Integrand size = 16, antiderivative size = 229 \[ \int x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=b^2 c^2 x \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )+\frac {1}{2} b c^3 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {1}{2} b c x^2 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2-\frac {1}{3} i c^3 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3+\frac {1}{3} x^3 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3+b c^3 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2 \log \left (2-\frac {2}{1-\frac {i c}{x}}\right )+\frac {1}{2} b^3 c^3 \log \left (1+\frac {c^2}{x^2}\right )+b^3 c^3 \log (x)-i b^2 c^3 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right ) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-\frac {i c}{x}}\right )+\frac {1}{2} b^3 c^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-\frac {i c}{x}}\right ) \]
b^2*c^2*x*(a+b*arccot(x/c))+1/2*b*c^3*(a+b*arccot(x/c))^2+1/2*b*c*x^2*(a+b *arccot(x/c))^2-1/3*I*c^3*(a+b*arccot(x/c))^3+1/3*x^3*(a+b*arccot(x/c))^3+ b*c^3*(a+b*arccot(x/c))^2*ln(2-2/(1-I*c/x))+1/2*b^3*c^3*ln(1+c^2/x^2)+b^3* c^3*ln(x)-I*b^2*c^3*(a+b*arccot(x/c))*polylog(2,-1+2/(1-I*c/x))+1/2*b^3*c^ 3*polylog(3,-1+2/(1-I*c/x))
Time = 0.96 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.47 \[ \int x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=\frac {1}{6} \left (3 a^2 b c x^2+2 a^3 x^3+6 a^2 b x^3 \arctan \left (\frac {c}{x}\right )-3 a^2 b c^3 \log \left (c^2+x^2\right )+6 a b^2 \left (c^2 x+\left (-i c^3+x^3\right ) \arctan \left (\frac {c}{x}\right )^2+c \arctan \left (\frac {c}{x}\right ) \left (c^2+x^2+2 c^2 \log \left (1-e^{2 i \arctan \left (\frac {c}{x}\right )}\right )\right )-i c^3 \operatorname {PolyLog}\left (2,e^{2 i \arctan \left (\frac {c}{x}\right )}\right )\right )+\frac {1}{4} b^3 \left (-i c^3 \pi ^3+24 c^2 x \arctan \left (\frac {c}{x}\right )+12 c^3 \arctan \left (\frac {c}{x}\right )^2+12 c x^2 \arctan \left (\frac {c}{x}\right )^2+8 i c^3 \arctan \left (\frac {c}{x}\right )^3+8 x^3 \arctan \left (\frac {c}{x}\right )^3+24 c^3 \arctan \left (\frac {c}{x}\right )^2 \log \left (1-e^{-2 i \arctan \left (\frac {c}{x}\right )}\right )-24 c^3 \log \left (\frac {1}{\sqrt {1+\frac {c^2}{x^2}}}\right )-24 c^3 \log \left (\frac {c}{x}\right )+24 i c^3 \arctan \left (\frac {c}{x}\right ) \operatorname {PolyLog}\left (2,e^{-2 i \arctan \left (\frac {c}{x}\right )}\right )+12 c^3 \operatorname {PolyLog}\left (3,e^{-2 i \arctan \left (\frac {c}{x}\right )}\right )\right )\right ) \]
(3*a^2*b*c*x^2 + 2*a^3*x^3 + 6*a^2*b*x^3*ArcTan[c/x] - 3*a^2*b*c^3*Log[c^2 + x^2] + 6*a*b^2*(c^2*x + ((-I)*c^3 + x^3)*ArcTan[c/x]^2 + c*ArcTan[c/x]* (c^2 + x^2 + 2*c^2*Log[1 - E^((2*I)*ArcTan[c/x])]) - I*c^3*PolyLog[2, E^(( 2*I)*ArcTan[c/x])]) + (b^3*((-I)*c^3*Pi^3 + 24*c^2*x*ArcTan[c/x] + 12*c^3* ArcTan[c/x]^2 + 12*c*x^2*ArcTan[c/x]^2 + (8*I)*c^3*ArcTan[c/x]^3 + 8*x^3*A rcTan[c/x]^3 + 24*c^3*ArcTan[c/x]^2*Log[1 - E^((-2*I)*ArcTan[c/x])] - 24*c ^3*Log[1/Sqrt[1 + c^2/x^2]] - 24*c^3*Log[c/x] + (24*I)*c^3*ArcTan[c/x]*Pol yLog[2, E^((-2*I)*ArcTan[c/x])] + 12*c^3*PolyLog[3, E^((-2*I)*ArcTan[c/x]) ]))/4)/6
Time = 1.64 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.04, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.938, Rules used = {5363, 5361, 5453, 5361, 5453, 5361, 243, 47, 14, 16, 5419, 5459, 5403, 5527, 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.
| \(\displaystyle \int x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx\) |
| \(\Big \downarrow \) 5363 |
| \(\displaystyle -\int x^4 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3d\frac {1}{x}\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {1}{3} x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3-b c \int \frac {x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{\frac {c^2}{x^2}+1}d\frac {1}{x}\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle \frac {1}{3} x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3-b c \left (\int x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2d\frac {1}{x}-c^2 \int \frac {x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{\frac {c^2}{x^2}+1}d\frac {1}{x}\right )\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {1}{3} x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3-b c \left (c^2 \left (-\int \frac {x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{\frac {c^2}{x^2}+1}d\frac {1}{x}\right )+b c \int \frac {x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )}{\frac {c^2}{x^2}+1}d\frac {1}{x}-\frac {1}{2} x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2\right )\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle \frac {1}{3} x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3-b c \left (c^2 \left (-\int \frac {x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{\frac {c^2}{x^2}+1}d\frac {1}{x}\right )+b c \left (\int x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )d\frac {1}{x}-c^2 \int \frac {a+b \arctan \left (\frac {c}{x}\right )}{\frac {c^2}{x^2}+1}d\frac {1}{x}\right )-\frac {1}{2} x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2\right )\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {1}{3} x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3-b c \left (c^2 \left (-\int \frac {x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{\frac {c^2}{x^2}+1}d\frac {1}{x}\right )+b c \left (c^2 \left (-\int \frac {a+b \arctan \left (\frac {c}{x}\right )}{\frac {c^2}{x^2}+1}d\frac {1}{x}\right )+b c \int \frac {x}{\frac {c^2}{x^2}+1}d\frac {1}{x}-x \left (a+b \arctan \left (\frac {c}{x}\right )\right )\right )-\frac {1}{2} x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {1}{3} x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3-b c \left (c^2 \left (-\int \frac {x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{\frac {c^2}{x^2}+1}d\frac {1}{x}\right )+b c \left (c^2 \left (-\int \frac {a+b \arctan \left (\frac {c}{x}\right )}{\frac {c^2}{x^2}+1}d\frac {1}{x}\right )+\frac {1}{2} b c \int \frac {x}{\frac {c^2}{x^2}+1}d\frac {1}{x^2}-x \left (a+b \arctan \left (\frac {c}{x}\right )\right )\right )-\frac {1}{2} x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2\right )\) |
| \(\Big \downarrow \) 47 |
| \(\displaystyle \frac {1}{3} x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3-b c \left (c^2 \left (-\int \frac {x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{\frac {c^2}{x^2}+1}d\frac {1}{x}\right )+b c \left (c^2 \left (-\int \frac {a+b \arctan \left (\frac {c}{x}\right )}{\frac {c^2}{x^2}+1}d\frac {1}{x}\right )+\frac {1}{2} b c \left (\int xd\frac {1}{x^2}-c^2 \int \frac {1}{\frac {c^2}{x^2}+1}d\frac {1}{x^2}\right )-x \left (a+b \arctan \left (\frac {c}{x}\right )\right )\right )-\frac {1}{2} x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2\right )\) |
| \(\Big \downarrow \) 14 |
| \(\displaystyle \frac {1}{3} x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3-b c \left (c^2 \left (-\int \frac {x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{\frac {c^2}{x^2}+1}d\frac {1}{x}\right )+b c \left (c^2 \left (-\int \frac {a+b \arctan \left (\frac {c}{x}\right )}{\frac {c^2}{x^2}+1}d\frac {1}{x}\right )+\frac {1}{2} b c \left (\log \left (\frac {1}{x^2}\right )-c^2 \int \frac {1}{\frac {c^2}{x^2}+1}d\frac {1}{x^2}\right )-x \left (a+b \arctan \left (\frac {c}{x}\right )\right )\right )-\frac {1}{2} x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{3} x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3-b c \left (c^2 \left (-\int \frac {x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{\frac {c^2}{x^2}+1}d\frac {1}{x}\right )+b c \left (c^2 \left (-\int \frac {a+b \arctan \left (\frac {c}{x}\right )}{\frac {c^2}{x^2}+1}d\frac {1}{x}\right )-x \left (a+b \arctan \left (\frac {c}{x}\right )\right )+\frac {1}{2} b c \left (\log \left (\frac {1}{x^2}\right )-\log \left (\frac {c^2}{x^2}+1\right )\right )\right )-\frac {1}{2} x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2\right )\) |
| \(\Big \downarrow \) 5419 |
| \(\displaystyle \frac {1}{3} x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3-b c \left (c^2 \left (-\int \frac {x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{\frac {c^2}{x^2}+1}d\frac {1}{x}\right )+b c \left (-\frac {c \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{2 b}-x \left (a+b \arctan \left (\frac {c}{x}\right )\right )+\frac {1}{2} b c \left (\log \left (\frac {1}{x^2}\right )-\log \left (\frac {c^2}{x^2}+1\right )\right )\right )-\frac {1}{2} x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2\right )\) |
| \(\Big \downarrow \) 5459 |
| \(\displaystyle \frac {1}{3} x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3-b c \left (-\left (c^2 \left (i \int \frac {x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{\frac {c}{x}+i}d\frac {1}{x}-\frac {i \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3}{3 b}\right )\right )+b c \left (-\frac {c \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{2 b}-x \left (a+b \arctan \left (\frac {c}{x}\right )\right )+\frac {1}{2} b c \left (\log \left (\frac {1}{x^2}\right )-\log \left (\frac {c^2}{x^2}+1\right )\right )\right )-\frac {1}{2} x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2\right )\) |
| \(\Big \downarrow \) 5403 |
| \(\displaystyle \frac {1}{3} x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3-b c \left (-\left (c^2 \left (i \left (2 i b c \int \frac {\left (a+b \arctan \left (\frac {c}{x}\right )\right ) \log \left (2-\frac {2}{1-\frac {i c}{x}}\right )}{\frac {c^2}{x^2}+1}d\frac {1}{x}-i \log \left (2-\frac {2}{1-\frac {i c}{x}}\right ) \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2\right )-\frac {i \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3}{3 b}\right )\right )+b c \left (-\frac {c \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{2 b}-x \left (a+b \arctan \left (\frac {c}{x}\right )\right )+\frac {1}{2} b c \left (\log \left (\frac {1}{x^2}\right )-\log \left (\frac {c^2}{x^2}+1\right )\right )\right )-\frac {1}{2} x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2\right )\) |
| \(\Big \downarrow \) 5527 |
| \(\displaystyle \frac {1}{3} x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3-b c \left (-\left (c^2 \left (i \left (2 i b c \left (\frac {i \operatorname {PolyLog}\left (2,\frac {2}{1-\frac {i c}{x}}-1\right ) \left (a+b \arctan \left (\frac {c}{x}\right )\right )}{2 c}-\frac {1}{2} i b \int \frac {\operatorname {PolyLog}\left (2,\frac {2}{1-\frac {i c}{x}}-1\right )}{\frac {c^2}{x^2}+1}d\frac {1}{x}\right )-i \log \left (2-\frac {2}{1-\frac {i c}{x}}\right ) \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2\right )-\frac {i \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3}{3 b}\right )\right )+b c \left (-\frac {c \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{2 b}-x \left (a+b \arctan \left (\frac {c}{x}\right )\right )+\frac {1}{2} b c \left (\log \left (\frac {1}{x^2}\right )-\log \left (\frac {c^2}{x^2}+1\right )\right )\right )-\frac {1}{2} x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2\right )\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle \frac {1}{3} x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3-b c \left (-\left (c^2 \left (i \left (2 i b c \left (\frac {i \operatorname {PolyLog}\left (2,\frac {2}{1-\frac {i c}{x}}-1\right ) \left (a+b \arctan \left (\frac {c}{x}\right )\right )}{2 c}-\frac {b \operatorname {PolyLog}\left (3,\frac {2}{1-\frac {i c}{x}}-1\right )}{4 c}\right )-i \log \left (2-\frac {2}{1-\frac {i c}{x}}\right ) \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2\right )-\frac {i \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3}{3 b}\right )\right )+b c \left (-\frac {c \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{2 b}-x \left (a+b \arctan \left (\frac {c}{x}\right )\right )+\frac {1}{2} b c \left (\log \left (\frac {1}{x^2}\right )-\log \left (\frac {c^2}{x^2}+1\right )\right )\right )-\frac {1}{2} x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2\right )\) |
(x^3*(a + b*ArcTan[c/x])^3)/3 - b*c*(-1/2*(x^2*(a + b*ArcTan[c/x])^2) + b* c*(-(x*(a + b*ArcTan[c/x])) - (c*(a + b*ArcTan[c/x])^2)/(2*b) + (b*c*(-Log [1 + c^2/x^2] + Log[x^(-2)]))/2) - c^2*(((-1/3*I)*(a + b*ArcTan[c/x])^3)/b + I*((-I)*(a + b*ArcTan[c/x])^2*Log[2 - 2/(1 - (I*c)/x)] + (2*I)*b*c*(((I /2)*(a + b*ArcTan[c/x])*PolyLog[2, -1 + 2/(1 - (I*c)/x)])/c - (b*PolyLog[3 , -1 + 2/(1 - (I*c)/x)])/(4*c)))))
3.2.48.3.1 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_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTan[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcTan[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_.) + ArcTan[(c_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*ArcTan[c*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 1] && IntegerQ[Simplif y[(m + 1)/n]]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_ Symbol] :> Simp[(a + b*ArcTan[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Si mp[b*c*(p/d) Int[(a + b*ArcTan[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_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbo l] :> Simp[(a + b*ArcTan[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_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e _.)*(x_)^2), x_Symbol] :> Simp[1/d Int[(f*x)^m*(a + b*ArcTan[c*x])^p, x], x] - Simp[e/(d*f^2) Int[(f*x)^(m + 2)*((a + b*ArcTan[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_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[(-I)*((a + b*ArcTan[c*x])^(p + 1)/(b*d*(p + 1))), x] + Si mp[I/d Int[(a + b*ArcTan[c*x])^p/(x*(I + c*x)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]
Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2 ), x_Symbol] :> Simp[I*(a + b*ArcTan[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)), x ] - Simp[b*p*(I/2) Int[(a + b*ArcTan[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 = 2.89 (sec) , antiderivative size = 2634, normalized size of antiderivative = 11.50
\[\text {Expression too large to display}\]
1/4*I*c^3*b^3*arctan(c/x)^2*csgn(I*((1+I*c/x)^2/(1+c^2/x^2)+1)^2)*csgn(I*( (1+I*c/x)^2/(1+c^2/x^2)+1))^2*Pi-1/2*I*c^3*b^3*arctan(c/x)^2*csgn(I*((1+I* c/x)^2/(1+c^2/x^2)-1)/((1+I*c/x)^2/(1+c^2/x^2)+1))*csgn(((1+I*c/x)^2/(1+c^ 2/x^2)-1)/((1+I*c/x)^2/(1+c^2/x^2)+1))^2*Pi+1/2*I*c^3*b^3*arctan(c/x)^2*cs gn(I*(1+I*c/x)/(1+c^2/x^2)^(1/2))*csgn(I*(1+I*c/x)^2/(1+c^2/x^2))^2*Pi+1/2 *I*c^3*b^3*arctan(c/x)^2*csgn(I*((1+I*c/x)^2/(1+c^2/x^2)-1)/((1+I*c/x)^2/( 1+c^2/x^2)+1))*csgn(((1+I*c/x)^2/(1+c^2/x^2)-1)/((1+I*c/x)^2/(1+c^2/x^2)+1 ))*Pi+1/4*I*c^3*b^3*arctan(c/x)^2*csgn(I*(1+I*c/x)^2/(1+c^2/x^2)/((1+I*c/x )^2/(1+c^2/x^2)+1)^2)^2*csgn(I/((1+I*c/x)^2/(1+c^2/x^2)+1)^2)*Pi-1/2*I*c^3 *b^3*arctan(c/x)^2*csgn(I*((1+I*c/x)^2/(1+c^2/x^2)-1)/((1+I*c/x)^2/(1+c^2/ x^2)+1))^2*csgn(I*((1+I*c/x)^2/(1+c^2/x^2)-1))*Pi+a*b^2*c^2*x+1/3*a^3*x^3- 1/2*I*c^3*b^3*arctan(c/x)^2*csgn(I*((1+I*c/x)^2/(1+c^2/x^2)-1)/((1+I*c/x)^ 2/(1+c^2/x^2)+1))^2*csgn(I/((1+I*c/x)^2/(1+c^2/x^2)+1))*Pi-1/4*I*c^3*b^3*a rctan(c/x)^2*csgn(I*(1+I*c/x)/(1+c^2/x^2)^(1/2))^2*csgn(I*(1+I*c/x)^2/(1+c ^2/x^2))*Pi+1/2*a^2*b*c*x^2-c^3*a*b^2*arctan(x/c)+c^3*a^2*b*ln(c/x)-1/2*c^ 3*a^2*b*ln(1+c^2/x^2)+I*c^3*b^3*arctan(c/x)+c^3*b^3*arctan(c/x)^2*ln(2)+c^ 3*b^3*ln(c/x)*arctan(c/x)^2-1/2*c^3*b^3*arctan(c/x)^2*ln(1+c^2/x^2)+c^3*b^ 3*arctan(c/x)^2*ln(1-(1+I*c/x)/(1+c^2/x^2)^(1/2))+c^3*b^3*arctan(c/x)^2*ln ((1+I*c/x)/(1+c^2/x^2)^(1/2))-c^3*b^3*arctan(c/x)^2*ln((1+I*c/x)^2/(1+c^2/ x^2)-1)+c^3*b^3*arctan(c/x)^2*ln((1+I*c/x)/(1+c^2/x^2)^(1/2)+1)-1/3*I*c...
\[ \int x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=\int { {\left (b \arctan \left (\frac {c}{x}\right ) + a\right )}^{3} x^{2} \,d x } \]
integral(b^3*x^2*arctan(c/x)^3 + 3*a*b^2*x^2*arctan(c/x)^2 + 3*a^2*b*x^2*a rctan(c/x) + a^3*x^2, x)
\[ \int x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=\int x^{2} \left (a + b \operatorname {atan}{\left (\frac {c}{x} \right )}\right )^{3}\, dx \]
\[ \int x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=\int { {\left (b \arctan \left (\frac {c}{x}\right ) + a\right )}^{3} x^{2} \,d x } \]
1/24*b^3*x^3*arctan2(c, x)^3 - 1/32*b^3*x^3*arctan2(c, x)*log(c^2 + x^2)^2 + 1/3*a^3*x^3 + 1/2*(2*x^3*arctan(c/x) - (c^2*log(c^2 + x^2) - x^2)*c)*a^ 2*b + integrate(1/32*(4*b^3*c*x^3*arctan2(c, x)^2 + 4*b^3*x^4*arctan2(c, x )*log(c^2 + x^2) + 4*(7*b^3*arctan2(c, x)^3 + 24*a*b^2*arctan2(c, x)^2)*x^ 4 + 4*(7*b^3*c^2*arctan2(c, x)^3 + 24*a*b^2*c^2*arctan2(c, x)^2)*x^2 + (3* b^3*c^2*x^2*arctan2(c, x) + 3*b^3*x^4*arctan2(c, x) - b^3*c*x^3)*log(c^2 + x^2)^2)/(c^2 + x^2), x)
\[ \int x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=\int { {\left (b \arctan \left (\frac {c}{x}\right ) + a\right )}^{3} x^{2} \,d x } \]
Timed out. \[ \int x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=\int x^2\,{\left (a+b\,\mathrm {atan}\left (\frac {c}{x}\right )\right )}^3 \,d x \]