Integrand size = 10, antiderivative size = 148 \[ \int x^3 \cot ^{-1}(a x)^3 \, dx=\frac {x}{4 a^3}+\frac {x^2 \cot ^{-1}(a x)}{4 a^2}-\frac {i \cot ^{-1}(a x)^2}{a^4}-\frac {3 x \cot ^{-1}(a x)^2}{4 a^3}+\frac {x^3 \cot ^{-1}(a x)^2}{4 a}-\frac {\cot ^{-1}(a x)^3}{4 a^4}+\frac {1}{4} x^4 \cot ^{-1}(a x)^3-\frac {\arctan (a x)}{4 a^4}+\frac {2 \cot ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{a^4}-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{a^4} \] Output:
1/4*x/a^3+1/4*x^2*arccot(a*x)/a^2-I*arccot(a*x)^2/a^4-3/4*x*arccot(a*x)^2/ a^3+1/4*x^3*arccot(a*x)^2/a-1/4*arccot(a*x)^3/a^4+1/4*x^4*arccot(a*x)^3-1/ 4*arctan(a*x)/a^4+2*arccot(a*x)*ln(2/(1+I*a*x))/a^4-I*polylog(2,1-2/(1+I*a *x))/a^4
Time = 0.26 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.65 \[ \int x^3 \cot ^{-1}(a x)^3 \, dx=\frac {a x+\left (-4 i-3 a x+a^3 x^3\right ) \cot ^{-1}(a x)^2+\left (-1+a^4 x^4\right ) \cot ^{-1}(a x)^3+\cot ^{-1}(a x) \left (1+a^2 x^2+8 \log \left (1-e^{2 i \cot ^{-1}(a x)}\right )\right )-4 i \operatorname {PolyLog}\left (2,e^{2 i \cot ^{-1}(a x)}\right )}{4 a^4} \] Input:
Integrate[x^3*ArcCot[a*x]^3,x]
Output:
(a*x + (-4*I - 3*a*x + a^3*x^3)*ArcCot[a*x]^2 + (-1 + a^4*x^4)*ArcCot[a*x] ^3 + ArcCot[a*x]*(1 + a^2*x^2 + 8*Log[1 - E^((2*I)*ArcCot[a*x])]) - (4*I)* PolyLog[2, E^((2*I)*ArcCot[a*x])])/(4*a^4)
Time = 1.66 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.73, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.300, Rules used = {5362, 5452, 5362, 5452, 5346, 5362, 262, 216, 5420, 5456, 5380, 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.
\(\displaystyle \int x^3 \cot ^{-1}(a x)^3 \, dx\) |
\(\Big \downarrow \) 5362 |
\(\displaystyle \frac {3}{4} a \int \frac {x^4 \cot ^{-1}(a x)^2}{a^2 x^2+1}dx+\frac {1}{4} x^4 \cot ^{-1}(a x)^3\) |
\(\Big \downarrow \) 5452 |
\(\displaystyle \frac {3}{4} a \left (\frac {\int x^2 \cot ^{-1}(a x)^2dx}{a^2}-\frac {\int \frac {x^2 \cot ^{-1}(a x)^2}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{4} x^4 \cot ^{-1}(a x)^3\) |
\(\Big \downarrow \) 5362 |
\(\displaystyle \frac {3}{4} a \left (\frac {\frac {2}{3} a \int \frac {x^3 \cot ^{-1}(a x)}{a^2 x^2+1}dx+\frac {1}{3} x^3 \cot ^{-1}(a x)^2}{a^2}-\frac {\int \frac {x^2 \cot ^{-1}(a x)^2}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{4} x^4 \cot ^{-1}(a x)^3\) |
\(\Big \downarrow \) 5452 |
\(\displaystyle \frac {3}{4} a \left (\frac {\frac {2}{3} a \left (\frac {\int x \cot ^{-1}(a x)dx}{a^2}-\frac {\int \frac {x \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{3} x^3 \cot ^{-1}(a x)^2}{a^2}-\frac {\frac {\int \cot ^{-1}(a x)^2dx}{a^2}-\frac {\int \frac {\cot ^{-1}(a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}\right )+\frac {1}{4} x^4 \cot ^{-1}(a x)^3\) |
\(\Big \downarrow \) 5346 |
\(\displaystyle \frac {3}{4} a \left (\frac {\frac {2}{3} a \left (\frac {\int x \cot ^{-1}(a x)dx}{a^2}-\frac {\int \frac {x \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{3} x^3 \cot ^{-1}(a x)^2}{a^2}-\frac {\frac {2 a \int \frac {x \cot ^{-1}(a x)}{a^2 x^2+1}dx+x \cot ^{-1}(a x)^2}{a^2}-\frac {\int \frac {\cot ^{-1}(a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}\right )+\frac {1}{4} x^4 \cot ^{-1}(a x)^3\) |
\(\Big \downarrow \) 5362 |
\(\displaystyle \frac {3}{4} a \left (\frac {\frac {2}{3} a \left (\frac {\frac {1}{2} a \int \frac {x^2}{a^2 x^2+1}dx+\frac {1}{2} x^2 \cot ^{-1}(a x)}{a^2}-\frac {\int \frac {x \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{3} x^3 \cot ^{-1}(a x)^2}{a^2}-\frac {\frac {2 a \int \frac {x \cot ^{-1}(a x)}{a^2 x^2+1}dx+x \cot ^{-1}(a x)^2}{a^2}-\frac {\int \frac {\cot ^{-1}(a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}\right )+\frac {1}{4} x^4 \cot ^{-1}(a x)^3\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {3}{4} a \left (\frac {\frac {2}{3} a \left (\frac {\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\int \frac {1}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)}{a^2}-\frac {\int \frac {x \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{3} x^3 \cot ^{-1}(a x)^2}{a^2}-\frac {\frac {2 a \int \frac {x \cot ^{-1}(a x)}{a^2 x^2+1}dx+x \cot ^{-1}(a x)^2}{a^2}-\frac {\int \frac {\cot ^{-1}(a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}\right )+\frac {1}{4} x^4 \cot ^{-1}(a x)^3\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {3}{4} a \left (\frac {\frac {2}{3} a \left (\frac {\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)}{a^2}-\frac {\int \frac {x \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{3} x^3 \cot ^{-1}(a x)^2}{a^2}-\frac {\frac {2 a \int \frac {x \cot ^{-1}(a x)}{a^2 x^2+1}dx+x \cot ^{-1}(a x)^2}{a^2}-\frac {\int \frac {\cot ^{-1}(a x)^2}{a^2 x^2+1}dx}{a^2}}{a^2}\right )+\frac {1}{4} x^4 \cot ^{-1}(a x)^3\) |
\(\Big \downarrow \) 5420 |
\(\displaystyle \frac {3}{4} a \left (\frac {\frac {2}{3} a \left (\frac {\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)}{a^2}-\frac {\int \frac {x \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{3} x^3 \cot ^{-1}(a x)^2}{a^2}-\frac {\frac {2 a \int \frac {x \cot ^{-1}(a x)}{a^2 x^2+1}dx+x \cot ^{-1}(a x)^2}{a^2}+\frac {\cot ^{-1}(a x)^3}{3 a^3}}{a^2}\right )+\frac {1}{4} x^4 \cot ^{-1}(a x)^3\) |
\(\Big \downarrow \) 5456 |
\(\displaystyle \frac {1}{4} x^4 \cot ^{-1}(a x)^3+\frac {3}{4} a \left (\frac {\frac {1}{3} x^3 \cot ^{-1}(a x)^2+\frac {2}{3} a \left (\frac {\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)}{a^2}-\frac {\frac {i \cot ^{-1}(a x)^2}{2 a^2}-\frac {\int \frac {\cot ^{-1}(a x)}{i-a x}dx}{a}}{a^2}\right )}{a^2}-\frac {\frac {\cot ^{-1}(a x)^3}{3 a^3}+\frac {x \cot ^{-1}(a x)^2+2 a \left (\frac {i \cot ^{-1}(a x)^2}{2 a^2}-\frac {\int \frac {\cot ^{-1}(a x)}{i-a x}dx}{a}\right )}{a^2}}{a^2}\right )\) |
\(\Big \downarrow \) 5380 |
\(\displaystyle \frac {1}{4} x^4 \cot ^{-1}(a x)^3+\frac {3}{4} a \left (\frac {\frac {1}{3} x^3 \cot ^{-1}(a x)^2+\frac {2}{3} a \left (\frac {\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)}{a^2}-\frac {\frac {i \cot ^{-1}(a x)^2}{2 a^2}-\frac {\int \frac {\log \left (\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx+\frac {\log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)}{a}}{a}}{a^2}\right )}{a^2}-\frac {\frac {\cot ^{-1}(a x)^3}{3 a^3}+\frac {x \cot ^{-1}(a x)^2+2 a \left (\frac {i \cot ^{-1}(a x)^2}{2 a^2}-\frac {\int \frac {\log \left (\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx+\frac {\log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)}{a}}{a}\right )}{a^2}}{a^2}\right )\) |
\(\Big \downarrow \) 2849 |
\(\displaystyle \frac {1}{4} x^4 \cot ^{-1}(a x)^3+\frac {3}{4} a \left (\frac {\frac {1}{3} x^3 \cot ^{-1}(a x)^2+\frac {2}{3} a \left (\frac {\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)}{a^2}-\frac {\frac {i \cot ^{-1}(a x)^2}{2 a^2}-\frac {\frac {\log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)}{a}-\frac {i \int \frac {\log \left (\frac {2}{i a x+1}\right )}{1-\frac {2}{i a x+1}}d\frac {1}{i a x+1}}{a}}{a}}{a^2}\right )}{a^2}-\frac {\frac {\cot ^{-1}(a x)^3}{3 a^3}+\frac {x \cot ^{-1}(a x)^2+2 a \left (\frac {i \cot ^{-1}(a x)^2}{2 a^2}-\frac {\frac {\log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)}{a}-\frac {i \int \frac {\log \left (\frac {2}{i a x+1}\right )}{1-\frac {2}{i a x+1}}d\frac {1}{i a x+1}}{a}}{a}\right )}{a^2}}{a^2}\right )\) |
\(\Big \downarrow \) 2752 |
\(\displaystyle \frac {1}{4} x^4 \cot ^{-1}(a x)^3+\frac {3}{4} a \left (\frac {\frac {1}{3} x^3 \cot ^{-1}(a x)^2+\frac {2}{3} a \left (\frac {\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)}{a^2}-\frac {\frac {i \cot ^{-1}(a x)^2}{2 a^2}-\frac {\frac {\log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)}{a}-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}}{a}}{a^2}\right )}{a^2}-\frac {\frac {\cot ^{-1}(a x)^3}{3 a^3}+\frac {x \cot ^{-1}(a x)^2+2 a \left (\frac {i \cot ^{-1}(a x)^2}{2 a^2}-\frac {\frac {\log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)}{a}-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}}{a}\right )}{a^2}}{a^2}\right )\) |
Input:
Int[x^3*ArcCot[a*x]^3,x]
Output:
(x^4*ArcCot[a*x]^3)/4 + (3*a*(((x^3*ArcCot[a*x]^2)/3 + (2*a*(((x^2*ArcCot[ a*x])/2 + (a*(x/a^2 - ArcTan[a*x]/a^3))/2)/a^2 - (((I/2)*ArcCot[a*x]^2)/a^ 2 - ((ArcCot[a*x]*Log[2/(1 + I*a*x)])/a - ((I/2)*PolyLog[2, 1 - 2/(1 + I*a *x)])/a)/a)/a^2))/3)/a^2 - (ArcCot[a*x]^3/(3*a^3) + (x*ArcCot[a*x]^2 + 2*a *(((I/2)*ArcCot[a*x]^2)/a^2 - ((ArcCot[a*x]*Log[2/(1 + I*a*x)])/a - ((I/2) *PolyLog[2, 1 - 2/(1 + I*a*x)])/a)/a))/a^2)/a^2))/4
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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_.) + 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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 856 vs. \(2 (130 ) = 260\).
Time = 6.48 (sec) , antiderivative size = 857, normalized size of antiderivative = 5.79
method | result | size |
risch | \(-\frac {i \ln \left (-i a x +1\right )^{3}}{32 a^{4}}+\frac {25 i \ln \left (-i a x +1\right )^{2}}{128 a^{4}}-\frac {3 i \ln \left (-i a x +1\right )^{2} x^{4}}{128}+\frac {3 i \ln \left (-i a x +1\right ) x^{4}}{256}-\frac {i \operatorname {dilog}\left (\frac {1}{2}-\frac {i a x}{2}\right )}{a^{4}}+\frac {23 i \ln \left (-i a x +1\right )}{48 a^{4}}+\frac {3 \ln \left (-i a x +1\right )^{2} x}{32 a^{3}}+\frac {7 \ln \left (-i a x +1\right ) x}{64 a^{3}}+\frac {3 \pi \ln \left (-i a x +1\right )^{2}}{32 a^{4}}-\frac {7 \pi \ln \left (-i a x +1\right )}{64 a^{4}}-\frac {3 \pi \ln \left (-i a x +1\right )^{2} x^{4}}{32}-\frac {\left (-3 i a^{4} \ln \left (-i a x +1\right ) x^{4}+3 \pi \,a^{4} x^{4}+2 a^{3} x^{3}+3 i \ln \left (-i a x +1\right )-6 a x -3 \pi +8 i\right ) \ln \left (i a x +1\right )^{2}}{32 a^{4}}-\frac {\ln \left (-i a x +1\right )^{2} x^{3}}{32 a}-\frac {\ln \left (-i a x +1\right ) x^{3}}{192 a}-\frac {i \pi ^{2}}{4 a^{4}}-\frac {319 i \ln \left (a^{2} x^{2}+1\right )}{1536 a^{4}}-\frac {\pi ^{3}}{32 a^{4}}+\frac {\pi }{8 a^{4}}-\frac {i \ln \left (\frac {1}{2}-\frac {i a x}{2}\right ) \ln \left (\frac {1}{2}+\frac {i a x}{2}\right )}{a^{4}}+\frac {x}{4 a^{3}}-\frac {511 \arctan \left (a x \right )}{768 a^{4}}+\left (-\frac {3 i \left (a^{4} x^{4}-1\right ) \ln \left (-i a x +1\right )^{2}}{32 a^{4}}+\frac {x \left (3 \pi \,a^{3} x^{3}+2 a^{2} x^{2}-6\right ) \ln \left (-i a x +1\right )}{16 a^{3}}-\frac {-3 i \pi ^{2} a^{4} x^{4}-4 i \pi \,a^{3} x^{3}-4 i x^{2} a^{2}+12 i \pi a x +6 \ln \left (-i a x +1\right ) \pi +16 i \ln \left (-i a x +1\right )}{32 a^{4}}\right ) \ln \left (i a x +1\right )+\frac {i}{4 a^{4}}+\frac {i \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (-i a x +1\right )}{a^{4}}+\frac {i \left (-i a x +1\right )^{3} \ln \left (-i a x +1\right )}{24 a^{4}}-\frac {i \left (-i a x +1\right )^{3} \ln \left (-i a x +1\right )^{2}}{16 a^{4}}-\frac {3 i \left (-i a x +1\right )^{4} \ln \left (-i a x +1\right )}{256 a^{4}}+\frac {3 i \left (-i a x +1\right )^{4} \ln \left (-i a x +1\right )^{2}}{128 a^{4}}-\frac {3 i \left (-i a x +1\right )^{2} \ln \left (-i a x +1\right )}{32 a^{4}}+\frac {3 i \left (-i a x +1\right )^{2} \ln \left (-i a x +1\right )^{2}}{32 a^{4}}-\frac {21 i \ln \left (-i a x +1\right ) x^{2}}{128 a^{2}}-\frac {7 i \pi \arctan \left (a x \right )}{64 a^{4}}-\frac {i \left (a^{4} x^{4}-1\right ) \ln \left (i a x +1\right )^{3}}{32 a^{4}}-\frac {3 i \pi ^{2} \ln \left (-i a x +1\right ) x^{4}}{32}+\frac {3 i \ln \left (-i a x +1\right )^{2} x^{2}}{64 a^{2}}+\frac {\pi ^{2} x^{3}}{16 a}+\frac {\pi \,x^{2}}{8 a^{2}}+\frac {3 i \pi \ln \left (-i a x +1\right ) x}{8 a^{3}}-\frac {i \pi \ln \left (-i a x +1\right ) x^{3}}{8 a}+\frac {3 \pi ^{2} \arctan \left (a x \right )}{16 a^{4}}-\frac {57 \pi \ln \left (a^{2} x^{2}+1\right )}{128 a^{4}}+\frac {i \ln \left (-i a x +1\right )^{3} x^{4}}{32}+\frac {x^{4} \pi ^{3}}{32}-\frac {3 \pi ^{2} x}{16 a^{3}}\) | \(857\) |
parts | \(\text {Expression too large to display}\) | \(1015\) |
derivativedivides | \(\text {Expression too large to display}\) | \(1017\) |
default | \(\text {Expression too large to display}\) | \(1017\) |
Input:
int(x^3*arccot(a*x)^3,x,method=_RETURNVERBOSE)
Output:
(-3/32*I*(a^4*x^4-1)/a^4*ln(1-I*a*x)^2+1/16*x*(3*Pi*a^3*x^3+2*a^2*x^2-6)/a ^3*ln(1-I*a*x)-1/32*(-3*I*Pi^2*a^4*x^4-4*I*Pi*a^3*x^3-4*I*x^2*a^2+12*I*Pi* a*x+6*ln(1-I*a*x)*Pi+16*I*ln(1-I*a*x))/a^4)*ln(1+I*a*x)-1/32/a^4*Pi^3+1/8/ a^4*Pi+1/4*I/a^4+I/a^4*ln(1/2+1/2*I*a*x)*ln(1-I*a*x)+1/4*x/a^3-511/768*arc tan(a*x)/a^4+1/24*I/a^4*(1-I*a*x)^3*ln(1-I*a*x)-1/16*I/a^4*(1-I*a*x)^3*ln( 1-I*a*x)^2-3/256*I/a^4*(1-I*a*x)^4*ln(1-I*a*x)+3/128*I/a^4*(1-I*a*x)^4*ln( 1-I*a*x)^2+1/32*I*ln(1-I*a*x)^3*x^4-3/32*I/a^4*(1-I*a*x)^2*ln(1-I*a*x)+3/3 2*I/a^4*(1-I*a*x)^2*ln(1-I*a*x)^2-1/32*I/a^4*ln(1-I*a*x)^3+25/128*I/a^4*ln (1-I*a*x)^2-3/128*I*ln(1-I*a*x)^2*x^4+3/256*I*ln(1-I*a*x)*x^4+1/16/a*Pi^2* x^3+1/8/a^2*Pi*x^2-21/128*I/a^2*ln(1-I*a*x)*x^2-7/64*I/a^4*Pi*arctan(a*x)+ 3/16/a^4*Pi^2*arctan(a*x)-57/128/a^4*Pi*ln(a^2*x^2+1)-I/a^4*dilog(1/2-1/2* I*a*x)+23/48*I/a^4*ln(1-I*a*x)-1/32*I*(a^4*x^4-1)/a^4*ln(1+I*a*x)^3-3/32*I *Pi^2*ln(1-I*a*x)*x^4+3/64*I/a^2*ln(1-I*a*x)^2*x^2+3/8*I/a^3*Pi*ln(1-I*a*x )*x-1/8*I/a*Pi*ln(1-I*a*x)*x^3+3/32/a^3*ln(1-I*a*x)^2*x+7/64/a^3*ln(1-I*a* x)*x+1/32*x^4*Pi^3-3/16/a^3*Pi^2*x+3/32/a^4*Pi*ln(1-I*a*x)^2-7/64/a^4*Pi*l n(1-I*a*x)-3/32*Pi*ln(1-I*a*x)^2*x^4-1/32*(-3*I*x^4*ln(1-I*a*x)*a^4+3*Pi*a ^4*x^4+2*a^3*x^3+3*I*ln(1-I*a*x)-6*a*x-3*Pi+8*I)/a^4*ln(1+I*a*x)^2-1/32/a* ln(1-I*a*x)^2*x^3-1/192/a*ln(1-I*a*x)*x^3-1/4*I/a^4*Pi^2-319/1536*I/a^4*ln (a^2*x^2+1)-I/a^4*ln(1/2-1/2*I*a*x)*ln(1/2+1/2*I*a*x)
\[ \int x^3 \cot ^{-1}(a x)^3 \, dx=\int { x^{3} \operatorname {arccot}\left (a x\right )^{3} \,d x } \] Input:
integrate(x^3*arccot(a*x)^3,x, algorithm="fricas")
Output:
integral(x^3*arccot(a*x)^3, x)
\[ \int x^3 \cot ^{-1}(a x)^3 \, dx=\int x^{3} \operatorname {acot}^{3}{\left (a x \right )}\, dx \] Input:
integrate(x**3*acot(a*x)**3,x)
Output:
Integral(x**3*acot(a*x)**3, x)
\[ \int x^3 \cot ^{-1}(a x)^3 \, dx=\int { x^{3} \operatorname {arccot}\left (a x\right )^{3} \,d x } \] Input:
integrate(x^3*arccot(a*x)^3,x, algorithm="maxima")
Output:
1/64*(8*a^4*x^4*arctan2(1, a*x)^3 + 4*a^3*x^3*arctan2(1, a*x)^2 + 4*(512*a ^5*integrate(1/64*x^5*arctan(1/(a*x))^3/(a^5*x^2 + a^3), x) + 192*a^4*inte grate(1/64*x^4*arctan(1/(a*x))^2/(a^5*x^2 + a^3), x) + 48*a^4*integrate(1/ 64*x^4*log(a^2*x^2 + 1)^2/(a^5*x^2 + a^3), x) + 64*a^4*integrate(1/64*x^4* log(a^2*x^2 + 1)/(a^5*x^2 + a^3), x) + 512*a^3*integrate(1/64*x^3*arctan(1 /(a*x))^3/(a^5*x^2 + a^3), x) + 128*a^3*integrate(1/64*x^3*arctan(1/(a*x)) /(a^5*x^2 + a^3), x) - 192*a^2*integrate(1/64*x^2*log(a^2*x^2 + 1)/(a^5*x^ 2 + a^3), x) - 384*a*integrate(1/64*x*arctan(1/(a*x))/(a^5*x^2 + a^3), x) - arctan(a*x)^3/a^4 - 3*arctan(a*x)^2*arctan(1/(a*x))/a^4 - 3*arctan(a*x)* arctan(1/(a*x))^2/a^4 - 48*integrate(1/64*log(a^2*x^2 + 1)^2/(a^5*x^2 + a^ 3), x))*a^4 - 12*a*x*arctan2(1, a*x)^2 - 8*arctan2(1, a*x)^3 - (a^3*x^3 - 3*a*x)*log(a^2*x^2 + 1)^2)/a^4
\[ \int x^3 \cot ^{-1}(a x)^3 \, dx=\int { x^{3} \operatorname {arccot}\left (a x\right )^{3} \,d x } \] Input:
integrate(x^3*arccot(a*x)^3,x, algorithm="giac")
Output:
integrate(x^3*arccot(a*x)^3, x)
Timed out. \[ \int x^3 \cot ^{-1}(a x)^3 \, dx=\int x^3\,{\mathrm {acot}\left (a\,x\right )}^3 \,d x \] Input:
int(x^3*acot(a*x)^3,x)
Output:
int(x^3*acot(a*x)^3, x)
\[ \int x^3 \cot ^{-1}(a x)^3 \, dx=\frac {\mathit {acot} \left (a x \right )^{3} a^{4} x^{4}-\mathit {acot} \left (a x \right )^{3}+\mathit {acot} \left (a x \right )^{2} a^{3} x^{3}-3 \mathit {acot} \left (a x \right )^{2} a x +\mathit {acot} \left (a x \right ) a^{2} x^{2}+\mathit {acot} \left (a x \right )-8 \left (\int \frac {\mathit {acot} \left (a x \right ) x}{a^{2} x^{2}+1}d x \right ) a^{2}+a x}{4 a^{4}} \] Input:
int(x^3*acot(a*x)^3,x)
Output:
(acot(a*x)**3*a**4*x**4 - acot(a*x)**3 + acot(a*x)**2*a**3*x**3 - 3*acot(a *x)**2*a*x + acot(a*x)*a**2*x**2 + acot(a*x) - 8*int((acot(a*x)*x)/(a**2*x **2 + 1),x)*a**2 + a*x)/(4*a**4)