Integrand size = 10, antiderivative size = 157 \[ \int x^2 \cot ^{-1}(a x)^3 \, dx=\frac {x \cot ^{-1}(a x)}{a^2}+\frac {\cot ^{-1}(a x)^2}{2 a^3}+\frac {x^2 \cot ^{-1}(a x)^2}{2 a}-\frac {i \cot ^{-1}(a x)^3}{3 a^3}+\frac {1}{3} x^3 \cot ^{-1}(a x)^3+\frac {\cot ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^3}+\frac {\log \left (1+a^2 x^2\right )}{2 a^3}-\frac {i \cot ^{-1}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{a^3}+\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{2 a^3} \]
x*arccot(a*x)/a^2+1/2*arccot(a*x)^2/a^3+1/2*x^2*arccot(a*x)^2/a-1/3*I*arcc ot(a*x)^3/a^3+1/3*x^3*arccot(a*x)^3+arccot(a*x)^2*ln(2/(1+I*a*x))/a^3+1/2* ln(a^2*x^2+1)/a^3-I*arccot(a*x)*polylog(2,1-2/(1+I*a*x))/a^3+1/2*polylog(3 ,1-2/(1+I*a*x))/a^3
Time = 0.36 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.97 \[ \int x^2 \cot ^{-1}(a x)^3 \, dx=\frac {-i \pi ^3+24 a x \cot ^{-1}(a x)+12 \cot ^{-1}(a x)^2+12 a^2 x^2 \cot ^{-1}(a x)^2+8 i \cot ^{-1}(a x)^3+8 a^3 x^3 \cot ^{-1}(a x)^3+24 \cot ^{-1}(a x)^2 \log \left (1-e^{-2 i \cot ^{-1}(a x)}\right )-24 \log \left (\frac {1}{\sqrt {1+\frac {1}{a^2 x^2}}}\right )-24 \log \left (\frac {1}{a x}\right )+24 i \cot ^{-1}(a x) \operatorname {PolyLog}\left (2,e^{-2 i \cot ^{-1}(a x)}\right )+12 \operatorname {PolyLog}\left (3,e^{-2 i \cot ^{-1}(a x)}\right )}{24 a^3} \]
((-I)*Pi^3 + 24*a*x*ArcCot[a*x] + 12*ArcCot[a*x]^2 + 12*a^2*x^2*ArcCot[a*x ]^2 + (8*I)*ArcCot[a*x]^3 + 8*a^3*x^3*ArcCot[a*x]^3 + 24*ArcCot[a*x]^2*Log [1 - E^((-2*I)*ArcCot[a*x])] - 24*Log[1/Sqrt[1 + 1/(a^2*x^2)]] - 24*Log[1/ (a*x)] + (24*I)*ArcCot[a*x]*PolyLog[2, E^((-2*I)*ArcCot[a*x])] + 12*PolyLo g[3, E^((-2*I)*ArcCot[a*x])])/(24*a^3)
Time = 1.36 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.17, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {5362, 5452, 5362, 5452, 5346, 240, 5420, 5456, 5380, 5530, 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.
\(\displaystyle \int x^2 \cot ^{-1}(a x)^3 \, dx\) |
\(\Big \downarrow \) 5362 |
\(\displaystyle a \int \frac {x^3 \cot ^{-1}(a x)^2}{a^2 x^2+1}dx+\frac {1}{3} x^3 \cot ^{-1}(a x)^3\) |
\(\Big \downarrow \) 5452 |
\(\displaystyle a \left (\frac {\int x \cot ^{-1}(a x)^2dx}{a^2}-\frac {\int \frac {x \cot ^{-1}(a x)^2}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{3} x^3 \cot ^{-1}(a x)^3\) |
\(\Big \downarrow \) 5362 |
\(\displaystyle a \left (\frac {a \int \frac {x^2 \cot ^{-1}(a x)}{a^2 x^2+1}dx+\frac {1}{2} x^2 \cot ^{-1}(a x)^2}{a^2}-\frac {\int \frac {x \cot ^{-1}(a x)^2}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{3} x^3 \cot ^{-1}(a x)^3\) |
\(\Big \downarrow \) 5452 |
\(\displaystyle a \left (\frac {a \left (\frac {\int \cot ^{-1}(a x)dx}{a^2}-\frac {\int \frac {\cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)^2}{a^2}-\frac {\int \frac {x \cot ^{-1}(a x)^2}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{3} x^3 \cot ^{-1}(a x)^3\) |
\(\Big \downarrow \) 5346 |
\(\displaystyle a \left (\frac {a \left (\frac {a \int \frac {x}{a^2 x^2+1}dx+x \cot ^{-1}(a x)}{a^2}-\frac {\int \frac {\cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)^2}{a^2}-\frac {\int \frac {x \cot ^{-1}(a x)^2}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{3} x^3 \cot ^{-1}(a x)^3\) |
\(\Big \downarrow \) 240 |
\(\displaystyle a \left (\frac {a \left (\frac {\frac {\log \left (a^2 x^2+1\right )}{2 a}+x \cot ^{-1}(a x)}{a^2}-\frac {\int \frac {\cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)^2}{a^2}-\frac {\int \frac {x \cot ^{-1}(a x)^2}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{3} x^3 \cot ^{-1}(a x)^3\) |
\(\Big \downarrow \) 5420 |
\(\displaystyle a \left (\frac {a \left (\frac {\cot ^{-1}(a x)^2}{2 a^3}+\frac {\frac {\log \left (a^2 x^2+1\right )}{2 a}+x \cot ^{-1}(a x)}{a^2}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)^2}{a^2}-\frac {\int \frac {x \cot ^{-1}(a x)^2}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{3} x^3 \cot ^{-1}(a x)^3\) |
\(\Big \downarrow \) 5456 |
\(\displaystyle \frac {1}{3} x^3 \cot ^{-1}(a x)^3+a \left (\frac {a \left (\frac {\cot ^{-1}(a x)^2}{2 a^3}+\frac {\frac {\log \left (a^2 x^2+1\right )}{2 a}+x \cot ^{-1}(a x)}{a^2}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)^2}{a^2}-\frac {\frac {i \cot ^{-1}(a x)^3}{3 a^2}-\frac {\int \frac {\cot ^{-1}(a x)^2}{i-a x}dx}{a}}{a^2}\right )\) |
\(\Big \downarrow \) 5380 |
\(\displaystyle \frac {1}{3} x^3 \cot ^{-1}(a x)^3+a \left (\frac {a \left (\frac {\cot ^{-1}(a x)^2}{2 a^3}+\frac {\frac {\log \left (a^2 x^2+1\right )}{2 a}+x \cot ^{-1}(a x)}{a^2}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)^2}{a^2}-\frac {\frac {i \cot ^{-1}(a x)^3}{3 a^2}-\frac {2 \int \frac {\cot ^{-1}(a x) \log \left (\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx+\frac {\log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)^2}{a}}{a}}{a^2}\right )\) |
\(\Big \downarrow \) 5530 |
\(\displaystyle \frac {1}{3} x^3 \cot ^{-1}(a x)^3+a \left (\frac {a \left (\frac {\cot ^{-1}(a x)^2}{2 a^3}+\frac {\frac {\log \left (a^2 x^2+1\right )}{2 a}+x \cot ^{-1}(a x)}{a^2}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)^2}{a^2}-\frac {\frac {i \cot ^{-1}(a x)^3}{3 a^2}-\frac {2 \left (-\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right ) \cot ^{-1}(a x)}{2 a}\right )+\frac {\log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)^2}{a}}{a}}{a^2}\right )\) |
\(\Big \downarrow \) 7164 |
\(\displaystyle \frac {1}{3} x^3 \cot ^{-1}(a x)^3+a \left (\frac {a \left (\frac {\cot ^{-1}(a x)^2}{2 a^3}+\frac {\frac {\log \left (a^2 x^2+1\right )}{2 a}+x \cot ^{-1}(a x)}{a^2}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)^2}{a^2}-\frac {\frac {i \cot ^{-1}(a x)^3}{3 a^2}-\frac {2 \left (\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )}{4 a}-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right ) \cot ^{-1}(a x)}{2 a}\right )+\frac {\log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)^2}{a}}{a}}{a^2}\right )\) |
(x^3*ArcCot[a*x]^3)/3 + a*(((x^2*ArcCot[a*x]^2)/2 + a*(ArcCot[a*x]^2/(2*a^ 3) + (x*ArcCot[a*x] + Log[1 + a^2*x^2]/(2*a))/a^2))/a^2 - (((I/3)*ArcCot[a *x]^3)/a^2 - ((ArcCot[a*x]^2*Log[2/(1 + I*a*x)])/a + 2*(((-1/2*I)*ArcCot[a *x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/a + PolyLog[3, 1 - 2/(1 + I*a*x)]/(4*a) ))/a)/a^2)
3.1.26.3.1 Defintions of rubi rules used
Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[((a_.) + ArcCot[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcCot[c*x^n])^p, x] + Simp[b*c*n*p Int[x^n*((a + b*ArcCot[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && (EqQ[n, 1] || EqQ[p, 1])
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_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcCot[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] - Simp[b*c*( p/e) Int[(a + b*ArcCot[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_.) + 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[f^2/e Int[(f*x)^(m - 2)*(a + b*ArcCot[c*x] )^p, x], x] - Simp[d*(f^2/e) 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] && GtQ[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*e*(p + 1))), x] - Simp[ 1/(c*d) Int[(a + b*ArcCot[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
Int[(Log[u_]*((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2 ), x_Symbol] :> Simp[(-I)*(a + b*ArcCot[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)) , x] - Simp[b*p*(I/2) Int[(a + b*ArcCot[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[e, c ^2*d] && EqQ[(1 - u)^2 - (1 - 2*(I/(I - 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 = 5.94 (sec) , antiderivative size = 1036, normalized size of antiderivative = 6.60
method | result | size |
parts | \(\text {Expression too large to display}\) | \(1036\) |
derivativedivides | \(\text {Expression too large to display}\) | \(1038\) |
default | \(\text {Expression too large to display}\) | \(1038\) |
1/3*x^3*arccot(a*x)^3+1/a^3*(1/2*a^2*x^2*arccot(a*x)^2-1/2*arccot(a*x)^2*l n(a^2*x^2+1)+arccot(a*x)^2*ln((I+a*x)/(a^2*x^2+1)^(1/2))-arccot(a*x)^2*ln( (I+a*x)^2/(a^2*x^2+1)-1)-1/12*I*arccot(a*x)*(-3*arccot(a*x)*Pi*csgn(I*((I+ a*x)^2/(a^2*x^2+1)-1))^2*csgn(I*((I+a*x)^2/(a^2*x^2+1)-1)^2)+6*arccot(a*x) *Pi*csgn(I*((I+a*x)^2/(a^2*x^2+1)-1))*csgn(I*((I+a*x)^2/(a^2*x^2+1)-1)^2)^ 2-3*arccot(a*x)*Pi*csgn(I*((I+a*x)^2/(a^2*x^2+1)-1)^2)^3-3*arccot(a*x)*Pi* csgn(I/((I+a*x)^2/(a^2*x^2+1)-1)^2)*csgn(I*(I+a*x)^2/(a^2*x^2+1)/((I+a*x)^ 2/(a^2*x^2+1)-1)^2)^2+3*arccot(a*x)*Pi*csgn(I/((I+a*x)^2/(a^2*x^2+1)-1)^2) *csgn(I*(I+a*x)^2/(a^2*x^2+1)/((I+a*x)^2/(a^2*x^2+1)-1)^2)*csgn(I*(I+a*x)^ 2/(a^2*x^2+1))-3*arccot(a*x)*Pi*csgn(I*(I+a*x)^2/(a^2*x^2+1)/((I+a*x)^2/(a ^2*x^2+1)-1)^2)^3-3*arccot(a*x)*Pi*csgn(I*(I+a*x)^2/(a^2*x^2+1)/((I+a*x)^2 /(a^2*x^2+1)-1)^2)^2*csgn(I*(I+a*x)^2/(a^2*x^2+1))+3*arccot(a*x)*Pi*csgn(I *(I+a*x)^2/(a^2*x^2+1))^3-6*arccot(a*x)*Pi*csgn(I*(I+a*x)^2/(a^2*x^2+1))^2 *csgn(I*(I+a*x)/(a^2*x^2+1)^(1/2))+3*arccot(a*x)*Pi*csgn(I*(I+a*x)^2/(a^2* x^2+1))*csgn(I*(I+a*x)/(a^2*x^2+1)^(1/2))^2+6*arccot(a*x)*Pi*csgn(I*(I+a*x )^2/(a^2*x^2+1)/((I+a*x)^2/(a^2*x^2+1)-1)^2)^2+4*arccot(a*x)^2-6*Pi*arccot (a*x)+6*I*arccot(a*x)+12*I*arccot(a*x)*ln(2)-12+12*I*a*x)-ln((I+a*x)/(a^2* x^2+1)^(1/2)-1)-ln(1+(I+a*x)/(a^2*x^2+1)^(1/2))+arccot(a*x)^2*ln(1-(I+a*x) /(a^2*x^2+1)^(1/2))-2*I*arccot(a*x)*polylog(2,(I+a*x)/(a^2*x^2+1)^(1/2))+2 *polylog(3,(I+a*x)/(a^2*x^2+1)^(1/2))+arccot(a*x)^2*ln(1+(I+a*x)/(a^2*x...
\[ \int x^2 \cot ^{-1}(a x)^3 \, dx=\int { x^{2} \operatorname {arccot}\left (a x\right )^{3} \,d x } \]
\[ \int x^2 \cot ^{-1}(a x)^3 \, dx=\int x^{2} \operatorname {acot}^{3}{\left (a x \right )}\, dx \]
\[ \int x^2 \cot ^{-1}(a x)^3 \, dx=\int { x^{2} \operatorname {arccot}\left (a x\right )^{3} \,d x } \]
1/24*x^3*arctan2(1, a*x)^3 - 1/32*x^3*arctan2(1, a*x)*log(a^2*x^2 + 1)^2 + integrate(1/32*(28*a^2*x^4*arctan2(1, a*x)^3 + 4*a^2*x^4*arctan2(1, a*x)* log(a^2*x^2 + 1) + 4*a*x^3*arctan2(1, a*x)^2 + 28*x^2*arctan2(1, a*x)^3 + (3*a^2*x^4*arctan2(1, a*x) - a*x^3 + 3*x^2*arctan2(1, a*x))*log(a^2*x^2 + 1)^2)/(a^2*x^2 + 1), x)
\[ \int x^2 \cot ^{-1}(a x)^3 \, dx=\int { x^{2} \operatorname {arccot}\left (a x\right )^{3} \,d x } \]
Timed out. \[ \int x^2 \cot ^{-1}(a x)^3 \, dx=\int x^2\,{\mathrm {acot}\left (a\,x\right )}^3 \,d x \]