Integrand size = 23, antiderivative size = 196 \[ \int \frac {x^3 (a+b \arctan (c x))}{d+i c d x} \, dx=\frac {i a x}{c^3 d}-\frac {b x}{2 c^3 d}+\frac {i b x^2}{6 c^2 d}+\frac {b \arctan (c x)}{2 c^4 d}+\frac {i b x \arctan (c x)}{c^3 d}+\frac {x^2 (a+b \arctan (c x))}{2 c^2 d}-\frac {i x^3 (a+b \arctan (c x))}{3 c d}+\frac {(a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^4 d}-\frac {2 i b \log \left (1+c^2 x^2\right )}{3 c^4 d}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c^4 d} \]
I*a*x/c^3/d-1/2*b*x/c^3/d+1/6*I*b*x^2/c^2/d+1/2*b*arctan(c*x)/c^4/d+I*b*x* arctan(c*x)/c^3/d+1/2*x^2*(a+b*arctan(c*x))/c^2/d-1/3*I*x^3*(a+b*arctan(c* x))/c/d+(a+b*arctan(c*x))*ln(2/(1+I*c*x))/c^4/d-2/3*I*b*ln(c^2*x^2+1)/c^4/ d+1/2*I*b*polylog(2,1-2/(1+I*c*x))/c^4/d
Time = 0.53 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.85 \[ \int \frac {x^3 (a+b \arctan (c x))}{d+i c d x} \, dx=-\frac {i \left (-b-6 a c x-3 i b c x+3 i a c^2 x^2-b c^2 x^2+2 a c^3 x^3+6 b \arctan (c x)^2+\arctan (c x) \left (6 a+b \left (3 i-6 c x+3 i c^2 x^2+2 c^3 x^3\right )+6 i b \log \left (1+e^{2 i \arctan (c x)}\right )\right )-3 i a \log \left (1+c^2 x^2\right )+4 b \log \left (1+c^2 x^2\right )+3 b \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )}{6 c^4 d} \]
((-1/6*I)*(-b - 6*a*c*x - (3*I)*b*c*x + (3*I)*a*c^2*x^2 - b*c^2*x^2 + 2*a* c^3*x^3 + 6*b*ArcTan[c*x]^2 + ArcTan[c*x]*(6*a + b*(3*I - 6*c*x + (3*I)*c^ 2*x^2 + 2*c^3*x^3) + (6*I)*b*Log[1 + E^((2*I)*ArcTan[c*x])]) - (3*I)*a*Log [1 + c^2*x^2] + 4*b*Log[1 + c^2*x^2] + 3*b*PolyLog[2, -E^((2*I)*ArcTan[c*x ])]))/(c^4*d)
Time = 1.02 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.10, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {5401, 27, 5361, 243, 49, 2009, 5401, 5361, 262, 216, 5401, 2009, 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.
\(\displaystyle \int \frac {x^3 (a+b \arctan (c x))}{d+i c d x} \, dx\) |
\(\Big \downarrow \) 5401 |
\(\displaystyle \frac {i \int \frac {x^2 (a+b \arctan (c x))}{d (i c x+1)}dx}{c}-\frac {i \int x^2 (a+b \arctan (c x))dx}{c d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {i \int \frac {x^2 (a+b \arctan (c x))}{i c x+1}dx}{c d}-\frac {i \int x^2 (a+b \arctan (c x))dx}{c d}\) |
\(\Big \downarrow \) 5361 |
\(\displaystyle \frac {i \int \frac {x^2 (a+b \arctan (c x))}{i c x+1}dx}{c d}-\frac {i \left (\frac {1}{3} x^3 (a+b \arctan (c x))-\frac {1}{3} b c \int \frac {x^3}{c^2 x^2+1}dx\right )}{c d}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {i \int \frac {x^2 (a+b \arctan (c x))}{i c x+1}dx}{c d}-\frac {i \left (\frac {1}{3} x^3 (a+b \arctan (c x))-\frac {1}{6} b c \int \frac {x^2}{c^2 x^2+1}dx^2\right )}{c d}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {i \int \frac {x^2 (a+b \arctan (c x))}{i c x+1}dx}{c d}-\frac {i \left (\frac {1}{3} x^3 (a+b \arctan (c x))-\frac {1}{6} b c \int \left (\frac {1}{c^2}-\frac {1}{c^2 \left (c^2 x^2+1\right )}\right )dx^2\right )}{c d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {i \int \frac {x^2 (a+b \arctan (c x))}{i c x+1}dx}{c d}-\frac {i \left (\frac {1}{3} x^3 (a+b \arctan (c x))-\frac {1}{6} b c \left (\frac {x^2}{c^2}-\frac {\log \left (c^2 x^2+1\right )}{c^4}\right )\right )}{c d}\) |
\(\Big \downarrow \) 5401 |
\(\displaystyle \frac {i \left (\frac {i \int \frac {x (a+b \arctan (c x))}{i c x+1}dx}{c}-\frac {i \int x (a+b \arctan (c x))dx}{c}\right )}{c d}-\frac {i \left (\frac {1}{3} x^3 (a+b \arctan (c x))-\frac {1}{6} b c \left (\frac {x^2}{c^2}-\frac {\log \left (c^2 x^2+1\right )}{c^4}\right )\right )}{c d}\) |
\(\Big \downarrow \) 5361 |
\(\displaystyle \frac {i \left (\frac {i \int \frac {x (a+b \arctan (c x))}{i c x+1}dx}{c}-\frac {i \left (\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \int \frac {x^2}{c^2 x^2+1}dx\right )}{c}\right )}{c d}-\frac {i \left (\frac {1}{3} x^3 (a+b \arctan (c x))-\frac {1}{6} b c \left (\frac {x^2}{c^2}-\frac {\log \left (c^2 x^2+1\right )}{c^4}\right )\right )}{c d}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {i \left (\frac {i \int \frac {x (a+b \arctan (c x))}{i c x+1}dx}{c}-\frac {i \left (\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\int \frac {1}{c^2 x^2+1}dx}{c^2}\right )\right )}{c}\right )}{c d}-\frac {i \left (\frac {1}{3} x^3 (a+b \arctan (c x))-\frac {1}{6} b c \left (\frac {x^2}{c^2}-\frac {\log \left (c^2 x^2+1\right )}{c^4}\right )\right )}{c d}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {i \left (\frac {i \int \frac {x (a+b \arctan (c x))}{i c x+1}dx}{c}-\frac {i \left (\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\arctan (c x)}{c^3}\right )\right )}{c}\right )}{c d}-\frac {i \left (\frac {1}{3} x^3 (a+b \arctan (c x))-\frac {1}{6} b c \left (\frac {x^2}{c^2}-\frac {\log \left (c^2 x^2+1\right )}{c^4}\right )\right )}{c d}\) |
\(\Big \downarrow \) 5401 |
\(\displaystyle \frac {i \left (\frac {i \left (\frac {i \int \frac {a+b \arctan (c x)}{i c x+1}dx}{c}-\frac {i \int (a+b \arctan (c x))dx}{c}\right )}{c}-\frac {i \left (\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\arctan (c x)}{c^3}\right )\right )}{c}\right )}{c d}-\frac {i \left (\frac {1}{3} x^3 (a+b \arctan (c x))-\frac {1}{6} b c \left (\frac {x^2}{c^2}-\frac {\log \left (c^2 x^2+1\right )}{c^4}\right )\right )}{c d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {i \left (\frac {i \left (\frac {i \int \frac {a+b \arctan (c x)}{i c x+1}dx}{c}-\frac {i \left (a x+b x \arctan (c x)-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )}{c}\right )}{c}-\frac {i \left (\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\arctan (c x)}{c^3}\right )\right )}{c}\right )}{c d}-\frac {i \left (\frac {1}{3} x^3 (a+b \arctan (c x))-\frac {1}{6} b c \left (\frac {x^2}{c^2}-\frac {\log \left (c^2 x^2+1\right )}{c^4}\right )\right )}{c d}\) |
\(\Big \downarrow \) 5379 |
\(\displaystyle \frac {i \left (\frac {i \left (\frac {i \left (\frac {i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}-i b \int \frac {\log \left (\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx\right )}{c}-\frac {i \left (a x+b x \arctan (c x)-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )}{c}\right )}{c}-\frac {i \left (\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\arctan (c x)}{c^3}\right )\right )}{c}\right )}{c d}-\frac {i \left (\frac {1}{3} x^3 (a+b \arctan (c x))-\frac {1}{6} b c \left (\frac {x^2}{c^2}-\frac {\log \left (c^2 x^2+1\right )}{c^4}\right )\right )}{c d}\) |
\(\Big \downarrow \) 2849 |
\(\displaystyle \frac {i \left (\frac {i \left (\frac {i \left (\frac {i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}-\frac {b \int \frac {\log \left (\frac {2}{i c x+1}\right )}{1-\frac {2}{i c x+1}}d\frac {1}{i c x+1}}{c}\right )}{c}-\frac {i \left (a x+b x \arctan (c x)-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )}{c}\right )}{c}-\frac {i \left (\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\arctan (c x)}{c^3}\right )\right )}{c}\right )}{c d}-\frac {i \left (\frac {1}{3} x^3 (a+b \arctan (c x))-\frac {1}{6} b c \left (\frac {x^2}{c^2}-\frac {\log \left (c^2 x^2+1\right )}{c^4}\right )\right )}{c d}\) |
\(\Big \downarrow \) 2752 |
\(\displaystyle \frac {i \left (\frac {i \left (\frac {i \left (\frac {i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{2 c}\right )}{c}-\frac {i \left (a x+b x \arctan (c x)-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )}{c}\right )}{c}-\frac {i \left (\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\arctan (c x)}{c^3}\right )\right )}{c}\right )}{c d}-\frac {i \left (\frac {1}{3} x^3 (a+b \arctan (c x))-\frac {1}{6} b c \left (\frac {x^2}{c^2}-\frac {\log \left (c^2 x^2+1\right )}{c^4}\right )\right )}{c d}\) |
((-I)*((x^3*(a + b*ArcTan[c*x]))/3 - (b*c*(x^2/c^2 - Log[1 + c^2*x^2]/c^4) )/6))/(c*d) + (I*(((-I)*((x^2*(a + b*ArcTan[c*x]))/2 - (b*c*(x/c^2 - ArcTa n[c*x]/c^3))/2))/c + (I*(((-I)*(a*x + b*x*ArcTan[c*x] - (b*Log[1 + c^2*x^2 ])/(2*c)))/c + (I*((I*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c - (b*PolyL og[2, 1 - 2/(1 + I*c*x)])/(2*c)))/c))/c))/(c*d)
3.1.43.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
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[(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[((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_)]*(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_)), x_Symbol] :> Simp[f/e Int[(f*x)^(m - 1)*(a + b*ArcTan[c*x])^p , x], x] - Simp[d*(f/e) Int[(f*x)^(m - 1)*((a + b*ArcTan[c*x])^p/(d + e*x )), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e ^2, 0] && GtQ[m, 0]
Time = 2.04 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.59
method | result | size |
derivativedivides | \(\frac {-\frac {5 i b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{48 d}-\frac {i b \arctan \left (c x \right ) c^{3} x^{3}}{3 d}+\frac {a \,c^{2} x^{2}}{2 d}-\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d}+\frac {i b \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{2 d}+\frac {i b \,c^{2} x^{2}}{6 d}+\frac {i b \ln \left (-\frac {i \left (c x +i\right )}{2}\right ) \ln \left (c x -i\right )}{2 d}+\frac {b \arctan \left (c x \right ) c^{2} x^{2}}{2 d}-\frac {b \arctan \left (c x \right ) \ln \left (c x -i\right )}{d}-\frac {11 i b \ln \left (c^{2} x^{2}+1\right )}{24 d}-\frac {i a \,c^{3} x^{3}}{3 d}-\frac {i b \ln \left (c x -i\right )^{2}}{4 d}-\frac {b c x}{2 d}+\frac {i a c x}{d}+\frac {2 i b}{3 d}-\frac {i a \arctan \left (c x \right )}{d}+\frac {5 b \arctan \left (\frac {c x}{2}\right )}{24 d}-\frac {5 b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{24 d}-\frac {5 b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{12 d}+\frac {i b \arctan \left (c x \right ) c x}{d}+\frac {11 b \arctan \left (c x \right )}{12 d}}{c^{4}}\) | \(312\) |
default | \(\frac {-\frac {5 i b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{48 d}-\frac {i b \arctan \left (c x \right ) c^{3} x^{3}}{3 d}+\frac {a \,c^{2} x^{2}}{2 d}-\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d}+\frac {i b \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{2 d}+\frac {i b \,c^{2} x^{2}}{6 d}+\frac {i b \ln \left (-\frac {i \left (c x +i\right )}{2}\right ) \ln \left (c x -i\right )}{2 d}+\frac {b \arctan \left (c x \right ) c^{2} x^{2}}{2 d}-\frac {b \arctan \left (c x \right ) \ln \left (c x -i\right )}{d}-\frac {11 i b \ln \left (c^{2} x^{2}+1\right )}{24 d}-\frac {i a \,c^{3} x^{3}}{3 d}-\frac {i b \ln \left (c x -i\right )^{2}}{4 d}-\frac {b c x}{2 d}+\frac {i a c x}{d}+\frac {2 i b}{3 d}-\frac {i a \arctan \left (c x \right )}{d}+\frac {5 b \arctan \left (\frac {c x}{2}\right )}{24 d}-\frac {5 b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{24 d}-\frac {5 b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{12 d}+\frac {i b \arctan \left (c x \right ) c x}{d}+\frac {11 b \arctan \left (c x \right )}{12 d}}{c^{4}}\) | \(312\) |
risch | \(\frac {i a x}{c^{3} d}-\frac {b \left (\frac {1}{3} c^{2} x^{3}+\frac {1}{2} i c \,x^{2}-x \right ) \ln \left (i c x +1\right )}{2 c^{3} d}+\frac {i b \,x^{2}}{6 c^{2} d}-\frac {5 a}{6 d \,c^{4}}+\frac {a \,x^{2}}{2 d \,c^{2}}-\frac {11 i b \ln \left (c^{2} x^{2}+1\right )}{24 c^{4} d}+\frac {b \,x^{3} \ln \left (-i c x +1\right )}{6 d c}+\frac {i b \ln \left (i c x +1\right )^{2}}{4 d \,c^{4}}-\frac {b x \ln \left (-i c x +1\right )}{2 d \,c^{3}}-\frac {i a \,x^{3}}{3 d c}-\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d \,c^{4}}-\frac {5 i \ln \left (-i c x +1\right ) b}{12 d \,c^{4}}-\frac {b x}{2 c^{3} d}-\frac {i a \arctan \left (c x \right )}{d \,c^{4}}-\frac {i b \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{2 d \,c^{4}}+\frac {i b \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{2 d \,c^{4}}+\frac {31 i b}{72 d \,c^{4}}+\frac {i x^{2} b \ln \left (-i c x +1\right )}{4 d \,c^{2}}+\frac {i b \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i c x}{2}\right )}{2 d \,c^{4}}+\frac {11 b \arctan \left (c x \right )}{12 c^{4} d}\) | \(349\) |
parts | \(\frac {i a x}{c^{3} d}+\frac {a \,x^{2}}{2 d \,c^{2}}-\frac {i b \arctan \left (c x \right ) x^{3}}{3 d c}-\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d \,c^{4}}+\frac {i b \,x^{2}}{6 c^{2} d}-\frac {5 i b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{48 d \,c^{4}}+\frac {i b \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{2 d \,c^{4}}+\frac {b \arctan \left (c x \right ) x^{2}}{2 d \,c^{2}}-\frac {b \arctan \left (c x \right ) \ln \left (c x -i\right )}{d \,c^{4}}-\frac {11 i b \ln \left (c^{2} x^{2}+1\right )}{24 c^{4} d}-\frac {i b \ln \left (c x -i\right )^{2}}{4 d \,c^{4}}-\frac {i a \,x^{3}}{3 d c}-\frac {b x}{2 c^{3} d}-\frac {i a \arctan \left (c x \right )}{d \,c^{4}}+\frac {i b \ln \left (-\frac {i \left (c x +i\right )}{2}\right ) \ln \left (c x -i\right )}{2 d \,c^{4}}+\frac {2 i b}{3 d \,c^{4}}+\frac {5 b \arctan \left (\frac {c x}{2}\right )}{24 d \,c^{4}}-\frac {5 b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{24 d \,c^{4}}-\frac {5 b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{12 d \,c^{4}}+\frac {i b x \arctan \left (c x \right )}{c^{3} d}+\frac {11 b \arctan \left (c x \right )}{12 c^{4} d}\) | \(353\) |
1/c^4*(-5/48*I/d*b*ln(c^4*x^4+10*c^2*x^2+9)-1/3*I/d*b*arctan(c*x)*c^3*x^3+ 1/2/d*a*c^2*x^2-1/2/d*a*ln(c^2*x^2+1)+1/2*I/d*b*dilog(-1/2*I*(c*x+I))+1/6* I/d*b*c^2*x^2+1/2*I/d*b*ln(-1/2*I*(c*x+I))*ln(c*x-I)+1/2/d*b*arctan(c*x)*c ^2*x^2-1/d*b*arctan(c*x)*ln(c*x-I)-11/24*I/d*b*ln(c^2*x^2+1)-1/3*I/d*a*c^3 *x^3-1/4*I/d*b*ln(c*x-I)^2-1/2/d*b*c*x+I/d*a*c*x+2/3*I/d*b-I/d*a*arctan(c* x)+5/24/d*b*arctan(1/2*c*x)-5/24/d*b*arctan(1/6*c^3*x^3+7/6*c*x)-5/12/d*b* arctan(1/2*c*x-1/2*I)+I/d*b*arctan(c*x)*c*x+11/12/d*b*arctan(c*x))
\[ \int \frac {x^3 (a+b \arctan (c x))}{d+i c d x} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{3}}{i \, c d x + d} \,d x } \]
\[ \int \frac {x^3 (a+b \arctan (c x))}{d+i c d x} \, dx=- \frac {i \left (\int \frac {6 i b \log {\left (i c x + 1 \right )}}{c^{2} x^{2} + 1}\, dx + \int \frac {12 a c^{4} x^{4}}{c^{2} x^{2} + 1}\, dx + \int \frac {6 b c x}{c^{2} x^{2} + 1}\, dx + \int \frac {b c^{3} x^{3}}{c^{2} x^{2} + 1}\, dx + \int \frac {12 i a c^{3} x^{3}}{c^{2} x^{2} + 1}\, dx + \int \frac {3 i b c^{2} x^{2}}{c^{2} x^{2} + 1}\, dx + \int \left (- \frac {2 i b c^{4} x^{4}}{c^{2} x^{2} + 1}\right )\, dx + \int \left (- \frac {6 b c x \log {\left (i c x + 1 \right )}}{c^{2} x^{2} + 1}\right )\, dx + \int \frac {6 b c^{3} x^{3} \log {\left (i c x + 1 \right )}}{c^{2} x^{2} + 1}\, dx + \int \left (- \frac {6 i b c^{4} x^{4} \log {\left (i c x + 1 \right )}}{c^{2} x^{2} + 1}\right )\, dx\right )}{12 c^{3} d} + \frac {\left (2 b c^{3} x^{3} + 3 i b c^{2} x^{2} - 6 b c x - 6 i b \log {\left (i c x + 1 \right )}\right ) \log {\left (- i c x + 1 \right )}}{12 c^{4} d} \]
-I*(Integral(6*I*b*log(I*c*x + 1)/(c**2*x**2 + 1), x) + Integral(12*a*c**4 *x**4/(c**2*x**2 + 1), x) + Integral(6*b*c*x/(c**2*x**2 + 1), x) + Integra l(b*c**3*x**3/(c**2*x**2 + 1), x) + Integral(12*I*a*c**3*x**3/(c**2*x**2 + 1), x) + Integral(3*I*b*c**2*x**2/(c**2*x**2 + 1), x) + Integral(-2*I*b*c **4*x**4/(c**2*x**2 + 1), x) + Integral(-6*b*c*x*log(I*c*x + 1)/(c**2*x**2 + 1), x) + Integral(6*b*c**3*x**3*log(I*c*x + 1)/(c**2*x**2 + 1), x) + In tegral(-6*I*b*c**4*x**4*log(I*c*x + 1)/(c**2*x**2 + 1), x))/(12*c**3*d) + (2*b*c**3*x**3 + 3*I*b*c**2*x**2 - 6*b*c*x - 6*I*b*log(I*c*x + 1))*log(-I* c*x + 1)/(12*c**4*d)
\[ \int \frac {x^3 (a+b \arctan (c x))}{d+i c d x} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{3}}{i \, c d x + d} \,d x } \]
-1/6*a*(I*(2*c^2*x^3 + 3*I*c*x^2 - 6*x)/(c^3*d) + 6*log(I*c*x + 1)/(c^4*d) ) - 1/72*(432*I*c^8*d*integrate(1/12*x^4*arctan(c*x)/(c^5*d*x^2 + c^3*d), x) + 216*c^8*d*integrate(1/12*x^4*log(c^2*x^2 + 1)/(c^5*d*x^2 + c^3*d), x) - 432*c^7*d*integrate(1/12*x^3*arctan(c*x)/(c^5*d*x^2 + c^3*d), x) + 216* I*c^7*d*integrate(1/12*x^3*log(c^2*x^2 + 1)/(c^5*d*x^2 + c^3*d), x) + 432* c^5*d*integrate(1/12*x*arctan(c*x)/(c^5*d*x^2 + c^3*d), x) - 216*I*c^5*d*i ntegrate(1/12*x*log(c^2*x^2 + 1)/(c^5*d*x^2 + c^3*d), x) + 4*c^3*x^3 - 216 *c^4*d*integrate(1/12*log(c^2*x^2 + 1)/(c^5*d*x^2 + c^3*d), x) + 3*I*c^2*x ^2 - 30*c*x - 6*(-2*I*c^3*x^3 + 3*c^2*x^2 + 6*I*c*x - 5)*arctan(c*x) + 18* I*arctan(c*x)^2 - 3*(2*c^3*x^3 + 3*I*c^2*x^2 - 6*c*x + I)*log(c^2*x^2 + 1) + 9*I*log(c^2*x^2 + 1)^2 + 18*I*log(12*c^5*d*x^2 + 12*c^3*d))*b/(c^4*d)
\[ \int \frac {x^3 (a+b \arctan (c x))}{d+i c d x} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{3}}{i \, c d x + d} \,d x } \]
Timed out. \[ \int \frac {x^3 (a+b \arctan (c x))}{d+i c d x} \, dx=\int \frac {x^3\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}{d+c\,d\,x\,1{}\mathrm {i}} \,d x \]