Integrand size = 16, antiderivative size = 149 \[ \int x^3 \left (a+b \arctan \left (c x^2\right )\right )^3 \, dx=-\frac {3 i b \left (a+b \arctan \left (c x^2\right )\right )^2}{4 c^2}-\frac {3 b x^2 \left (a+b \arctan \left (c x^2\right )\right )^2}{4 c}+\frac {\left (a+b \arctan \left (c x^2\right )\right )^3}{4 c^2}+\frac {1}{4} x^4 \left (a+b \arctan \left (c x^2\right )\right )^3-\frac {3 b^2 \left (a+b \arctan \left (c x^2\right )\right ) \log \left (\frac {2}{1+i c x^2}\right )}{2 c^2}-\frac {3 i b^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x^2}\right )}{4 c^2} \]
-3/4*I*b*(a+b*arctan(c*x^2))^2/c^2-3/4*b*x^2*(a+b*arctan(c*x^2))^2/c+1/4*( a+b*arctan(c*x^2))^3/c^2+1/4*x^4*(a+b*arctan(c*x^2))^3-3/2*b^2*(a+b*arctan (c*x^2))*ln(2/(1+I*c*x^2))/c^2-3/4*I*b^3*polylog(2,1-2/(1+I*c*x^2))/c^2
Time = 0.51 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.14 \[ \int x^3 \left (a+b \arctan \left (c x^2\right )\right )^3 \, dx=\frac {3 b^2 \left (a+a c^2 x^4+b \left (i-c x^2\right )\right ) \arctan \left (c x^2\right )^2+b^3 \left (1+c^2 x^4\right ) \arctan \left (c x^2\right )^3+3 b \arctan \left (c x^2\right ) \left (a \left (a-2 b c x^2+a c^2 x^4\right )-2 b^2 \log \left (1+e^{2 i \arctan \left (c x^2\right )}\right )\right )+a \left (a c x^2 \left (-3 b+a c x^2\right )+3 b^2 \log \left (1+c^2 x^4\right )\right )+3 i b^3 \operatorname {PolyLog}\left (2,-e^{2 i \arctan \left (c x^2\right )}\right )}{4 c^2} \]
(3*b^2*(a + a*c^2*x^4 + b*(I - c*x^2))*ArcTan[c*x^2]^2 + b^3*(1 + c^2*x^4) *ArcTan[c*x^2]^3 + 3*b*ArcTan[c*x^2]*(a*(a - 2*b*c*x^2 + a*c^2*x^4) - 2*b^ 2*Log[1 + E^((2*I)*ArcTan[c*x^2])]) + a*(a*c*x^2*(-3*b + a*c*x^2) + 3*b^2* Log[1 + c^2*x^4]) + (3*I)*b^3*PolyLog[2, -E^((2*I)*ArcTan[c*x^2])])/(4*c^2 )
Time = 0.90 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.11, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {5363, 5361, 5451, 5345, 5419, 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.
| \(\displaystyle \int x^3 \left (a+b \arctan \left (c x^2\right )\right )^3 \, dx\) |
| \(\Big \downarrow \) 5363 |
| \(\displaystyle \frac {1}{2} \int x^2 \left (a+b \arctan \left (c x^2\right )\right )^3dx^2\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} x^4 \left (a+b \arctan \left (c x^2\right )\right )^3-\frac {3}{2} b c \int \frac {x^4 \left (a+b \arctan \left (c x^2\right )\right )^2}{c^2 x^4+1}dx^2\right )\) |
| \(\Big \downarrow \) 5451 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} x^4 \left (a+b \arctan \left (c x^2\right )\right )^3-\frac {3}{2} b c \left (\frac {\int \left (a+b \arctan \left (c x^2\right )\right )^2dx^2}{c^2}-\frac {\int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{c^2 x^4+1}dx^2}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 5345 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} x^4 \left (a+b \arctan \left (c x^2\right )\right )^3-\frac {3}{2} b c \left (\frac {x^2 \left (a+b \arctan \left (c x^2\right )\right )^2-2 b c \int \frac {x^2 \left (a+b \arctan \left (c x^2\right )\right )}{c^2 x^4+1}dx^2}{c^2}-\frac {\int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{c^2 x^4+1}dx^2}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 5419 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} x^4 \left (a+b \arctan \left (c x^2\right )\right )^3-\frac {3}{2} b c \left (\frac {x^2 \left (a+b \arctan \left (c x^2\right )\right )^2-2 b c \int \frac {x^2 \left (a+b \arctan \left (c x^2\right )\right )}{c^2 x^4+1}dx^2}{c^2}-\frac {\left (a+b \arctan \left (c x^2\right )\right )^3}{3 b c^3}\right )\right )\) |
| \(\Big \downarrow \) 5455 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} x^4 \left (a+b \arctan \left (c x^2\right )\right )^3-\frac {3}{2} b c \left (-\frac {\left (a+b \arctan \left (c x^2\right )\right )^3}{3 b c^3}+\frac {x^2 \left (a+b \arctan \left (c x^2\right )\right )^2-2 b c \left (-\frac {\int \frac {a+b \arctan \left (c x^2\right )}{i-c x^2}dx^2}{c}-\frac {i \left (a+b \arctan \left (c x^2\right )\right )^2}{2 b c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 5379 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} x^4 \left (a+b \arctan \left (c x^2\right )\right )^3-\frac {3}{2} b c \left (-\frac {\left (a+b \arctan \left (c x^2\right )\right )^3}{3 b c^3}+\frac {x^2 \left (a+b \arctan \left (c x^2\right )\right )^2-2 b c \left (-\frac {\frac {\log \left (\frac {2}{1+i c x^2}\right ) \left (a+b \arctan \left (c x^2\right )\right )}{c}-b \int \frac {\log \left (\frac {2}{i c x^2+1}\right )}{c^2 x^4+1}dx^2}{c}-\frac {i \left (a+b \arctan \left (c x^2\right )\right )^2}{2 b c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} x^4 \left (a+b \arctan \left (c x^2\right )\right )^3-\frac {3}{2} b c \left (-\frac {\left (a+b \arctan \left (c x^2\right )\right )^3}{3 b c^3}+\frac {x^2 \left (a+b \arctan \left (c x^2\right )\right )^2-2 b c \left (-\frac {\frac {i b \int \frac {\log \left (\frac {2}{i c x^2+1}\right )}{1-\frac {2}{i c x^2+1}}d\frac {1}{i c x^2+1}}{c}+\frac {\log \left (\frac {2}{1+i c x^2}\right ) \left (a+b \arctan \left (c x^2\right )\right )}{c}}{c}-\frac {i \left (a+b \arctan \left (c x^2\right )\right )^2}{2 b c^2}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} x^4 \left (a+b \arctan \left (c x^2\right )\right )^3-\frac {3}{2} b c \left (-\frac {\left (a+b \arctan \left (c x^2\right )\right )^3}{3 b c^3}+\frac {x^2 \left (a+b \arctan \left (c x^2\right )\right )^2-2 b c \left (-\frac {i \left (a+b \arctan \left (c x^2\right )\right )^2}{2 b c^2}-\frac {\frac {\log \left (\frac {2}{1+i c x^2}\right ) \left (a+b \arctan \left (c x^2\right )\right )}{c}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x^2+1}\right )}{2 c}}{c}\right )}{c^2}\right )\right )\) |
((x^4*(a + b*ArcTan[c*x^2])^3)/2 - (3*b*c*(-1/3*(a + b*ArcTan[c*x^2])^3/(b *c^3) + (x^2*(a + b*ArcTan[c*x^2])^2 - 2*b*c*(((-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))/2)/2
3.1.86.3.1 Defintions of rubi rules used
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_Symbol] :> Simp[x*(a + b*ArcTan[c*x^n])^p, x] - Simp[b*c*n*p Int[x^n*((a + b*ArcTan[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_.) + 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_.)/((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[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 = 4.66 (sec) , antiderivative size = 399, normalized size of antiderivative = 2.68
| method | result | size |
| default | \(\frac {a^{3} x^{4}}{4}+\frac {b^{3} x^{4} \arctan \left (c \,x^{2}\right )^{3}}{4}-\frac {3 b^{3} \arctan \left (c \,x^{2}\right )^{2} x^{2}}{4 c}+\frac {b^{3} \arctan \left (c \,x^{2}\right )^{3}}{4 c^{2}}+\frac {3 b^{3} \ln \left (c^{2} x^{4}+1\right ) \arctan \left (c \,x^{2}\right )}{4 c^{2}}-\frac {3 b^{3} \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 )}{16 c^{3}}+\frac {3 a \,b^{2} x^{4} \arctan \left (c \,x^{2}\right )^{2}}{4}-\frac {3 a \,b^{2} \arctan \left (c \,x^{2}\right ) x^{2}}{2 c}+\frac {3 a \,b^{2} \arctan \left (c \,x^{2}\right )^{2}}{4 c^{2}}+\frac {3 a \,b^{2} \ln \left (c^{2} x^{4}+1\right )}{4 c^{2}}+\frac {3 a^{2} b \,x^{4} \arctan \left (c \,x^{2}\right )}{4}-\frac {3 a^{2} b \,x^{2}}{4 c}+\frac {3 a^{2} b \arctan \left (c \,x^{2}\right )}{4 c^{2}}\) | \(399\) |
| parts | \(\frac {a^{3} x^{4}}{4}+\frac {b^{3} x^{4} \arctan \left (c \,x^{2}\right )^{3}}{4}-\frac {3 b^{3} \arctan \left (c \,x^{2}\right )^{2} x^{2}}{4 c}+\frac {b^{3} \arctan \left (c \,x^{2}\right )^{3}}{4 c^{2}}+\frac {3 b^{3} \ln \left (c^{2} x^{4}+1\right ) \arctan \left (c \,x^{2}\right )}{4 c^{2}}-\frac {3 b^{3} \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 )}{16 c^{3}}+\frac {3 a \,b^{2} x^{4} \arctan \left (c \,x^{2}\right )^{2}}{4}-\frac {3 a \,b^{2} \arctan \left (c \,x^{2}\right ) x^{2}}{2 c}+\frac {3 a \,b^{2} \arctan \left (c \,x^{2}\right )^{2}}{4 c^{2}}+\frac {3 a \,b^{2} \ln \left (c^{2} x^{4}+1\right )}{4 c^{2}}+\frac {3 a^{2} b \,x^{4} \arctan \left (c \,x^{2}\right )}{4}-\frac {3 a^{2} b \,x^{2}}{4 c}+\frac {3 a^{2} b \arctan \left (c \,x^{2}\right )}{4 c^{2}}\) | \(399\) |
| risch | \(\frac {3 a \,b^{2} \ln \left (c^{2} x^{4}+1\right )}{4 c^{2}}+\frac {3 i b^{2} \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (c \,\textit {\_Z}^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right )\right )}{\sum }\frac {\left (\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (-i c \,x^{2}+1\right )+2 c \left (-\frac {\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \left (\ln \left (\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \left (i \sqrt {\frac {i}{c}}+\sqrt {\frac {i}{c}}+x -\underline {\hspace {1.25 ex}}\alpha \right )}{\sqrt {\frac {i}{c}}}\right )+\ln \left (\frac {\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (i \sqrt {\frac {i}{c}}-\sqrt {\frac {i}{c}}-x +\underline {\hspace {1.25 ex}}\alpha \right )}{\sqrt {\frac {i}{c}}}\right )\right )}{2 c}-\frac {\operatorname {dilog}\left (\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \left (i \sqrt {\frac {i}{c}}+\sqrt {\frac {i}{c}}+x -\underline {\hspace {1.25 ex}}\alpha \right )}{\sqrt {\frac {i}{c}}}\right )+\operatorname {dilog}\left (\frac {\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (i \sqrt {\frac {i}{c}}-\sqrt {\frac {i}{c}}-x +\underline {\hspace {1.25 ex}}\alpha \right )}{\sqrt {\frac {i}{c}}}\right )}{2 c}\right )\right ) b}{c}\right )}{4 c}+\frac {a^{3} x^{4}}{4}-\frac {3 a \,b^{2} x^{4} \ln \left (-i c \,x^{2}+1\right )^{2}}{16}-\frac {3 a \,b^{2} \ln \left (-i c \,x^{2}+1\right )^{2}}{16 c^{2}}+\frac {3 i b \,a^{2} x^{4} \ln \left (-i c \,x^{2}+1\right )}{8}+\frac {3 i b^{3} \ln \left (-i c \,x^{2}+1\right )^{2}}{16 c^{2}}+\frac {3 b^{3} x^{2} \ln \left (-i c \,x^{2}+1\right )^{2}}{16 c}+\frac {i b^{3} \left (c^{2} x^{4}+1\right ) \ln \left (i c \,x^{2}+1\right )^{3}}{32 c^{2}}+\frac {3 i b^{3} \ln \left (c^{2} x^{4}+1\right )}{16 c^{2}}-\frac {3 a^{2} b \,x^{2}}{4 c}+\frac {3 a^{2} b \arctan \left (c \,x^{2}\right )}{4 c^{2}}-\frac {3 b^{3} \arctan \left (c \,x^{2}\right )}{8 c^{2}}-\frac {3 b^{2} \left (i b \,c^{2} x^{4} \ln \left (-i c \,x^{2}+1\right )+2 a \,c^{2} x^{4}-2 b c \,x^{2}+i b \ln \left (-i c \,x^{2}+1\right )+2 i b +2 a \right ) \ln \left (i c \,x^{2}+1\right )^{2}}{32 c^{2}}-\frac {3 i a \,b^{2} x^{2} \ln \left (-i c \,x^{2}+1\right )}{4 c}-\frac {i b^{3} \ln \left (-i c \,x^{2}+1\right )^{3}}{32 c^{2}}-\frac {i b^{3} x^{4} \ln \left (-i c \,x^{2}+1\right )^{3}}{32}+\left (\frac {3 i b^{3} \left (c^{2} x^{4}+1\right ) \ln \left (-i c \,x^{2}+1\right )^{2}}{32 c^{2}}+\frac {3 b^{2} \left (2 c \,x^{2} a -b \right )^{2} \ln \left (-i c \,x^{2}+1\right )}{32 c^{2} a}-\frac {3 b \left (4 i a^{3} c^{2} x^{4}-8 i a^{2} b c \,x^{2}+4 i \ln \left (-i c \,x^{2}+1\right ) a \,b^{2}+4 i a \,b^{2}-4 \ln \left (-i c \,x^{2}+1\right ) a^{2} b +\ln \left (-i c \,x^{2}+1\right ) b^{3}\right )}{32 a \,c^{2}}\right ) \ln \left (i c \,x^{2}+1\right )\) | \(758\) |
1/4*a^3*x^4+1/4*b^3*x^4*arctan(c*x^2)^3-3/4*b^3*arctan(c*x^2)^2/c*x^2+1/4* b^3*arctan(c*x^2)^3/c^2+3/4*b^3/c^2*ln(c^2*x^4+1)*arctan(c*x^2)-3/16*b^3/c ^3*sum(1/_alpha^2*(2*ln(x-_alpha)*ln(c^2*x^4+1)-c*(1/c/_alpha^3*ln(x-_alph a)^2+2/_alpha*ln(x-_alpha)*(_alpha^2*ln(1/2*(x+_alpha)/_alpha)*c-ln((_alph a^3*c+x)/_alpha/(_alpha^2*c+1))+ln((_alpha^3*c-x)/_alpha/(_alpha^2*c-1)))+ 2/_alpha*(_alpha^2*dilog(1/2*(x+_alpha)/_alpha)*c-dilog((_alpha^3*c+x)/_al pha/(_alpha^2*c+1))+dilog((_alpha^3*c-x)/_alpha/(_alpha^2*c-1))))),_alpha= RootOf(_Z^4*c^2+1))+3/4*a*b^2*x^4*arctan(c*x^2)^2-3/2*a*b^2*arctan(c*x^2)/ c*x^2+3/4*a*b^2/c^2*arctan(c*x^2)^2+3/4*a*b^2/c^2*ln(c^2*x^4+1)+3/4*a^2*b* x^4*arctan(c*x^2)-3/4*a^2*b/c*x^2+3/4*a^2*b/c^2*arctan(c*x^2)
\[ \int x^3 \left (a+b \arctan \left (c x^2\right )\right )^3 \, dx=\int { {\left (b \arctan \left (c x^{2}\right ) + a\right )}^{3} x^{3} \,d x } \]
integral(b^3*x^3*arctan(c*x^2)^3 + 3*a*b^2*x^3*arctan(c*x^2)^2 + 3*a^2*b*x ^3*arctan(c*x^2) + a^3*x^3, x)
\[ \int x^3 \left (a+b \arctan \left (c x^2\right )\right )^3 \, dx=\int x^{3} \left (a + b \operatorname {atan}{\left (c x^{2} \right )}\right )^{3}\, dx \]
\[ \int x^3 \left (a+b \arctan \left (c x^2\right )\right )^3 \, dx=\int { {\left (b \arctan \left (c x^{2}\right ) + a\right )}^{3} x^{3} \,d x } \]
3/4*a*b^2*x^4*arctan(c*x^2)^2 + 1/4*a^3*x^4 + 3/4*(x^4*arctan(c*x^2) - c*( x^2/c^2 - arctan(c*x^2)/c^3))*a^2*b - 3/4*(2*c*(x^2/c^2 - arctan(c*x^2)/c^ 3)*arctan(c*x^2) + (arctan(c*x^2)^2 - log(4*c^5*x^4 + 4*c^3))/c^2)*a*b^2 + 1/128*(4*x^4*arctan(c*x^2)^3 - 3*x^4*arctan(c*x^2)*log(c^2*x^4 + 1)^2 + 1 28*integrate(1/64*(12*c^2*x^7*arctan(c*x^2)*log(c^2*x^4 + 1) - 12*c*x^5*ar ctan(c*x^2)^2 + 56*(c^2*x^7 + x^3)*arctan(c*x^2)^3 + 3*(c*x^5 + 2*(c^2*x^7 + x^3)*arctan(c*x^2))*log(c^2*x^4 + 1)^2)/(c^2*x^4 + 1), x))*b^3
\[ \int x^3 \left (a+b \arctan \left (c x^2\right )\right )^3 \, dx=\int { {\left (b \arctan \left (c x^{2}\right ) + a\right )}^{3} x^{3} \,d x } \]
Timed out. \[ \int x^3 \left (a+b \arctan \left (c x^2\right )\right )^3 \, dx=\int x^3\,{\left (a+b\,\mathrm {atan}\left (c\,x^2\right )\right )}^3 \,d x \]