Integrand size = 16, antiderivative size = 154 \[ \int x^5 \left (a+b \arctan \left (c x^2\right )\right )^2 \, dx=\frac {b^2 x^2}{6 c^2}-\frac {b^2 \arctan \left (c x^2\right )}{6 c^3}-\frac {b x^4 \left (a+b \arctan \left (c x^2\right )\right )}{6 c}-\frac {i \left (a+b \arctan \left (c x^2\right )\right )^2}{6 c^3}+\frac {1}{6} x^6 \left (a+b \arctan \left (c x^2\right )\right )^2-\frac {b \left (a+b \arctan \left (c x^2\right )\right ) \log \left (\frac {2}{1+i c x^2}\right )}{3 c^3}-\frac {i b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x^2}\right )}{6 c^3} \]
1/6*b^2*x^2/c^2-1/6*b^2*arctan(c*x^2)/c^3-1/6*b*x^4*(a+b*arctan(c*x^2))/c- 1/6*I*(a+b*arctan(c*x^2))^2/c^3+1/6*x^6*(a+b*arctan(c*x^2))^2-1/3*b*(a+b*a rctan(c*x^2))*ln(2/(1+I*c*x^2))/c^3-1/6*I*b^2*polylog(2,1-2/(1+I*c*x^2))/c ^3
Time = 0.36 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.92 \[ \int x^5 \left (a+b \arctan \left (c x^2\right )\right )^2 \, dx=\frac {b^2 c x^2-a b c^2 x^4+a^2 c^3 x^6+b^2 \left (i+c^3 x^6\right ) \arctan \left (c x^2\right )^2-b \arctan \left (c x^2\right ) \left (b+b c^2 x^4-2 a c^3 x^6+2 b \log \left (1+e^{2 i \arctan \left (c x^2\right )}\right )\right )+a b \log \left (1+c^2 x^4\right )+i b^2 \operatorname {PolyLog}\left (2,-e^{2 i \arctan \left (c x^2\right )}\right )}{6 c^3} \]
(b^2*c*x^2 - a*b*c^2*x^4 + a^2*c^3*x^6 + b^2*(I + c^3*x^6)*ArcTan[c*x^2]^2 - b*ArcTan[c*x^2]*(b + b*c^2*x^4 - 2*a*c^3*x^6 + 2*b*Log[1 + E^((2*I)*Arc Tan[c*x^2])]) + a*b*Log[1 + c^2*x^4] + I*b^2*PolyLog[2, -E^((2*I)*ArcTan[c *x^2])])/(6*c^3)
Time = 0.79 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.11, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5363, 5361, 5451, 5361, 262, 216, 5455, 5379, 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.
| \(\displaystyle \int x^5 \left (a+b \arctan \left (c x^2\right )\right )^2 \, dx\) |
| \(\Big \downarrow \) 5363 |
| \(\displaystyle \frac {1}{2} \int x^4 \left (a+b \arctan \left (c x^2\right )\right )^2dx^2\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} x^6 \left (a+b \arctan \left (c x^2\right )\right )^2-\frac {2}{3} b c \int \frac {x^6 \left (a+b \arctan \left (c x^2\right )\right )}{c^2 x^4+1}dx^2\right )\) |
| \(\Big \downarrow \) 5451 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} x^6 \left (a+b \arctan \left (c x^2\right )\right )^2-\frac {2}{3} b c \left (\frac {\int x^2 \left (a+b \arctan \left (c x^2\right )\right )dx^2}{c^2}-\frac {\int \frac {x^2 \left (a+b \arctan \left (c x^2\right )\right )}{c^2 x^4+1}dx^2}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} x^6 \left (a+b \arctan \left (c x^2\right )\right )^2-\frac {2}{3} b c \left (\frac {\frac {1}{2} x^4 \left (a+b \arctan \left (c x^2\right )\right )-\frac {1}{2} b c \int \frac {x^4}{c^2 x^4+1}dx^2}{c^2}-\frac {\int \frac {x^2 \left (a+b \arctan \left (c x^2\right )\right )}{c^2 x^4+1}dx^2}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} x^6 \left (a+b \arctan \left (c x^2\right )\right )^2-\frac {2}{3} b c \left (\frac {\frac {1}{2} x^4 \left (a+b \arctan \left (c x^2\right )\right )-\frac {1}{2} b c \left (\frac {x^2}{c^2}-\frac {\int \frac {1}{c^2 x^4+1}dx^2}{c^2}\right )}{c^2}-\frac {\int \frac {x^2 \left (a+b \arctan \left (c x^2\right )\right )}{c^2 x^4+1}dx^2}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} x^6 \left (a+b \arctan \left (c x^2\right )\right )^2-\frac {2}{3} b c \left (\frac {\frac {1}{2} x^4 \left (a+b \arctan \left (c x^2\right )\right )-\frac {1}{2} b c \left (\frac {x^2}{c^2}-\frac {\arctan \left (c x^2\right )}{c^3}\right )}{c^2}-\frac {\int \frac {x^2 \left (a+b \arctan \left (c x^2\right )\right )}{c^2 x^4+1}dx^2}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 5455 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} x^6 \left (a+b \arctan \left (c x^2\right )\right )^2-\frac {2}{3} b c \left (\frac {\frac {1}{2} x^4 \left (a+b \arctan \left (c x^2\right )\right )-\frac {1}{2} b c \left (\frac {x^2}{c^2}-\frac {\arctan \left (c x^2\right )}{c^3}\right )}{c^2}-\frac {-\frac {\int \frac {a+b \arctan \left (c x^2\right )}{i-c x^2}dx^2}{c}-\frac {i \left (a+b \arctan \left (c x^2\right )\right )^2}{2 b c^2}}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 5379 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} x^6 \left (a+b \arctan \left (c x^2\right )\right )^2-\frac {2}{3} b c \left (\frac {\frac {1}{2} x^4 \left (a+b \arctan \left (c x^2\right )\right )-\frac {1}{2} b c \left (\frac {x^2}{c^2}-\frac {\arctan \left (c x^2\right )}{c^3}\right )}{c^2}-\frac {-\frac {\frac {\log \left (\frac {2}{1+i c x^2}\right ) \left (a+b \arctan \left (c x^2\right )\right )}{c}-b \int \frac {\log \left (\frac {2}{i c x^2+1}\right )}{c^2 x^4+1}dx^2}{c}-\frac {i \left (a+b \arctan \left (c x^2\right )\right )^2}{2 b c^2}}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} x^6 \left (a+b \arctan \left (c x^2\right )\right )^2-\frac {2}{3} b c \left (\frac {\frac {1}{2} x^4 \left (a+b \arctan \left (c x^2\right )\right )-\frac {1}{2} b c \left (\frac {x^2}{c^2}-\frac {\arctan \left (c x^2\right )}{c^3}\right )}{c^2}-\frac {-\frac {\frac {i b \int \frac {\log \left (\frac {2}{i c x^2+1}\right )}{1-\frac {2}{i c x^2+1}}d\frac {1}{i c x^2+1}}{c}+\frac {\log \left (\frac {2}{1+i c x^2}\right ) \left (a+b \arctan \left (c x^2\right )\right )}{c}}{c}-\frac {i \left (a+b \arctan \left (c x^2\right )\right )^2}{2 b c^2}}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} x^6 \left (a+b \arctan \left (c x^2\right )\right )^2-\frac {2}{3} b c \left (\frac {\frac {1}{2} x^4 \left (a+b \arctan \left (c x^2\right )\right )-\frac {1}{2} b c \left (\frac {x^2}{c^2}-\frac {\arctan \left (c x^2\right )}{c^3}\right )}{c^2}-\frac {-\frac {i \left (a+b \arctan \left (c x^2\right )\right )^2}{2 b c^2}-\frac {\frac {\log \left (\frac {2}{1+i c x^2}\right ) \left (a+b \arctan \left (c x^2\right )\right )}{c}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x^2+1}\right )}{2 c}}{c}}{c^2}\right )\right )\) |
((x^6*(a + b*ArcTan[c*x^2])^2)/3 - (2*b*c*(((x^4*(a + b*ArcTan[c*x^2]))/2 - (b*c*(x^2/c^2 - ArcTan[c*x^2]/c^3))/2)/c^2 - (((-1/2*I)*(a + b*ArcTan[c* x^2])^2)/(b*c^2) - (((a + b*ArcTan[c*x^2])*Log[2/(1 + I*c*x^2)])/c + ((I/2 )*b*PolyLog[2, 1 - 2/(1 + I*c*x^2)])/c)/c)/c^2))/3)/2
3.1.75.3.1 Defintions of rubi rules used
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_.) + 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_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c*( p/e) Int[(a + b*ArcTan[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_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e _.)*(x_)^2), x_Symbol] :> Simp[f^2/e Int[(f*x)^(m - 2)*(a + b*ArcTan[c*x] )^p, x], x] - Simp[d*(f^2/e) 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] && GtQ[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*e*(p + 1))), x] - Si mp[1/(c*d) Int[(a + b*ArcTan[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 3.56 (sec) , antiderivative size = 333, normalized size of antiderivative = 2.16
| method | result | size |
| default | \(\frac {x^{6} a^{2}}{6}+\frac {b^{2} x^{6} \arctan \left (c \,x^{2}\right )^{2}}{6}-\frac {b^{2} \arctan \left (c \,x^{2}\right ) x^{4}}{6 c}+\frac {b^{2} \arctan \left (c \,x^{2}\right ) \ln \left (c^{2} x^{4}+1\right )}{6 c^{3}}+\frac {b^{2} x^{2}}{6 c^{2}}-\frac {b^{2} \arctan \left (c \,x^{2}\right )}{6 c^{3}}-\frac {b^{2} \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (c^{2} \textit {\_Z}^{4}+1\right )}{\sum }\frac {2 \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (c^{2} x^{4}+1\right )-c \left (\frac {\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{c \,\underline {\hspace {1.25 ex}}\alpha ^{3}}+\frac {2 \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2} \ln \left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right ) c -\ln \left (\frac {c \,\underline {\hspace {1.25 ex}}\alpha ^{3}+x}{\underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2} c +1\right )}\right )+\ln \left (\frac {c \,\underline {\hspace {1.25 ex}}\alpha ^{3}-x}{\underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2} c -1\right )}\right )\right )}{\underline {\hspace {1.25 ex}}\alpha }+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{2} \operatorname {dilog}\left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right ) c -2 \operatorname {dilog}\left (\frac {c \,\underline {\hspace {1.25 ex}}\alpha ^{3}+x}{\underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2} c +1\right )}\right )+2 \operatorname {dilog}\left (\frac {c \,\underline {\hspace {1.25 ex}}\alpha ^{3}-x}{\underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2} c -1\right )}\right )}{\underline {\hspace {1.25 ex}}\alpha }\right )}{\underline {\hspace {1.25 ex}}\alpha ^{2}}\right )}{24 c^{4}}+\frac {a b \,x^{6} \arctan \left (c \,x^{2}\right )}{3}-\frac {a b \,x^{4}}{6 c}+\frac {a b \ln \left (c^{2} x^{4}+1\right )}{6 c^{3}}\) | \(333\) |
| parts | \(\frac {x^{6} a^{2}}{6}+\frac {b^{2} x^{6} \arctan \left (c \,x^{2}\right )^{2}}{6}-\frac {b^{2} \arctan \left (c \,x^{2}\right ) x^{4}}{6 c}+\frac {b^{2} \arctan \left (c \,x^{2}\right ) \ln \left (c^{2} x^{4}+1\right )}{6 c^{3}}+\frac {b^{2} x^{2}}{6 c^{2}}-\frac {b^{2} \arctan \left (c \,x^{2}\right )}{6 c^{3}}-\frac {b^{2} \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (c^{2} \textit {\_Z}^{4}+1\right )}{\sum }\frac {2 \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (c^{2} x^{4}+1\right )-c \left (\frac {\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{c \,\underline {\hspace {1.25 ex}}\alpha ^{3}}+\frac {2 \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2} \ln \left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right ) c -\ln \left (\frac {c \,\underline {\hspace {1.25 ex}}\alpha ^{3}+x}{\underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2} c +1\right )}\right )+\ln \left (\frac {c \,\underline {\hspace {1.25 ex}}\alpha ^{3}-x}{\underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2} c -1\right )}\right )\right )}{\underline {\hspace {1.25 ex}}\alpha }+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{2} \operatorname {dilog}\left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right ) c -2 \operatorname {dilog}\left (\frac {c \,\underline {\hspace {1.25 ex}}\alpha ^{3}+x}{\underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2} c +1\right )}\right )+2 \operatorname {dilog}\left (\frac {c \,\underline {\hspace {1.25 ex}}\alpha ^{3}-x}{\underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2} c -1\right )}\right )}{\underline {\hspace {1.25 ex}}\alpha }\right )}{\underline {\hspace {1.25 ex}}\alpha ^{2}}\right )}{24 c^{4}}+\frac {a b \,x^{6} \arctan \left (c \,x^{2}\right )}{3}-\frac {a b \,x^{4}}{6 c}+\frac {a b \ln \left (c^{2} x^{4}+1\right )}{6 c^{3}}\) | \(333\) |
| risch | \(-\frac {i b^{2} \ln \left (\frac {1}{2}+\frac {i c \,x^{2}}{2}\right ) \ln \left (\frac {1}{2}-\frac {i c \,x^{2}}{2}\right )}{6 c^{3}}+\frac {a b \ln \left (c^{2} x^{4}+1\right )}{6 c^{3}}-\frac {a b \,x^{4}}{6 c}+\frac {i b^{2} x^{4} \ln \left (i c \,x^{2}+1\right )}{12 c}-\frac {b^{2} x^{6} \ln \left (i c \,x^{2}+1\right )^{2}}{24}-\frac {b^{2} \arctan \left (c \,x^{2}\right )}{12 c^{3}}+\frac {b^{2} x^{2}}{6 c^{2}}-\frac {b^{2} x^{6} \ln \left (-i c \,x^{2}+1\right )^{2}}{24}-\frac {i b a \,x^{6} \ln \left (i c \,x^{2}+1\right )}{6}-\frac {i b^{2} x^{4} \ln \left (-i c \,x^{2}+1\right )}{12 c}+\frac {i b^{2} \ln \left (\frac {1}{2}+\frac {i c \,x^{2}}{2}\right ) \ln \left (-i c \,x^{2}+1\right )}{6 c^{3}}-\frac {i b^{2} \ln \left (i c \,x^{2}+1\right ) \ln \left (-i c \,x^{2}+1\right )}{12 c^{3}}+\frac {x^{6} a^{2}}{6}+\frac {i b^{2} \ln \left (-i c \,x^{2}+1\right )^{2}}{24 c^{3}}+\frac {5 i b^{2} \ln \left (-i c \,x^{2}+1\right )}{72 c^{3}}+\frac {11 i b^{2} \ln \left (i c \,x^{2}+1\right )}{72 c^{3}}-\frac {i b^{2} \ln \left (i c \,x^{2}+1\right )^{2}}{24 c^{3}}+\frac {b^{2} \ln \left (i c \,x^{2}+1\right ) \ln \left (-i c \,x^{2}+1\right ) x^{6}}{12}-\frac {i b^{2} \ln \left (c^{2} x^{4}+1\right )}{9 c^{3}}-\frac {i b^{2} \operatorname {dilog}\left (\frac {1}{2}-\frac {i c \,x^{2}}{2}\right )}{6 c^{3}}-\frac {17 i b^{2}}{108 c^{3}}+\frac {i a b \,x^{6} \ln \left (-i c \,x^{2}+1\right )}{6}\) | \(424\) |
1/6*x^6*a^2+1/6*b^2*x^6*arctan(c*x^2)^2-1/6*b^2*arctan(c*x^2)/c*x^4+1/6*b^ 2*arctan(c*x^2)/c^3*ln(c^2*x^4+1)+1/6*b^2*x^2/c^2-1/6*b^2*arctan(c*x^2)/c^ 3-1/24*b^2/c^4*sum(1/_alpha^2*(2*ln(x-_alpha)*ln(c^2*x^4+1)-c*(1/c/_alpha^ 3*ln(x-_alpha)^2+2/_alpha*ln(x-_alpha)*(_alpha^2*ln(1/2*(x+_alpha)/_alpha) *c-ln((_alpha^3*c+x)/_alpha/(_alpha^2*c+1))+ln((_alpha^3*c-x)/_alpha/(_alp ha^2*c-1)))+2/_alpha*(_alpha^2*dilog(1/2*(x+_alpha)/_alpha)*c-dilog((_alph a^3*c+x)/_alpha/(_alpha^2*c+1))+dilog((_alpha^3*c-x)/_alpha/(_alpha^2*c-1) )))),_alpha=RootOf(_Z^4*c^2+1))+1/3*a*b*x^6*arctan(c*x^2)-1/6/c*a*b*x^4+1/ 6*a*b/c^3*ln(c^2*x^4+1)
\[ \int x^5 \left (a+b \arctan \left (c x^2\right )\right )^2 \, dx=\int { {\left (b \arctan \left (c x^{2}\right ) + a\right )}^{2} x^{5} \,d x } \]
\[ \int x^5 \left (a+b \arctan \left (c x^2\right )\right )^2 \, dx=\int x^{5} \left (a + b \operatorname {atan}{\left (c x^{2} \right )}\right )^{2}\, dx \]
\[ \int x^5 \left (a+b \arctan \left (c x^2\right )\right )^2 \, dx=\int { {\left (b \arctan \left (c x^{2}\right ) + a\right )}^{2} x^{5} \,d x } \]
1/6*a^2*x^6 + 1/6*(2*x^6*arctan(c*x^2) - (x^4/c^2 - log(c^2*x^4 + 1)/c^4)* c)*a*b + 1/96*(4*x^6*arctan(c*x^2)^2 - x^6*log(c^2*x^4 + 1)^2 + 96*integra te(1/48*(4*c^2*x^9*log(c^2*x^4 + 1) - 8*c*x^7*arctan(c*x^2) + 36*(c^2*x^9 + x^5)*arctan(c*x^2)^2 + 3*(c^2*x^9 + x^5)*log(c^2*x^4 + 1)^2)/(c^2*x^4 + 1), x))*b^2
\[ \int x^5 \left (a+b \arctan \left (c x^2\right )\right )^2 \, dx=\int { {\left (b \arctan \left (c x^{2}\right ) + a\right )}^{2} x^{5} \,d x } \]
Timed out. \[ \int x^5 \left (a+b \arctan \left (c x^2\right )\right )^2 \, dx=\int x^5\,{\left (a+b\,\mathrm {atan}\left (c\,x^2\right )\right )}^2 \,d x \]