Integrand size = 18, antiderivative size = 270 \[ \int (d+e x)^2 (a+b \arctan (c x))^2 \, dx=-\frac {2 a b d e x}{c}+\frac {b^2 e^2 x}{3 c^2}-\frac {b^2 e^2 \arctan (c x)}{3 c^3}-\frac {2 b^2 d e x \arctan (c x)}{c}-\frac {b e^2 x^2 (a+b \arctan (c x))}{3 c}+\frac {i \left (3 c^2 d^2-e^2\right ) (a+b \arctan (c x))^2}{3 c^3}-\frac {d \left (d^2-\frac {3 e^2}{c^2}\right ) (a+b \arctan (c x))^2}{3 e}+\frac {(d+e x)^3 (a+b \arctan (c x))^2}{3 e}+\frac {2 b \left (3 c^2 d^2-e^2\right ) (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}+\frac {b^2 d e \log \left (1+c^2 x^2\right )}{c^2}+\frac {i b^2 \left (3 c^2 d^2-e^2\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{3 c^3} \] Output:
-2*a*b*d*e*x/c+1/3*b^2*e^2*x/c^2-1/3*b^2*e^2*arctan(c*x)/c^3-2*b^2*d*e*x*a rctan(c*x)/c-1/3*b*e^2*x^2*(a+b*arctan(c*x))/c+1/3*I*(3*c^2*d^2-e^2)*(a+b* arctan(c*x))^2/c^3-1/3*d*(d^2-3*e^2/c^2)*(a+b*arctan(c*x))^2/e+1/3*(e*x+d) ^3*(a+b*arctan(c*x))^2/e+2/3*b*(3*c^2*d^2-e^2)*(a+b*arctan(c*x))*ln(2/(1+I *c*x))/c^3+b^2*d*e*ln(c^2*x^2+1)/c^2+1/3*I*b^2*(3*c^2*d^2-e^2)*polylog(2,1 -2/(1+I*c*x))/c^3
Time = 0.36 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.16 \[ \int (d+e x)^2 (a+b \arctan (c x))^2 \, dx=\frac {3 a^2 c^3 d^2 x-6 a b c^2 d e x+b^2 c e^2 x+3 a^2 c^3 d e x^2-a b c^2 e^2 x^2+a^2 c^3 e^2 x^3+b^2 \left (-3 i c^2 d^2+3 c d e+i e^2+c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )\right ) \arctan (c x)^2+b \arctan (c x) \left (6 a c d e-b e \left (e+6 c^2 d x+c^2 e x^2\right )+2 a c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )+2 b \left (3 c^2 d^2-e^2\right ) \log \left (1+e^{2 i \arctan (c x)}\right )\right )-3 a b c^2 d^2 \log \left (1+c^2 x^2\right )+3 b^2 c d e \log \left (1+c^2 x^2\right )+a b e^2 \log \left (1+c^2 x^2\right )-i b^2 \left (3 c^2 d^2-e^2\right ) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )}{3 c^3} \] Input:
Integrate[(d + e*x)^2*(a + b*ArcTan[c*x])^2,x]
Output:
(3*a^2*c^3*d^2*x - 6*a*b*c^2*d*e*x + b^2*c*e^2*x + 3*a^2*c^3*d*e*x^2 - a*b *c^2*e^2*x^2 + a^2*c^3*e^2*x^3 + b^2*((-3*I)*c^2*d^2 + 3*c*d*e + I*e^2 + c ^3*x*(3*d^2 + 3*d*e*x + e^2*x^2))*ArcTan[c*x]^2 + b*ArcTan[c*x]*(6*a*c*d*e - b*e*(e + 6*c^2*d*x + c^2*e*x^2) + 2*a*c^3*x*(3*d^2 + 3*d*e*x + e^2*x^2) + 2*b*(3*c^2*d^2 - e^2)*Log[1 + E^((2*I)*ArcTan[c*x])]) - 3*a*b*c^2*d^2*L og[1 + c^2*x^2] + 3*b^2*c*d*e*Log[1 + c^2*x^2] + a*b*e^2*Log[1 + c^2*x^2] - I*b^2*(3*c^2*d^2 - e^2)*PolyLog[2, -E^((2*I)*ArcTan[c*x])])/(3*c^3)
Time = 0.62 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.05, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5389, 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.
| \(\displaystyle \int (d+e x)^2 (a+b \arctan (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5389 |
| \(\displaystyle \frac {(d+e x)^3 (a+b \arctan (c x))^2}{3 e}-\frac {2 b c \int \left (\frac {x (a+b \arctan (c x)) e^3}{c^2}+\frac {3 d (a+b \arctan (c x)) e^2}{c^2}+\frac {\left (d \left (c^2 d^2-3 e^2\right )+e \left (3 c^2 d^2-e^2\right ) x\right ) (a+b \arctan (c x))}{c^2 \left (c^2 x^2+1\right )}\right )dx}{3 e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(d+e x)^3 (a+b \arctan (c x))^2}{3 e}-\frac {2 b c \left (\frac {e^3 x^2 (a+b \arctan (c x))}{2 c^2}-\frac {i e \left (3 c^2 d^2-e^2\right ) (a+b \arctan (c x))^2}{2 b c^4}-\frac {e \left (3 c^2 d^2-e^2\right ) \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c^4}+\frac {d \left (c^2 d^2-3 e^2\right ) (a+b \arctan (c x))^2}{2 b c^3}+\frac {3 a d e^2 x}{c^2}+\frac {b e^3 \arctan (c x)}{2 c^4}+\frac {3 b d e^2 x \arctan (c x)}{c^2}-\frac {b e^3 x}{2 c^3}-\frac {i b e \left (3 c^2 d^2-e^2\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{2 c^4}-\frac {3 b d e^2 \log \left (c^2 x^2+1\right )}{2 c^3}\right )}{3 e}\) |
Input:
Int[(d + e*x)^2*(a + b*ArcTan[c*x])^2,x]
Output:
((d + e*x)^3*(a + b*ArcTan[c*x])^2)/(3*e) - (2*b*c*((3*a*d*e^2*x)/c^2 - (b *e^3*x)/(2*c^3) + (b*e^3*ArcTan[c*x])/(2*c^4) + (3*b*d*e^2*x*ArcTan[c*x])/ c^2 + (e^3*x^2*(a + b*ArcTan[c*x]))/(2*c^2) + (d*(c^2*d^2 - 3*e^2)*(a + b* ArcTan[c*x])^2)/(2*b*c^3) - ((I/2)*e*(3*c^2*d^2 - e^2)*(a + b*ArcTan[c*x]) ^2)/(b*c^4) - (e*(3*c^2*d^2 - e^2)*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)]) /c^4 - (3*b*d*e^2*Log[1 + c^2*x^2])/(2*c^3) - ((I/2)*b*e*(3*c^2*d^2 - e^2) *PolyLog[2, 1 - 2/(1 + I*c*x)])/c^4))/(3*e)
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(q_.), x_Sy mbol] :> Simp[(d + e*x)^(q + 1)*((a + b*ArcTan[c*x])^p/(e*(q + 1))), x] - S imp[b*c*(p/(e*(q + 1))) Int[ExpandIntegrand[(a + b*ArcTan[c*x])^(p - 1), (d + e*x)^(q + 1)/(1 + c^2*x^2), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 1] && IntegerQ[q] && NeQ[q, -1]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 502 vs. \(2 (250 ) = 500\).
Time = 0.56 (sec) , antiderivative size = 503, normalized size of antiderivative = 1.86
| method | result | size |
| parts | \(\frac {a^{2} \left (e x +d \right )^{3}}{3 e}+\frac {b^{2} \left (\frac {c \,e^{2} \arctan \left (c x \right )^{2} x^{3}}{3}+c e \arctan \left (c x \right )^{2} x^{2} d +\arctan \left (c x \right )^{2} c x \,d^{2}+\frac {c \arctan \left (c x \right )^{2} d^{3}}{3 e}-\frac {2 \left (3 \arctan \left (c x \right ) c^{2} d \,e^{2} x +\frac {\arctan \left (c x \right ) e^{3} c^{2} x^{2}}{2}+\frac {3 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{2} d^{2} e}{2}-\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) e^{3}}{2}+\arctan \left (c x \right )^{2} c^{3} d^{3}-3 \arctan \left (c x \right )^{2} c d \,e^{2}-\frac {e \left (3 c^{2} d^{2}-e^{2}\right ) \left (-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{2}\right )}{2}-\frac {3 \ln \left (c^{2} x^{2}+1\right ) c d \,e^{2}}{2}+\frac {\arctan \left (c x \right ) e^{3}}{2}-\frac {e^{3} c x}{2}-\frac {d c \left (c^{2} d^{2}-3 e^{2}\right ) \arctan \left (c x \right )^{2}}{2}\right )}{3 c^{2} e}\right )}{c}+\frac {2 a b \,e^{2} \arctan \left (c x \right ) x^{3}}{3}+2 a b e \arctan \left (c x \right ) x^{2} d +2 a b \arctan \left (c x \right ) x \,d^{2}-\frac {e^{2} b a \,x^{2}}{3 c}-\frac {2 a b d e x}{c}-\frac {b \ln \left (c^{2} x^{2}+1\right ) a \,d^{2}}{c}+\frac {b \ln \left (c^{2} x^{2}+1\right ) a \,e^{2}}{3 c^{3}}+\frac {2 d e b a \arctan \left (c x \right )}{c^{2}}\) | \(503\) |
| derivativedivides | \(\frac {\frac {a^{2} \left (c e x +c d \right )^{3}}{3 c^{2} e}+\frac {b^{2} \left (\frac {\arctan \left (c x \right )^{2} c^{3} d^{3}}{3 e}+\arctan \left (c x \right )^{2} c^{3} d^{2} x +e \arctan \left (c x \right )^{2} c^{3} d \,x^{2}+\frac {e^{2} \arctan \left (c x \right )^{2} c^{3} x^{3}}{3}-\frac {2 \left (3 \arctan \left (c x \right ) c^{2} d \,e^{2} x +\frac {\arctan \left (c x \right ) e^{3} c^{2} x^{2}}{2}+\frac {3 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{2} d^{2} e}{2}-\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) e^{3}}{2}+\arctan \left (c x \right )^{2} c^{3} d^{3}-3 \arctan \left (c x \right )^{2} c d \,e^{2}-\frac {e \left (3 c^{2} d^{2}-e^{2}\right ) \left (-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{2}\right )}{2}-\frac {3 \ln \left (c^{2} x^{2}+1\right ) c d \,e^{2}}{2}+\frac {\arctan \left (c x \right ) e^{3}}{2}-\frac {e^{3} c x}{2}-\frac {d c \left (c^{2} d^{2}-3 e^{2}\right ) \arctan \left (c x \right )^{2}}{2}\right )}{3 e}\right )}{c^{2}}+2 a b \arctan \left (c x \right ) d^{2} c x +2 a b c e \arctan \left (c x \right ) d \,x^{2}+\frac {2 a b c \,e^{2} \arctan \left (c x \right ) x^{3}}{3}-2 a b d e x -\frac {a b \,e^{2} x^{2}}{3}-a b \ln \left (c^{2} x^{2}+1\right ) d^{2}+\frac {a b \,e^{2} \ln \left (c^{2} x^{2}+1\right )}{3 c^{2}}+\frac {2 a b e \arctan \left (c x \right ) d}{c}}{c}\) | \(512\) |
| default | \(\frac {\frac {a^{2} \left (c e x +c d \right )^{3}}{3 c^{2} e}+\frac {b^{2} \left (\frac {\arctan \left (c x \right )^{2} c^{3} d^{3}}{3 e}+\arctan \left (c x \right )^{2} c^{3} d^{2} x +e \arctan \left (c x \right )^{2} c^{3} d \,x^{2}+\frac {e^{2} \arctan \left (c x \right )^{2} c^{3} x^{3}}{3}-\frac {2 \left (3 \arctan \left (c x \right ) c^{2} d \,e^{2} x +\frac {\arctan \left (c x \right ) e^{3} c^{2} x^{2}}{2}+\frac {3 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{2} d^{2} e}{2}-\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) e^{3}}{2}+\arctan \left (c x \right )^{2} c^{3} d^{3}-3 \arctan \left (c x \right )^{2} c d \,e^{2}-\frac {e \left (3 c^{2} d^{2}-e^{2}\right ) \left (-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{2}\right )}{2}-\frac {3 \ln \left (c^{2} x^{2}+1\right ) c d \,e^{2}}{2}+\frac {\arctan \left (c x \right ) e^{3}}{2}-\frac {e^{3} c x}{2}-\frac {d c \left (c^{2} d^{2}-3 e^{2}\right ) \arctan \left (c x \right )^{2}}{2}\right )}{3 e}\right )}{c^{2}}+2 a b \arctan \left (c x \right ) d^{2} c x +2 a b c e \arctan \left (c x \right ) d \,x^{2}+\frac {2 a b c \,e^{2} \arctan \left (c x \right ) x^{3}}{3}-2 a b d e x -\frac {a b \,e^{2} x^{2}}{3}-a b \ln \left (c^{2} x^{2}+1\right ) d^{2}+\frac {a b \,e^{2} \ln \left (c^{2} x^{2}+1\right )}{3 c^{2}}+\frac {2 a b e \arctan \left (c x \right ) d}{c}}{c}\) | \(512\) |
| risch | \(\frac {x^{3} e^{2} a^{2}}{3}+x \,d^{2} a^{2}+\frac {i e^{2} b^{2} \ln \left (-i c x +1\right )^{2}}{12 c^{3}}-\frac {i \ln \left (-i c x +1\right )^{2} b^{2} d^{2}}{4 c}-\frac {i b^{2} \ln \left (\frac {1}{2}-\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) e^{2}}{3 c^{3}}-\frac {i d e \,b^{2} \ln \left (-i c x +1\right ) x}{c}+\frac {b^{2} d^{2} \arctan \left (c x \right )}{2 c}+x^{2} e d \,a^{2}-\frac {b \ln \left (c^{2} x^{2}+1\right ) a \,d^{2}}{c}+\left (\frac {\left (e x +d \right )^{3} b^{2} \ln \left (-i c x +1\right )}{6 e}+\frac {b \left (-2 i a \,c^{3} e^{3} x^{3}-6 i a \,c^{3} d \,e^{2} x^{2}-6 i a \,c^{3} d^{2} e x +i b \,c^{2} e^{3} x^{2}+3 i \ln \left (-i c x +1\right ) b \,c^{2} d^{2} e +6 i b \,c^{2} d \,e^{2} x -\ln \left (-i c x +1\right ) b \,c^{3} d^{3}-i \ln \left (-i c x +1\right ) b \,e^{3}+3 \ln \left (-i c x +1\right ) b c d \,e^{2}\right )}{6 c^{3} e}\right ) \ln \left (i c x +1\right )+\frac {b^{2} e^{2} x}{3 c^{2}}-\frac {17 b^{2} e^{2} \arctan \left (c x \right )}{36 c^{3}}-\frac {2 i d e b a}{c^{2}}+\frac {i b^{2} \arctan \left (c x \right ) d e}{4 c^{2}}-\frac {i e^{2} b^{2} \ln \left (-i c x +1\right ) x^{2}}{6 c}+i \ln \left (-i c x +1\right ) x a b \,d^{2}+\frac {i b^{2} \ln \left (\frac {1}{2}-\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) d^{2}}{c}-\frac {i b^{2} \ln \left (-i c x +1\right ) \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) d^{2}}{c}+\frac {i b^{2} \ln \left (-i c x +1\right ) \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) e^{2}}{3 c^{3}}+i d e b a \ln \left (-i c x +1\right ) x^{2}+\frac {b \ln \left (c^{2} x^{2}+1\right ) a \,e^{2}}{3 c^{3}}-\frac {e^{2} b a \,x^{2}}{3 c}+\frac {e d \,a^{2}}{c^{2}}-\frac {2 a b d e x}{c}+\frac {i e^{2} b a \ln \left (-i c x +1\right ) x^{3}}{3}-\frac {e^{2} b a}{3 c^{3}}+\frac {7 b^{2} d e \ln \left (c^{2} x^{2}+1\right )}{8 c^{2}}+\frac {i e^{2} b^{2}}{3 c^{3}}+\frac {i d^{2} a^{2}}{c}-\frac {i e^{2} a^{2}}{3 c^{3}}-\frac {b^{2} \left (c^{3} e^{2} x^{3}+3 c^{3} d e \,x^{2}+3 c^{3} d^{2} x -3 i c^{2} d^{2}+3 c e d +i e^{2}\right ) \ln \left (i c x +1\right )^{2}}{12 c^{3}}-\frac {e^{2} b^{2} \ln \left (-i c x +1\right )^{2} x^{3}}{12}-\frac {\ln \left (-i c x +1\right )^{2} x \,b^{2} d^{2}}{4}+\frac {2 d e b a \arctan \left (c x \right )}{c^{2}}-\frac {5 i e^{2} b^{2} \ln \left (c^{2} x^{2}+1\right )}{72 c^{3}}+\frac {i b^{2} d^{2} \ln \left (c^{2} x^{2}+1\right )}{4 c}+\frac {i b^{2} \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right ) d^{2}}{c}-\frac {i b^{2} \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right ) e^{2}}{3 c^{3}}+\frac {5 i b^{2} e^{2} \ln \left (-i c x +1\right )}{36 c^{3}}-\frac {i b^{2} \ln \left (-i c x +1\right ) d^{2}}{2 c}-\frac {d e \,b^{2} \ln \left (-i c x +1\right )^{2} x^{2}}{4}-\frac {d e \,b^{2} \ln \left (-i c x +1\right )^{2}}{4 c^{2}}+\frac {d e \,b^{2} \ln \left (-i c x +1\right )}{4 c^{2}}\) | \(965\) |
Input:
int((e*x+d)^2*(a+b*arctan(c*x))^2,x,method=_RETURNVERBOSE)
Output:
1/3*a^2*(e*x+d)^3/e+b^2/c*(1/3*c*e^2*arctan(c*x)^2*x^3+c*e*arctan(c*x)^2*x ^2*d+arctan(c*x)^2*c*x*d^2+1/3*c/e*arctan(c*x)^2*d^3-2/3/c^2/e*(3*arctan(c *x)*c^2*d*e^2*x+1/2*arctan(c*x)*e^3*c^2*x^2+3/2*arctan(c*x)*ln(c^2*x^2+1)* c^2*d^2*e-1/2*arctan(c*x)*ln(c^2*x^2+1)*e^3+arctan(c*x)^2*c^3*d^3-3*arctan (c*x)^2*c*d*e^2-1/2*e*(3*c^2*d^2-e^2)*(-1/2*I*(ln(c*x-I)*ln(c^2*x^2+1)-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)))+1/2*I*(ln (c*x+I)*ln(c^2*x^2+1)-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))))-3/2*ln(c^2*x^2+1)*c*d*e^2+1/2*arctan(c*x)*e^3-1/2*e^3*c*x-1 /2*d*c*(c^2*d^2-3*e^2)*arctan(c*x)^2))+2/3*a*b*e^2*arctan(c*x)*x^3+2*a*b*e *arctan(c*x)*x^2*d+2*a*b*arctan(c*x)*x*d^2-1/3/c*e^2*b*a*x^2-2*a*b*d*e*x/c -1/c*b*ln(c^2*x^2+1)*a*d^2+1/3/c^3*b*ln(c^2*x^2+1)*a*e^2+2/c^2*d*e*b*a*arc tan(c*x)
\[ \int (d+e x)^2 (a+b \arctan (c x))^2 \, dx=\int { {\left (e x + d\right )}^{2} {\left (b \arctan \left (c x\right ) + a\right )}^{2} \,d x } \] Input:
integrate((e*x+d)^2*(a+b*arctan(c*x))^2,x, algorithm="fricas")
Output:
integral(a^2*e^2*x^2 + 2*a^2*d*e*x + a^2*d^2 + (b^2*e^2*x^2 + 2*b^2*d*e*x + b^2*d^2)*arctan(c*x)^2 + 2*(a*b*e^2*x^2 + 2*a*b*d*e*x + a*b*d^2)*arctan( c*x), x)
\[ \int (d+e x)^2 (a+b \arctan (c x))^2 \, dx=\int \left (a + b \operatorname {atan}{\left (c x \right )}\right )^{2} \left (d + e x\right )^{2}\, dx \] Input:
integrate((e*x+d)**2*(a+b*atan(c*x))**2,x)
Output:
Integral((a + b*atan(c*x))**2*(d + e*x)**2, x)
\[ \int (d+e x)^2 (a+b \arctan (c x))^2 \, dx=\int { {\left (e x + d\right )}^{2} {\left (b \arctan \left (c x\right ) + a\right )}^{2} \,d x } \] Input:
integrate((e*x+d)^2*(a+b*arctan(c*x))^2,x, algorithm="maxima")
Output:
1/3*a^2*e^2*x^3 + 36*b^2*c^2*e^2*integrate(1/48*x^4*arctan(c*x)^2/(c^2*x^2 + 1), x) + 3*b^2*c^2*e^2*integrate(1/48*x^4*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 72*b^2*c^2*d*e*integrate(1/48*x^3*arctan(c*x)^2/(c^2*x^2 + 1), x ) + 4*b^2*c^2*e^2*integrate(1/48*x^4*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) + 6*b^2*c^2*d*e*integrate(1/48*x^3*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 36 *b^2*c^2*d^2*integrate(1/48*x^2*arctan(c*x)^2/(c^2*x^2 + 1), x) + 12*b^2*c ^2*d*e*integrate(1/48*x^3*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) + 3*b^2*c^2*d ^2*integrate(1/48*x^2*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 12*b^2*c^2*d^ 2*integrate(1/48*x^2*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) + a^2*d*e*x^2 + 1/ 4*b^2*d^2*arctan(c*x)^3/c - 8*b^2*c*e^2*integrate(1/48*x^3*arctan(c*x)/(c^ 2*x^2 + 1), x) - 24*b^2*c*d*e*integrate(1/48*x^2*arctan(c*x)/(c^2*x^2 + 1) , x) - 24*b^2*c*d^2*integrate(1/48*x*arctan(c*x)/(c^2*x^2 + 1), x) + 2*(x^ 2*arctan(c*x) - c*(x/c^2 - arctan(c*x)/c^3))*a*b*d*e + 1/3*(2*x^3*arctan(c *x) - c*(x^2/c^2 - log(c^2*x^2 + 1)/c^4))*a*b*e^2 + a^2*d^2*x + 36*b^2*e^2 *integrate(1/48*x^2*arctan(c*x)^2/(c^2*x^2 + 1), x) + 3*b^2*e^2*integrate( 1/48*x^2*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 72*b^2*d*e*integrate(1/48* x*arctan(c*x)^2/(c^2*x^2 + 1), x) + 6*b^2*d*e*integrate(1/48*x*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 3*b^2*d^2*integrate(1/48*log(c^2*x^2 + 1)^2/(c ^2*x^2 + 1), x) + (2*c*x*arctan(c*x) - log(c^2*x^2 + 1))*a*b*d^2/c + 1/12* (b^2*e^2*x^3 + 3*b^2*d*e*x^2 + 3*b^2*d^2*x)*arctan(c*x)^2 - 1/48*(b^2*e...
\[ \int (d+e x)^2 (a+b \arctan (c x))^2 \, dx=\int { {\left (e x + d\right )}^{2} {\left (b \arctan \left (c x\right ) + a\right )}^{2} \,d x } \] Input:
integrate((e*x+d)^2*(a+b*arctan(c*x))^2,x, algorithm="giac")
Output:
integrate((e*x + d)^2*(b*arctan(c*x) + a)^2, x)
Timed out. \[ \int (d+e x)^2 (a+b \arctan (c x))^2 \, dx=\int {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (d+e\,x\right )}^2 \,d x \] Input:
int((a + b*atan(c*x))^2*(d + e*x)^2,x)
Output:
int((a + b*atan(c*x))^2*(d + e*x)^2, x)
\[ \int (d+e x)^2 (a+b \arctan (c x))^2 \, dx=\frac {3 \mathit {atan} \left (c x \right )^{2} b^{2} c^{3} d^{2} x +3 \mathit {atan} \left (c x \right )^{2} b^{2} c^{3} d e \,x^{2}+\mathit {atan} \left (c x \right )^{2} b^{2} c^{3} e^{2} x^{3}+3 \mathit {atan} \left (c x \right )^{2} b^{2} c d e +6 \mathit {atan} \left (c x \right ) a b \,c^{3} d^{2} x +6 \mathit {atan} \left (c x \right ) a b \,c^{3} d e \,x^{2}+2 \mathit {atan} \left (c x \right ) a b \,c^{3} e^{2} x^{3}+6 \mathit {atan} \left (c x \right ) a b c d e -6 \mathit {atan} \left (c x \right ) b^{2} c^{2} d e x -\mathit {atan} \left (c x \right ) b^{2} c^{2} e^{2} x^{2}-\mathit {atan} \left (c x \right ) b^{2} e^{2}-6 \left (\int \frac {\mathit {atan} \left (c x \right ) x}{c^{2} x^{2}+1}d x \right ) b^{2} c^{4} d^{2}+2 \left (\int \frac {\mathit {atan} \left (c x \right ) x}{c^{2} x^{2}+1}d x \right ) b^{2} c^{2} e^{2}-3 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) a b \,c^{2} d^{2}+\mathrm {log}\left (c^{2} x^{2}+1\right ) a b \,e^{2}+3 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) b^{2} c d e +3 a^{2} c^{3} d^{2} x +3 a^{2} c^{3} d e \,x^{2}+a^{2} c^{3} e^{2} x^{3}-6 a b \,c^{2} d e x -a b \,c^{2} e^{2} x^{2}+b^{2} c \,e^{2} x}{3 c^{3}} \] Input:
int((e*x+d)^2*(a+b*atan(c*x))^2,x)
Output:
(3*atan(c*x)**2*b**2*c**3*d**2*x + 3*atan(c*x)**2*b**2*c**3*d*e*x**2 + ata n(c*x)**2*b**2*c**3*e**2*x**3 + 3*atan(c*x)**2*b**2*c*d*e + 6*atan(c*x)*a* b*c**3*d**2*x + 6*atan(c*x)*a*b*c**3*d*e*x**2 + 2*atan(c*x)*a*b*c**3*e**2* x**3 + 6*atan(c*x)*a*b*c*d*e - 6*atan(c*x)*b**2*c**2*d*e*x - atan(c*x)*b** 2*c**2*e**2*x**2 - atan(c*x)*b**2*e**2 - 6*int((atan(c*x)*x)/(c**2*x**2 + 1),x)*b**2*c**4*d**2 + 2*int((atan(c*x)*x)/(c**2*x**2 + 1),x)*b**2*c**2*e* *2 - 3*log(c**2*x**2 + 1)*a*b*c**2*d**2 + log(c**2*x**2 + 1)*a*b*e**2 + 3* log(c**2*x**2 + 1)*b**2*c*d*e + 3*a**2*c**3*d**2*x + 3*a**2*c**3*d*e*x**2 + a**2*c**3*e**2*x**3 - 6*a*b*c**2*d*e*x - a*b*c**2*e**2*x**2 + b**2*c*e** 2*x)/(3*c**3)