Integrand size = 21, antiderivative size = 211 \[ \int x (d+i c d x) (a+b \arctan (c x))^2 \, dx=-\frac {a b d x}{c}+\frac {i b^2 d x}{3 c}-\frac {i b^2 d \arctan (c x)}{3 c^2}-\frac {b^2 d x \arctan (c x)}{c}-\frac {1}{3} i b d x^2 (a+b \arctan (c x))+\frac {5 d (a+b \arctan (c x))^2}{6 c^2}+\frac {1}{2} d x^2 (a+b \arctan (c x))^2+\frac {1}{3} i c d x^3 (a+b \arctan (c x))^2-\frac {2 i b d (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^2}+\frac {b^2 d \log \left (1+c^2 x^2\right )}{2 c^2}+\frac {b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{3 c^2} \] Output:
-a*b*d*x/c+1/3*I*b^2*d*x/c-1/3*I*b^2*d*arctan(c*x)/c^2-b^2*d*x*arctan(c*x) /c-1/3*I*b*d*x^2*(a+b*arctan(c*x))+5/6*d*(a+b*arctan(c*x))^2/c^2+1/2*d*x^2 *(a+b*arctan(c*x))^2+1/3*I*c*d*x^3*(a+b*arctan(c*x))^2-2/3*I*b*d*(a+b*arct an(c*x))*ln(2/(1+I*c*x))/c^2+1/2*b^2*d*ln(c^2*x^2+1)/c^2+1/3*b^2*d*polylog (2,1-2/(1+I*c*x))/c^2
Time = 0.49 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.99 \[ \int x (d+i c d x) (a+b \arctan (c x))^2 \, dx=\frac {d \left (-6 a b c x+2 i b^2 c x+3 a^2 c^2 x^2-2 i a b c^2 x^2+2 i a^2 c^3 x^3+b^2 \left (1+3 c^2 x^2+2 i c^3 x^3\right ) \arctan (c x)^2+2 b \arctan (c x) \left (-i b \left (1-3 i c x+c^2 x^2\right )+a \left (3+3 c^2 x^2+2 i c^3 x^3\right )-2 i b \log \left (1+e^{2 i \arctan (c x)}\right )\right )+2 i a b \log \left (1+c^2 x^2\right )+3 b^2 \log \left (1+c^2 x^2\right )-2 b^2 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )}{6 c^2} \] Input:
Integrate[x*(d + I*c*d*x)*(a + b*ArcTan[c*x])^2,x]
Output:
(d*(-6*a*b*c*x + (2*I)*b^2*c*x + 3*a^2*c^2*x^2 - (2*I)*a*b*c^2*x^2 + (2*I) *a^2*c^3*x^3 + b^2*(1 + 3*c^2*x^2 + (2*I)*c^3*x^3)*ArcTan[c*x]^2 + 2*b*Arc Tan[c*x]*((-I)*b*(1 - (3*I)*c*x + c^2*x^2) + a*(3 + 3*c^2*x^2 + (2*I)*c^3* x^3) - (2*I)*b*Log[1 + E^((2*I)*ArcTan[c*x])]) + (2*I)*a*b*Log[1 + c^2*x^2 ] + 3*b^2*Log[1 + c^2*x^2] - 2*b^2*PolyLog[2, -E^((2*I)*ArcTan[c*x])]))/(6 *c^2)
Time = 0.59 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {5411, 2009}
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.
| \(\displaystyle \int x (d+i c d x) (a+b \arctan (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5411 |
| \(\displaystyle \int \left (d x (a+b \arctan (c x))^2+i c d x^2 (a+b \arctan (c x))^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5 d (a+b \arctan (c x))^2}{6 c^2}-\frac {2 i b d \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{3 c^2}+\frac {1}{3} i c d x^3 (a+b \arctan (c x))^2+\frac {1}{2} d x^2 (a+b \arctan (c x))^2-\frac {1}{3} i b d x^2 (a+b \arctan (c x))-\frac {a b d x}{c}-\frac {i b^2 d \arctan (c x)}{3 c^2}-\frac {b^2 d x \arctan (c x)}{c}+\frac {b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{3 c^2}+\frac {b^2 d \log \left (c^2 x^2+1\right )}{2 c^2}+\frac {i b^2 d x}{3 c}\) |
Input:
Int[x*(d + I*c*d*x)*(a + b*ArcTan[c*x])^2,x]
Output:
-((a*b*d*x)/c) + ((I/3)*b^2*d*x)/c - ((I/3)*b^2*d*ArcTan[c*x])/c^2 - (b^2* d*x*ArcTan[c*x])/c - (I/3)*b*d*x^2*(a + b*ArcTan[c*x]) + (5*d*(a + b*ArcTa n[c*x])^2)/(6*c^2) + (d*x^2*(a + b*ArcTan[c*x])^2)/2 + (I/3)*c*d*x^3*(a + b*ArcTan[c*x])^2 - (((2*I)/3)*b*d*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/ c^2 + (b^2*d*Log[1 + c^2*x^2])/(2*c^2) + (b^2*d*PolyLog[2, 1 - 2/(1 + I*c* x)])/(3*c^2)
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_ .)*(x_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcTan[c*x])^p, (f* x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] & & IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])
Time = 0.81 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.50
| method | result | size |
| parts | \(d \,a^{2} \left (\frac {1}{3} i c \,x^{3}+\frac {1}{2} x^{2}\right )+\frac {d \,b^{2} \left (-\frac {i \arctan \left (c x \right )}{3}+\frac {c^{2} x^{2} \arctan \left (c x \right )^{2}}{2}+\frac {i \arctan \left (c x \right )^{2} c^{3} x^{3}}{3}+\frac {i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {\arctan \left (c x \right )^{2}}{2}-c x \arctan \left (c x \right )-\frac {\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{6}+\frac {\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{6}+\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{6}+\frac {\ln \left (c x -i\right )^{2}}{12}+\frac {\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{6}-\frac {\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )}{6}-\frac {\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{6}-\frac {\ln \left (c x +i\right )^{2}}{12}-\frac {i \arctan \left (c x \right ) c^{2} x^{2}}{3}+\frac {\ln \left (c^{2} x^{2}+1\right )}{2}+\frac {i c x}{3}\right )}{c^{2}}+\frac {2 d a b \left (\frac {i \arctan \left (c x \right ) c^{3} x^{3}}{3}+\frac {c^{2} x^{2} \arctan \left (c x \right )}{2}-\frac {i c^{2} x^{2}}{6}-\frac {c x}{2}+\frac {i \ln \left (c^{2} x^{2}+1\right )}{6}+\frac {\arctan \left (c x \right )}{2}\right )}{c^{2}}\) | \(316\) |
| derivativedivides | \(\frac {d \,a^{2} \left (\frac {1}{3} i c^{3} x^{3}+\frac {1}{2} c^{2} x^{2}\right )+d \,b^{2} \left (-\frac {i \arctan \left (c x \right )}{3}+\frac {c^{2} x^{2} \arctan \left (c x \right )^{2}}{2}+\frac {i \arctan \left (c x \right )^{2} c^{3} x^{3}}{3}+\frac {i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {\arctan \left (c x \right )^{2}}{2}-c x \arctan \left (c x \right )-\frac {\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{6}+\frac {\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{6}+\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{6}+\frac {\ln \left (c x -i\right )^{2}}{12}+\frac {\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{6}-\frac {\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )}{6}-\frac {\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{6}-\frac {\ln \left (c x +i\right )^{2}}{12}-\frac {i \arctan \left (c x \right ) c^{2} x^{2}}{3}+\frac {\ln \left (c^{2} x^{2}+1\right )}{2}+\frac {i c x}{3}\right )+2 d a b \left (\frac {i \arctan \left (c x \right ) c^{3} x^{3}}{3}+\frac {c^{2} x^{2} \arctan \left (c x \right )}{2}-\frac {i c^{2} x^{2}}{6}-\frac {c x}{2}+\frac {i \ln \left (c^{2} x^{2}+1\right )}{6}+\frac {\arctan \left (c x \right )}{2}\right )}{c^{2}}\) | \(319\) |
| default | \(\frac {d \,a^{2} \left (\frac {1}{3} i c^{3} x^{3}+\frac {1}{2} c^{2} x^{2}\right )+d \,b^{2} \left (-\frac {i \arctan \left (c x \right )}{3}+\frac {c^{2} x^{2} \arctan \left (c x \right )^{2}}{2}+\frac {i \arctan \left (c x \right )^{2} c^{3} x^{3}}{3}+\frac {i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {\arctan \left (c x \right )^{2}}{2}-c x \arctan \left (c x \right )-\frac {\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{6}+\frac {\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{6}+\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{6}+\frac {\ln \left (c x -i\right )^{2}}{12}+\frac {\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{6}-\frac {\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )}{6}-\frac {\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{6}-\frac {\ln \left (c x +i\right )^{2}}{12}-\frac {i \arctan \left (c x \right ) c^{2} x^{2}}{3}+\frac {\ln \left (c^{2} x^{2}+1\right )}{2}+\frac {i c x}{3}\right )+2 d a b \left (\frac {i \arctan \left (c x \right ) c^{3} x^{3}}{3}+\frac {c^{2} x^{2} \arctan \left (c x \right )}{2}-\frac {i c^{2} x^{2}}{6}-\frac {c x}{2}+\frac {i \ln \left (c^{2} x^{2}+1\right )}{6}+\frac {\arctan \left (c x \right )}{2}\right )}{c^{2}}\) | \(319\) |
| risch | \(-\frac {d \,b^{2}}{3 c^{2}}+\frac {d \,x^{2} a^{2}}{2}+\frac {5 d \,a^{2}}{6 c^{2}}-\frac {5 d \,b^{2} \ln \left (-i c x +1\right )^{2}}{24 c^{2}}-\frac {a b d x}{c}-\frac {i d \,b^{2} \left (2 c^{3} x^{3}-3 i c^{2} x^{2}-i\right ) \ln \left (i c x +1\right )^{2}}{24 c^{2}}-\frac {d c a b \ln \left (-i c x +1\right ) x^{3}}{3}+\frac {i d b a \ln \left (-i c x +1\right ) x^{2}}{2}+\left (\frac {i d \,b^{2} \left (2 c \,x^{3}-3 i x^{2}\right ) \ln \left (-i c x +1\right )}{12}+\frac {d b \left (4 c^{3} x^{3} a -6 i a \,c^{2} x^{2}-2 b \,c^{2} x^{2}+6 i x b c +5 b \ln \left (-i c x +1\right )\right )}{12 c^{2}}\right ) \ln \left (i c x +1\right )+\frac {i b^{2} d x}{3 c}-\frac {d \,b^{2} \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{3 c^{2}}+\frac {d \,b^{2} \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i c x}{2}\right )}{3 c^{2}}+\frac {i d c \,a^{2} x^{3}}{3}-\frac {i d a b \,x^{2}}{3}-\frac {4 i d b a}{3 c^{2}}-\frac {25 i d \,b^{2} \arctan \left (c x \right )}{72 c^{2}}+\frac {d b a \arctan \left (c x \right )}{c^{2}}-\frac {i d c \,b^{2} \ln \left (-i c x +1\right )^{2} x^{3}}{12}-\frac {i d \,b^{2} \ln \left (-i c x +1\right ) x}{2 c}+\frac {i d b a \ln \left (c^{2} x^{2}+1\right )}{3 c^{2}}-\frac {d \,b^{2} \ln \left (-i c x +1\right )}{72 c^{2}}-\frac {d \,b^{2} \ln \left (-i c x +1\right )^{2} x^{2}}{8}+\frac {d \,b^{2} \ln \left (-i c x +1\right ) x^{2}}{6}+\frac {d \,b^{2} \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{3 c^{2}}+\frac {73 b^{2} d \ln \left (c^{2} x^{2}+1\right )}{144 c^{2}}\) | \(485\) |
Input:
int(x*(d+I*c*d*x)*(a+b*arctan(c*x))^2,x,method=_RETURNVERBOSE)
Output:
d*a^2*(1/3*I*c*x^3+1/2*x^2)+d*b^2/c^2*(-1/3*I*arctan(c*x)+1/2*c^2*x^2*arct an(c*x)^2+1/3*I*arctan(c*x)^2*c^3*x^3+1/3*I*ln(c^2*x^2+1)*arctan(c*x)+1/2* arctan(c*x)^2-c*x*arctan(c*x)-1/6*ln(c*x-I)*ln(c^2*x^2+1)+1/6*dilog(-1/2*I *(c*x+I))+1/6*ln(c*x-I)*ln(-1/2*I*(c*x+I))+1/12*ln(c*x-I)^2+1/6*ln(c*x+I)* ln(c^2*x^2+1)-1/6*dilog(1/2*I*(c*x-I))-1/6*ln(c*x+I)*ln(1/2*I*(c*x-I))-1/1 2*ln(c*x+I)^2-1/3*I*arctan(c*x)*c^2*x^2+1/2*ln(c^2*x^2+1)+1/3*I*c*x)+2*d*a *b/c^2*(1/3*I*arctan(c*x)*c^3*x^3+1/2*c^2*x^2*arctan(c*x)-1/6*I*c^2*x^2-1/ 2*c*x+1/6*I*ln(c^2*x^2+1)+1/2*arctan(c*x))
\[ \int x (d+i c d x) (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x \,d x } \] Input:
integrate(x*(d+I*c*d*x)*(a+b*arctan(c*x))^2,x, algorithm="fricas")
Output:
1/24*(-2*I*b^2*c*d*x^3 - 3*b^2*d*x^2)*log(-(c*x + I)/(c*x - I))^2 + integr al(1/6*(6*I*a^2*c^3*d*x^4 + 6*a^2*c^2*d*x^3 + 6*I*a^2*c*d*x^2 + 6*a^2*d*x - (6*a*b*c^3*d*x^4 + 2*(-3*I*a*b - b^2)*c^2*d*x^3 + 3*(2*a*b + I*b^2)*c*d* x^2 - 6*I*a*b*d*x)*log(-(c*x + I)/(c*x - I)))/(c^2*x^2 + 1), x)
Timed out. \[ \int x (d+i c d x) (a+b \arctan (c x))^2 \, dx=\text {Timed out} \] Input:
integrate(x*(d+I*c*d*x)*(a+b*atan(c*x))**2,x)
Output:
Timed out
\[ \int x (d+i c d x) (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x \,d x } \] Input:
integrate(x*(d+I*c*d*x)*(a+b*arctan(c*x))^2,x, algorithm="maxima")
Output:
1/3*I*a^2*c*d*x^3 + 1/2*b^2*d*x^2*arctan(c*x)^2 + 1/3*I*(2*x^3*arctan(c*x) - c*(x^2/c^2 - log(c^2*x^2 + 1)/c^4))*a*b*c*d + 1/48*I*(4*x^3*arctan(c*x) ^2 - x^3*log(c^2*x^2 + 1)^2 + 48*integrate(1/48*(4*c^2*x^4*log(c^2*x^2 + 1 ) - 8*c*x^3*arctan(c*x) + 36*(c^2*x^4 + x^2)*arctan(c*x)^2 + 3*(c^2*x^4 + x^2)*log(c^2*x^2 + 1)^2)/(c^2*x^2 + 1), x))*b^2*c*d + 1/2*a^2*d*x^2 + (x^2 *arctan(c*x) - c*(x/c^2 - arctan(c*x)/c^3))*a*b*d - 1/2*(2*c*(x/c^2 - arct an(c*x)/c^3)*arctan(c*x) + (arctan(c*x)^2 - log(c^2*x^2 + 1))/c^2)*b^2*d
\[ \int x (d+i c d x) (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x \,d x } \] Input:
integrate(x*(d+I*c*d*x)*(a+b*arctan(c*x))^2,x, algorithm="giac")
Output:
integrate((I*c*d*x + d)*(b*arctan(c*x) + a)^2*x, x)
Timed out. \[ \int x (d+i c d x) (a+b \arctan (c x))^2 \, dx=\int x\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,\left (d+c\,d\,x\,1{}\mathrm {i}\right ) \,d x \] Input:
int(x*(a + b*atan(c*x))^2*(d + c*d*x*1i),x)
Output:
int(x*(a + b*atan(c*x))^2*(d + c*d*x*1i), x)
\[ \int x (d+i c d x) (a+b \arctan (c x))^2 \, dx=\frac {d \left (2 \mathit {atan} \left (c x \right )^{2} b^{2} c^{3} i \,x^{3}+3 \mathit {atan} \left (c x \right )^{2} b^{2} c^{2} x^{2}+2 \mathit {atan} \left (c x \right )^{2} b^{2} c i x +3 \mathit {atan} \left (c x \right )^{2} b^{2}+4 \mathit {atan} \left (c x \right ) a b \,c^{3} i \,x^{3}+6 \mathit {atan} \left (c x \right ) a b \,c^{2} x^{2}+6 \mathit {atan} \left (c x \right ) a b -2 \mathit {atan} \left (c x \right ) b^{2} c^{2} i \,x^{2}-6 \mathit {atan} \left (c x \right ) b^{2} c x -2 \mathit {atan} \left (c x \right ) b^{2} i -2 \left (\int \mathit {atan} \left (c x \right )^{2}d x \right ) b^{2} c i +2 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) a b i +3 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) b^{2}+2 a^{2} c^{3} i \,x^{3}+3 a^{2} c^{2} x^{2}-2 a b \,c^{2} i \,x^{2}-6 a b c x +2 b^{2} c i x \right )}{6 c^{2}} \] Input:
int(x*(d+I*c*d*x)*(a+b*atan(c*x))^2,x)
Output:
(d*(2*atan(c*x)**2*b**2*c**3*i*x**3 + 3*atan(c*x)**2*b**2*c**2*x**2 + 2*at an(c*x)**2*b**2*c*i*x + 3*atan(c*x)**2*b**2 + 4*atan(c*x)*a*b*c**3*i*x**3 + 6*atan(c*x)*a*b*c**2*x**2 + 6*atan(c*x)*a*b - 2*atan(c*x)*b**2*c**2*i*x* *2 - 6*atan(c*x)*b**2*c*x - 2*atan(c*x)*b**2*i - 2*int(atan(c*x)**2,x)*b** 2*c*i + 2*log(c**2*x**2 + 1)*a*b*i + 3*log(c**2*x**2 + 1)*b**2 + 2*a**2*c* *3*i*x**3 + 3*a**2*c**2*x**2 - 2*a*b*c**2*i*x**2 - 6*a*b*c*x + 2*b**2*c*i* x))/(6*c**2)