Integrand size = 14, antiderivative size = 145 \[ \int x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=\frac {3}{2} i b c^2 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {3}{2} b c x \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {1}{2} c^2 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3+\frac {1}{2} x^2 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3-3 b^2 c^2 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right ) \log \left (2-\frac {2}{1-\frac {i c}{x}}\right )+\frac {3}{2} i b^3 c^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-\frac {i c}{x}}\right ) \]
3/2*I*b*c^2*(a+b*arccot(x/c))^2+3/2*b*c*x*(a+b*arccot(x/c))^2+1/2*c^2*(a+b *arccot(x/c))^3+1/2*x^2*(a+b*arccot(x/c))^3-3*b^2*c^2*(a+b*arccot(x/c))*ln (2-2/(1-I*c/x))+3/2*I*b^3*c^2*polylog(2,-1+2/(1-I*c/x))
Time = 0.29 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.20 \[ \int x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=\frac {1}{2} \left (3 b^2 \left (b c (i c+x)+a \left (c^2+x^2\right )\right ) \arctan \left (\frac {c}{x}\right )^2+b^3 \left (c^2+x^2\right ) \arctan \left (\frac {c}{x}\right )^3+3 b \arctan \left (\frac {c}{x}\right ) \left (a \left (2 b c x+a \left (c^2+x^2\right )\right )-2 b^2 c^2 \log \left (1-e^{2 i \arctan \left (\frac {c}{x}\right )}\right )\right )+a \left (a x (3 b c+a x)-6 b^2 c^2 \log \left (\frac {c}{\sqrt {1+\frac {c^2}{x^2}} x}\right )\right )+3 i b^3 c^2 \operatorname {PolyLog}\left (2,e^{2 i \arctan \left (\frac {c}{x}\right )}\right )\right ) \]
(3*b^2*(b*c*(I*c + x) + a*(c^2 + x^2))*ArcTan[c/x]^2 + b^3*(c^2 + x^2)*Arc Tan[c/x]^3 + 3*b*ArcTan[c/x]*(a*(2*b*c*x + a*(c^2 + x^2)) - 2*b^2*c^2*Log[ 1 - E^((2*I)*ArcTan[c/x])]) + a*(a*x*(3*b*c + a*x) - 6*b^2*c^2*Log[c/(Sqrt [1 + c^2/x^2]*x)]) + (3*I)*b^3*c^2*PolyLog[2, E^((2*I)*ArcTan[c/x])])/2
Time = 0.87 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5363, 5361, 5453, 5361, 5419, 5459, 5403, 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.
| \(\displaystyle \int x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx\) |
| \(\Big \downarrow \) 5363 |
| \(\displaystyle -\int x^3 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3d\frac {1}{x}\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {1}{2} x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3-\frac {3}{2} b c \int \frac {x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{\frac {c^2}{x^2}+1}d\frac {1}{x}\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle \frac {1}{2} x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3-\frac {3}{2} b c \left (\int x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2d\frac {1}{x}-c^2 \int \frac {\left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{\frac {c^2}{x^2}+1}d\frac {1}{x}\right )\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {1}{2} x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3-\frac {3}{2} b c \left (c^2 \left (-\int \frac {\left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{\frac {c^2}{x^2}+1}d\frac {1}{x}\right )+2 b c \int \frac {x \left (a+b \arctan \left (\frac {c}{x}\right )\right )}{\frac {c^2}{x^2}+1}d\frac {1}{x}-x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2\right )\) |
| \(\Big \downarrow \) 5419 |
| \(\displaystyle \frac {1}{2} x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3-\frac {3}{2} b c \left (2 b c \int \frac {x \left (a+b \arctan \left (\frac {c}{x}\right )\right )}{\frac {c^2}{x^2}+1}d\frac {1}{x}-\frac {c \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3}{3 b}-x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2\right )\) |
| \(\Big \downarrow \) 5459 |
| \(\displaystyle \frac {1}{2} x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3-\frac {3}{2} b c \left (2 b c \left (i \int \frac {x \left (a+b \arctan \left (\frac {c}{x}\right )\right )}{\frac {c}{x}+i}d\frac {1}{x}-\frac {i \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{2 b}\right )-\frac {c \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3}{3 b}-x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2\right )\) |
| \(\Big \downarrow \) 5403 |
| \(\displaystyle \frac {1}{2} x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3-\frac {3}{2} b c \left (2 b c \left (i \left (i b c \int \frac {\log \left (2-\frac {2}{1-\frac {i c}{x}}\right )}{\frac {c^2}{x^2}+1}d\frac {1}{x}-i \log \left (2-\frac {2}{1-\frac {i c}{x}}\right ) \left (a+b \arctan \left (\frac {c}{x}\right )\right )\right )-\frac {i \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{2 b}\right )-\frac {c \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3}{3 b}-x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2897 |
| \(\displaystyle \frac {1}{2} x^2 \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3-\frac {3}{2} b c \left (2 b c \left (i \left (-i \log \left (2-\frac {2}{1-\frac {i c}{x}}\right ) \left (a+b \arctan \left (\frac {c}{x}\right )\right )-\frac {1}{2} b \operatorname {PolyLog}\left (2,\frac {2}{1-\frac {i c}{x}}-1\right )\right )-\frac {i \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2}{2 b}\right )-\frac {c \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3}{3 b}-x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^2\right )\) |
(x^2*(a + b*ArcTan[c/x])^3)/2 - (3*b*c*(-(x*(a + b*ArcTan[c/x])^2) - (c*(a + b*ArcTan[c/x])^3)/(3*b) + 2*b*c*(((-1/2*I)*(a + b*ArcTan[c/x])^2)/b + I *((-I)*(a + b*ArcTan[c/x])*Log[2 - 2/(1 - (I*c)/x)] - (b*PolyLog[2, -1 + 2 /(1 - (I*c)/x)])/2))))/2
3.2.49.3.1 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_.) + 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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 404 vs. \(2 (131 ) = 262\).
Time = 8.16 (sec) , antiderivative size = 405, normalized size of antiderivative = 2.79
| method | result | size |
| derivativedivides | \(-c^{2} \left (-\frac {a^{3} x^{2}}{2 c^{2}}+b^{3} \left (-\frac {x^{2} \arctan \left (\frac {c}{x}\right )^{3}}{2 c^{2}}-\frac {3 x \arctan \left (\frac {c}{x}\right )^{2}}{2 c}-\frac {\arctan \left (\frac {c}{x}\right )^{3}}{2}+3 \ln \left (\frac {c}{x}\right ) \arctan \left (\frac {c}{x}\right )-\frac {3 \arctan \left (\frac {c}{x}\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )}{2}+\frac {3 i \ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {i c}{x}\right )}{2}-\frac {3 i \ln \left (\frac {c}{x}\right ) \ln \left (1-\frac {i c}{x}\right )}{2}+\frac {3 i \operatorname {dilog}\left (1+\frac {i c}{x}\right )}{2}-\frac {3 i \operatorname {dilog}\left (1-\frac {i c}{x}\right )}{2}-\frac {3 i \left (\ln \left (\frac {c}{x}-i\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )-\frac {\ln \left (\frac {c}{x}-i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (\frac {c}{x}+i\right )}{2}\right )-\ln \left (\frac {c}{x}-i\right ) \ln \left (-\frac {i \left (\frac {c}{x}+i\right )}{2}\right )\right )}{4}+\frac {3 i \left (\ln \left (\frac {c}{x}+i\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )-\frac {\ln \left (\frac {c}{x}+i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )-\ln \left (\frac {c}{x}+i\right ) \ln \left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )\right )}{4}\right )+3 a \,b^{2} \left (-\frac {x^{2} \arctan \left (\frac {c}{x}\right )^{2}}{2 c^{2}}-\frac {\arctan \left (\frac {c}{x}\right )^{2}}{2}-\frac {x \arctan \left (\frac {c}{x}\right )}{c}-\frac {\ln \left (1+\frac {c^{2}}{x^{2}}\right )}{2}+\ln \left (\frac {c}{x}\right )\right )+3 a^{2} b \left (-\frac {x^{2} \arctan \left (\frac {c}{x}\right )}{2 c^{2}}-\frac {x}{2 c}-\frac {\arctan \left (\frac {c}{x}\right )}{2}\right )\right )\) | \(405\) |
| default | \(-c^{2} \left (-\frac {a^{3} x^{2}}{2 c^{2}}+b^{3} \left (-\frac {x^{2} \arctan \left (\frac {c}{x}\right )^{3}}{2 c^{2}}-\frac {3 x \arctan \left (\frac {c}{x}\right )^{2}}{2 c}-\frac {\arctan \left (\frac {c}{x}\right )^{3}}{2}+3 \ln \left (\frac {c}{x}\right ) \arctan \left (\frac {c}{x}\right )-\frac {3 \arctan \left (\frac {c}{x}\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )}{2}+\frac {3 i \ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {i c}{x}\right )}{2}-\frac {3 i \ln \left (\frac {c}{x}\right ) \ln \left (1-\frac {i c}{x}\right )}{2}+\frac {3 i \operatorname {dilog}\left (1+\frac {i c}{x}\right )}{2}-\frac {3 i \operatorname {dilog}\left (1-\frac {i c}{x}\right )}{2}-\frac {3 i \left (\ln \left (\frac {c}{x}-i\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )-\frac {\ln \left (\frac {c}{x}-i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (\frac {c}{x}+i\right )}{2}\right )-\ln \left (\frac {c}{x}-i\right ) \ln \left (-\frac {i \left (\frac {c}{x}+i\right )}{2}\right )\right )}{4}+\frac {3 i \left (\ln \left (\frac {c}{x}+i\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )-\frac {\ln \left (\frac {c}{x}+i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )-\ln \left (\frac {c}{x}+i\right ) \ln \left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )\right )}{4}\right )+3 a \,b^{2} \left (-\frac {x^{2} \arctan \left (\frac {c}{x}\right )^{2}}{2 c^{2}}-\frac {\arctan \left (\frac {c}{x}\right )^{2}}{2}-\frac {x \arctan \left (\frac {c}{x}\right )}{c}-\frac {\ln \left (1+\frac {c^{2}}{x^{2}}\right )}{2}+\ln \left (\frac {c}{x}\right )\right )+3 a^{2} b \left (-\frac {x^{2} \arctan \left (\frac {c}{x}\right )}{2 c^{2}}-\frac {x}{2 c}-\frac {\arctan \left (\frac {c}{x}\right )}{2}\right )\right )\) | \(405\) |
| parts | \(\frac {a^{3} x^{2}}{2}-b^{3} c^{2} \left (-\frac {x^{2} \arctan \left (\frac {c}{x}\right )^{3}}{2 c^{2}}-\frac {3 x \arctan \left (\frac {c}{x}\right )^{2}}{2 c}-\frac {\arctan \left (\frac {c}{x}\right )^{3}}{2}+3 \ln \left (\frac {c}{x}\right ) \arctan \left (\frac {c}{x}\right )-\frac {3 \arctan \left (\frac {c}{x}\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )}{2}+\frac {3 i \ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {i c}{x}\right )}{2}-\frac {3 i \ln \left (\frac {c}{x}\right ) \ln \left (1-\frac {i c}{x}\right )}{2}+\frac {3 i \operatorname {dilog}\left (1+\frac {i c}{x}\right )}{2}-\frac {3 i \operatorname {dilog}\left (1-\frac {i c}{x}\right )}{2}-\frac {3 i \left (\ln \left (\frac {c}{x}-i\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )-\frac {\ln \left (\frac {c}{x}-i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (\frac {c}{x}+i\right )}{2}\right )-\ln \left (\frac {c}{x}-i\right ) \ln \left (-\frac {i \left (\frac {c}{x}+i\right )}{2}\right )\right )}{4}+\frac {3 i \left (\ln \left (\frac {c}{x}+i\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )-\frac {\ln \left (\frac {c}{x}+i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )-\ln \left (\frac {c}{x}+i\right ) \ln \left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )\right )}{4}\right )-3 a \,b^{2} c^{2} \left (-\frac {x^{2} \arctan \left (\frac {c}{x}\right )^{2}}{2 c^{2}}-\frac {\arctan \left (\frac {c}{x}\right )^{2}}{2}-\frac {x \arctan \left (\frac {c}{x}\right )}{c}-\frac {\ln \left (1+\frac {c^{2}}{x^{2}}\right )}{2}+\ln \left (\frac {c}{x}\right )\right )+\frac {3 x^{2} a^{2} b \arctan \left (\frac {c}{x}\right )}{2}-\frac {3 a^{2} b \,c^{2} \arctan \left (\frac {x}{c}\right )}{2}+\frac {3 a^{2} b c x}{2}\) | \(407\) |
| risch | \(\text {Expression too large to display}\) | \(233342\) |
-c^2*(-1/2*a^3/c^2*x^2+b^3*(-1/2/c^2*x^2*arctan(c/x)^3-3/2/c*x*arctan(c/x) ^2-1/2*arctan(c/x)^3+3*ln(c/x)*arctan(c/x)-3/2*arctan(c/x)*ln(1+c^2/x^2)+3 /2*I*ln(c/x)*ln(1+I*c/x)-3/2*I*ln(c/x)*ln(1-I*c/x)+3/2*I*dilog(1+I*c/x)-3/ 2*I*dilog(1-I*c/x)-3/4*I*(ln(c/x-I)*ln(1+c^2/x^2)-1/2*ln(c/x-I)^2-dilog(-1 /2*I*(c/x+I))-ln(c/x-I)*ln(-1/2*I*(c/x+I)))+3/4*I*(ln(c/x+I)*ln(1+c^2/x^2) -1/2*ln(c/x+I)^2-dilog(1/2*I*(c/x-I))-ln(c/x+I)*ln(1/2*I*(c/x-I))))+3*a*b^ 2*(-1/2/c^2*x^2*arctan(c/x)^2-1/2*arctan(c/x)^2-1/c*x*arctan(c/x)-1/2*ln(1 +c^2/x^2)+ln(c/x))+3*a^2*b*(-1/2/c^2*x^2*arctan(c/x)-1/2*x/c-1/2*arctan(c/ x)))
\[ \int x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=\int { {\left (b \arctan \left (\frac {c}{x}\right ) + a\right )}^{3} x \,d x } \]
\[ \int x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=\int x \left (a + b \operatorname {atan}{\left (\frac {c}{x} \right )}\right )^{3}\, dx \]
\[ \int x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=\int { {\left (b \arctan \left (\frac {c}{x}\right ) + a\right )}^{3} x \,d x } \]
3/2*a*b^2*x^2*arctan(c/x)^2 + 1/2*a^3*x^2 + 3/2*(x^2*arctan(c/x) - (c*arct an(x/c) - x)*c)*a^2*b - 3/2*((arctan(x/c)^2 - log(c^2 + x^2))*c^2 + 2*(c*a rctan(x/c) - x)*c*arctan(c/x))*a*b^2 + 1/32*(12*c^2*arctan(c/x)^2*arctan(x /c) + 8*c^2*arctan2(c, x)^3 + 8*x^2*arctan2(c, x)^3 + 4*(3*arctan(c/x)*arc tan(x/c)^2/c + arctan(x/c)^3/c)*c^3 + 12*c*x*arctan2(c, x)^2 + 96*c^3*inte grate(1/32*log(c^2 + x^2)^2/(c^2 + x^2), x) - 3*c*x*log(c^2 + x^2)^2 + 512 *c^2*integrate(1/32*x*arctan(c/x)^3/(c^2 + x^2), x) + 768*c^2*integrate(1/ 32*x*arctan(c/x)/(c^2 + x^2), x) + 384*c*integrate(1/32*x^2*arctan(c/x)^2/ (c^2 + x^2), x) + 96*c*integrate(1/32*x^2*log(c^2 + x^2)^2/(c^2 + x^2), x) + 384*c*integrate(1/32*x^2*log(c^2 + x^2)/(c^2 + x^2), x) + 512*integrate (1/32*x^3*arctan(c/x)^3/(c^2 + x^2), x))*b^3
\[ \int x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=\int { {\left (b \arctan \left (\frac {c}{x}\right ) + a\right )}^{3} x \,d x } \]
Timed out. \[ \int x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=\int x\,{\left (a+b\,\mathrm {atan}\left (\frac {c}{x}\right )\right )}^3 \,d x \]