccoth(a*x)*Pi*csgn(I*(a*x+1)/(a*x-1))*csgn(I/(a*x-1)*(a*x+1)/(( a*x+1)/(a*x-1)-1))*a*x+6*I*csgn(I/((a*x+1)/(a*x-1)-1))*csgn(I*(1+(a*x+1)/( a*x-1)))*arccoth(a*x)*Pi*csgn(I/((a*x+1)/(a*x-1)-1)*(1+(a*x+1)/(a*x-1)))*a *x+3*I*arccoth(a*x)*Pi*csgn(I*(a*x+1)/(a*x-1))*csgn(I/(a*x-1)*(a*x+1)/((a* x+1)/(a*x-1)-1))^2*a*x-3*I*arccoth(a*x)*Pi*csgn(I/(a*x-1)*(a*x+1)/((a*x+1) /(a*x-1)-1))^3*a*x+6*I*arccoth(a*x)*Pi*csgn(I/((a*x+1)/(a*x-1)-1)*(1+(a*x+ 1)/(a*x-1)))^3*a*x-3*I*arccoth(a*x)*Pi*csgn(I*(a*x+1)/(a*x-1))^3*a*x+6*I*a rccoth(a*x)*Pi*csgn(I*(a*x+1)/(a*x-1))^2*csgn(I/((a*x-1)/(a*x+1))^(1/2))*a *x-6*I*csgn(I/((a*x+1)/(a*x-1)-1))*arccoth(a*x)*Pi*csgn(I/((a*x+1)/(a*x-1) -1)*(1+(a*x+1)/(a*x-1)))^2*a*x-6*I*csgn(I*(1+(a*x+1)/(a*x-1)))*arccoth(a*x )*Pi*csgn(I/((a*x+1)/(a*x-1)-1)*(1+(a*x+1)/(a*x-1)))^2*a*x-3*I*arccoth(a*x )*Pi*csgn(I*(a*x+1)/(a*x-1))*csgn(I/((a*x-1)/(a*x+1))^(1/2))^2*a*x+3*I*csg n(I/((a*x+1)/(a*x-1)-1))*arccoth(a*x)*Pi*csgn(I/(a*x-1)*(a*x+1)/((a*x+1)/( a*x-1)-1))^2*a*x-4*arccoth(a*x)^2*a*x+12*arccoth(a*x)*ln(2)*a*x+6*a*x*arcc oth(a*x)-12*a*x-12)/a/x+ln(1+(a*x+1)/(a*x-1)))
\[ \int \frac {\coth ^{-1}(a x)^3}{x^4} \, dx=\int { \frac {\operatorname {arcoth}\left (a x\right )^{3}}{x^{4}} \,d x } \]
\[ \int \frac {\coth ^{-1}(a x)^3}{x^4} \, dx=\int \frac {\operatorname {acoth}^{3}{\left (a x \right )}}{x^{4}}\, dx \]
\[ \int \frac {\coth ^{-1}(a x)^3}{x^4} \, dx=\int { \frac {\operatorname {arcoth}\left (a x\right )^{3}}{x^{4}} \,d x } \]
1/4*a^4*integrate(x^4*log(a*x + 1)*log(a*x - 1)/(a*x^5 + x^4), x) - 1/2*a^ 4*integrate(x^4*log(a*x + 1)*log(x)/(a*x^5 + x^4), x) + 1/16*(2*a^2*log(a* x + 1) - 2*a^2*log(x) - (2*a*x - 1)/x^2)*a*log(a)^3 + 3/8*a*integrate(x*lo g(a*x - 1)/(a*x^5 + x^4), x)*log(a)^2 - 3/8*a*integrate(x*log(x)/(a*x^5 + x^4), x)*log(a)^2 - 1/48*(6*a^3*log(a*x + 1) - 6*a^3*log(x) - (6*a^2*x^2 - 3*a*x + 2)/x^3)*log(a)^3 + 1/4*a^2*integrate(x^2*log(a*x + 1)/(a*x^5 + x^ 4), x) + 3/4*a*integrate(x*log(a*x - 1)*log(x)/(a*x^5 + x^4), x)*log(a) - 3/8*a*integrate(x*log(x)^2/(a*x^5 + x^4), x)*log(a) + 3/8*integrate(log(a* x - 1)/(a*x^5 + x^4), x)*log(a)^2 - 3/8*integrate(log(x)/(a*x^5 + x^4), x) *log(a)^2 + 3/8*a*integrate(x*log(a*x + 1)*log(a*x - 1)^2/(a*x^5 + x^4), x ) - 3/8*a*integrate(x*log(a*x - 1)^2*log(x)/(a*x^5 + x^4), x) + 3/8*a*inte grate(x*log(a*x - 1)*log(x)^2/(a*x^5 + x^4), x) - 1/8*a*integrate(x*log(x) ^3/(a*x^5 + x^4), x) - 1/4*a*integrate(x*log(a*x + 1)*log(a*x - 1)/(a*x^5 + x^4), x) - 3/8*integrate(a*x*log(a*x - 1)^2/(a*x^5 + x^4), x)*log(a) - 3 /8*integrate(log(a*x - 1)^2/(a*x^5 + x^4), x)*log(a) + 3/4*integrate(log(a *x - 1)*log(x)/(a*x^5 + x^4), x)*log(a) - 3/8*integrate(log(x)^2/(a*x^5 + x^4), x)*log(a) - 1/48*(6*a^4*log(a*x - 1) - 6*a^4*log(x) + (6*a^3*x^2 + 3 *a^2*x + 2*a)/x^3)*log(-1/(a*x) + 1)^2/a + 1/864*(6*(18*a^5*x^3*log(a*x - 1)^2 + 18*a^5*x^3*log(x)^2 - 66*a^5*x^3*log(x) + 66*a^4*x^2 + 15*a^3*x + 4 *a^2 - 6*(6*a^5*x^3*log(x) - 11*a^5*x^3)*log(a*x - 1))*log(-1/(a*x) + 1...
\[ \int \frac {\coth ^{-1}(a x)^3}{x^4} \, dx=\int { \frac {\operatorname {arcoth}\left (a x\right )^{3}}{x^{4}} \,d x } \]
Timed out. \[ \int \frac {\coth ^{-1}(a x)^3}{x^4} \, dx=\int \frac {{\mathrm {acoth}\left (a\,x\right )}^3}{x^4} \,d x \]