Integrand size = 10, antiderivative size = 105 \[ \int \frac {\cot ^{-1}(a x)^3}{x^3} \, dx=\frac {3}{2} i a^2 \cot ^{-1}(a x)^2+\frac {3 a \cot ^{-1}(a x)^2}{2 x}-\frac {1}{2} a^2 \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^3}{2 x^2}+3 a^2 \cot ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )+\frac {3}{2} i a^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right ) \]
3/2*I*a^2*arccot(a*x)^2+3/2*a*arccot(a*x)^2/x-1/2*a^2*arccot(a*x)^3-1/2*ar ccot(a*x)^3/x^2+3*a^2*arccot(a*x)*ln(2-2/(1-I*a*x))+3/2*I*a^2*polylog(2,-1 +2/(1-I*a*x))
Time = 0.12 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.86 \[ \int \frac {\cot ^{-1}(a x)^3}{x^3} \, dx=-\frac {\cot ^{-1}(a x) \left (3 i a x (i+a x) \cot ^{-1}(a x)+\left (1+a^2 x^2\right ) \cot ^{-1}(a x)^2-6 a^2 x^2 \log \left (1+e^{2 i \cot ^{-1}(a x)}\right )\right )}{2 x^2}-\frac {3}{2} i a^2 \operatorname {PolyLog}\left (2,-e^{2 i \cot ^{-1}(a x)}\right ) \]
-1/2*(ArcCot[a*x]*((3*I)*a*x*(I + a*x)*ArcCot[a*x] + (1 + a^2*x^2)*ArcCot[ a*x]^2 - 6*a^2*x^2*Log[1 + E^((2*I)*ArcCot[a*x])]))/x^2 - ((3*I)/2)*a^2*Po lyLog[2, -E^((2*I)*ArcCot[a*x])]
Time = 0.72 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5362, 5454, 5362, 5420, 5460, 5404, 2897}
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.
\(\displaystyle \int \frac {\cot ^{-1}(a x)^3}{x^3} \, dx\) |
\(\Big \downarrow \) 5362 |
\(\displaystyle -\frac {3}{2} a \int \frac {\cot ^{-1}(a x)^2}{x^2 \left (a^2 x^2+1\right )}dx-\frac {\cot ^{-1}(a x)^3}{2 x^2}\) |
\(\Big \downarrow \) 5454 |
\(\displaystyle -\frac {3}{2} a \left (\int \frac {\cot ^{-1}(a x)^2}{x^2}dx-a^2 \int \frac {\cot ^{-1}(a x)^2}{a^2 x^2+1}dx\right )-\frac {\cot ^{-1}(a x)^3}{2 x^2}\) |
\(\Big \downarrow \) 5362 |
\(\displaystyle -\frac {3}{2} a \left (a^2 \left (-\int \frac {\cot ^{-1}(a x)^2}{a^2 x^2+1}dx\right )-2 a \int \frac {\cot ^{-1}(a x)}{x \left (a^2 x^2+1\right )}dx-\frac {\cot ^{-1}(a x)^2}{x}\right )-\frac {\cot ^{-1}(a x)^3}{2 x^2}\) |
\(\Big \downarrow \) 5420 |
\(\displaystyle -\frac {3}{2} a \left (-2 a \int \frac {\cot ^{-1}(a x)}{x \left (a^2 x^2+1\right )}dx+\frac {1}{3} a \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^2}{x}\right )-\frac {\cot ^{-1}(a x)^3}{2 x^2}\) |
\(\Big \downarrow \) 5460 |
\(\displaystyle -\frac {\cot ^{-1}(a x)^3}{2 x^2}-\frac {3}{2} a \left (-2 a \left (i \int \frac {\cot ^{-1}(a x)}{x (a x+i)}dx+\frac {1}{2} i \cot ^{-1}(a x)^2\right )+\frac {1}{3} a \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^2}{x}\right )\) |
\(\Big \downarrow \) 5404 |
\(\displaystyle -\frac {\cot ^{-1}(a x)^3}{2 x^2}-\frac {3}{2} a \left (-2 a \left (i \left (-i a \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \log \left (2-\frac {2}{1-i a x}\right ) \cot ^{-1}(a x)\right )+\frac {1}{2} i \cot ^{-1}(a x)^2\right )+\frac {1}{3} a \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^2}{x}\right )\) |
\(\Big \downarrow \) 2897 |
\(\displaystyle -\frac {\cot ^{-1}(a x)^3}{2 x^2}-\frac {3}{2} a \left (-2 a \left (i \left (\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )-i \log \left (2-\frac {2}{1-i a x}\right ) \cot ^{-1}(a x)\right )+\frac {1}{2} i \cot ^{-1}(a x)^2\right )+\frac {1}{3} a \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^2}{x}\right )\) |
-1/2*ArcCot[a*x]^3/x^2 - (3*a*(-(ArcCot[a*x]^2/x) + (a*ArcCot[a*x]^3)/3 - 2*a*((I/2)*ArcCot[a*x]^2 + I*((-I)*ArcCot[a*x]*Log[2 - 2/(1 - I*a*x)] + Po lyLog[2, -1 + 2/(1 - I*a*x)]/2))))/2
3.1.31.3.1 Defintions of rubi rules used
Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/ D[u, x])]}, Simp[C*PolyLog[2, 1 - u], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponents[u, x][[2]], Expon[Pq, x]]
Int[((a_.) + ArcCot[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcCot[c*x^n])^p/(m + 1)), x] + Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcCot[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_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_ Symbol] :> Simp[(a + b*ArcCot[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] + Si mp[b*c*(p/d) Int[(a + b*ArcCot[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_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbo l] :> Simp[-(a + b*ArcCot[c*x])^(p + 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]
Int[(((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e _.)*(x_)^2), x_Symbol] :> Simp[1/d Int[(f*x)^m*(a + b*ArcCot[c*x])^p, x], x] - Simp[e/(d*f^2) Int[(f*x)^(m + 2)*((a + b*ArcCot[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_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[I*((a + b*ArcCot[c*x])^(p + 1)/(b*d*(p + 1))), x] + Simp[ I/d Int[(a + b*ArcCot[c*x])^p/(x*(I + c*x)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 6.76 (sec) , antiderivative size = 2956, normalized size of antiderivative = 28.15
method | result | size |
parts | \(\text {Expression too large to display}\) | \(2956\) |
derivativedivides | \(\text {Expression too large to display}\) | \(2957\) |
default | \(\text {Expression too large to display}\) | \(2957\) |
-1/2*arccot(a*x)^3/x^2-3/2*a^2*(1/2*Pi*arccot(a*x)^2-1/2*I*Pi*arccot(a*x)* ln(1+I*(I+a*x)/(a^2*x^2+1)^(1/2))-1/2*I*Pi*arccot(a*x)*ln(1-I*(I+a*x)/(a^2 *x^2+1)^(1/2))+1/2*I*Pi*arccot(a*x)*ln(1+(I+a*x)^2/(a^2*x^2+1))+1/8*Pi*csg n(I/((I+a*x)^2/(a^2*x^2+1)-1))*csgn(I*(I+a*x)^2/(a^2*x^2+1))*csgn(I*(I+a*x )^2/(a^2*x^2+1)/((I+a*x)^2/(a^2*x^2+1)-1))*(2*I*arccot(a*x)*ln(1+(I+a*x)^2 /(a^2*x^2+1))+2*arccot(a*x)^2+polylog(2,-(I+a*x)^2/(a^2*x^2+1)))-1/4*Pi*cs gn(I/((I+a*x)^2/(a^2*x^2+1)-1))*csgn(I*(I+a*x)^2/(a^2*x^2+1))*csgn(I*(I+a* x)^2/(a^2*x^2+1)/((I+a*x)^2/(a^2*x^2+1)-1))*(I*arccot(a*x)*ln(1+I*(I+a*x)/ (a^2*x^2+1)^(1/2))+I*arccot(a*x)*ln(1-I*(I+a*x)/(a^2*x^2+1)^(1/2))+dilog(1 +I*(I+a*x)/(a^2*x^2+1)^(1/2))+dilog(1-I*(I+a*x)/(a^2*x^2+1)^(1/2)))+1/8*Pi *csgn(I*(I+a*x)^2/(a^2*x^2+1)/((I+a*x)^2/(a^2*x^2+1)-1))^3*(2*I*arccot(a*x )*ln(1+(I+a*x)^2/(a^2*x^2+1))+2*arccot(a*x)^2+polylog(2,-(I+a*x)^2/(a^2*x^ 2+1)))-1/4*Pi*csgn(I*(I+a*x)^2/(a^2*x^2+1)/((I+a*x)^2/(a^2*x^2+1)-1))^3*(I *arccot(a*x)*ln(1+I*(I+a*x)/(a^2*x^2+1)^(1/2))+I*arccot(a*x)*ln(1-I*(I+a*x )/(a^2*x^2+1)^(1/2))+dilog(1+I*(I+a*x)/(a^2*x^2+1)^(1/2))+dilog(1-I*(I+a*x )/(a^2*x^2+1)^(1/2)))+1/2*Pi*csgn(I/((I+a*x)^2/(a^2*x^2+1)-1))^2*(I*arccot (a*x)*ln(1+I*(I+a*x)/(a^2*x^2+1)^(1/2))+I*arccot(a*x)*ln(1-I*(I+a*x)/(a^2* x^2+1)^(1/2))+dilog(1+I*(I+a*x)/(a^2*x^2+1)^(1/2))+dilog(1-I*(I+a*x)/(a^2* x^2+1)^(1/2)))+1/4*Pi*csgn(I/((I+a*x)^2/(a^2*x^2+1)-1))^3*(2*I*arccot(a*x) *ln(1+(I+a*x)^2/(a^2*x^2+1))+2*arccot(a*x)^2+polylog(2,-(I+a*x)^2/(a^2*...
\[ \int \frac {\cot ^{-1}(a x)^3}{x^3} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )^{3}}{x^{3}} \,d x } \]
\[ \int \frac {\cot ^{-1}(a x)^3}{x^3} \, dx=\int \frac {\operatorname {acot}^{3}{\left (a x \right )}}{x^{3}}\, dx \]
\[ \int \frac {\cot ^{-1}(a x)^3}{x^3} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )^{3}}{x^{3}} \,d x } \]
-1/32*(8*a^2*x^2*arctan2(1, a*x)^3 - 12*a*x*arctan2(1, a*x)^2 + 3*a*x*log( a^2*x^2 + 1)^2 + 4*(3*a^2*arctan(a*x)*arctan(1/(a*x))^2 + (arctan(a*x)^3/a + 3*arctan(a*x)^2*arctan(1/(a*x))/a)*a^3 + 24*a^3*integrate(1/32*x^3*log( a^2*x^2 + 1)^2/(a^2*x^5 + x^3), x) - 96*a^3*integrate(1/32*x^3*log(a^2*x^2 + 1)/(a^2*x^5 + x^3), x) - 128*a^2*integrate(1/32*x^2*arctan(1/(a*x))^3/( a^2*x^5 + x^3), x) - 192*a^2*integrate(1/32*x^2*arctan(1/(a*x))/(a^2*x^5 + x^3), x) + 96*a*integrate(1/32*x*arctan(1/(a*x))^2/(a^2*x^5 + x^3), x) + 24*a*integrate(1/32*x*log(a^2*x^2 + 1)^2/(a^2*x^5 + x^3), x) - 128*integra te(1/32*arctan(1/(a*x))^3/(a^2*x^5 + x^3), x))*x^2 + 8*arctan2(1, a*x)^3)/ x^2
Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.28 \[ \int \frac {\cot ^{-1}(a x)^3}{x^3} \, dx=-\frac {1}{2} \, a \arctan \left (\frac {1}{a x}\right )^{3} - \frac {\arctan \left (\frac {1}{a x}\right )^{3}}{2 \, x^{2}} \]
Timed out. \[ \int \frac {\cot ^{-1}(a x)^3}{x^3} \, dx=\int \frac {{\mathrm {acot}\left (a\,x\right )}^3}{x^3} \,d x \]