Integrand size = 18, antiderivative size = 376 \[ \int (d+e x)^3 (a+b \arctan (c x))^2 \, dx=\frac {b^2 d e^2 x}{c^2}-\frac {a b e \left (6 c^2 d^2-e^2\right ) x}{2 c^3}+\frac {b^2 e^3 x^2}{12 c^2}-\frac {b^2 d e^2 \arctan (c x)}{c^3}-\frac {b^2 e \left (6 c^2 d^2-e^2\right ) x \arctan (c x)}{2 c^3}-\frac {b d e^2 x^2 (a+b \arctan (c x))}{c}-\frac {b e^3 x^3 (a+b \arctan (c x))}{6 c}+\frac {i d (c d-e) (c d+e) (a+b \arctan (c x))^2}{c^3}-\frac {\left (c^4 d^4-6 c^2 d^2 e^2+e^4\right ) (a+b \arctan (c x))^2}{4 c^4 e}+\frac {(d+e x)^4 (a+b \arctan (c x))^2}{4 e}+\frac {2 b d (c d-e) (c d+e) (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^3}-\frac {b^2 e^3 \log \left (1+c^2 x^2\right )}{12 c^4}+\frac {b^2 e \left (6 c^2 d^2-e^2\right ) \log \left (1+c^2 x^2\right )}{4 c^4}+\frac {i b^2 d (c d-e) (c d+e) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^3} \] Output:
b^2*d*e^2*x/c^2-1/2*a*b*e*(6*c^2*d^2-e^2)*x/c^3+1/12*b^2*e^3*x^2/c^2-b^2*d *e^2*arctan(c*x)/c^3-1/2*b^2*e*(6*c^2*d^2-e^2)*x*arctan(c*x)/c^3-b*d*e^2*x ^2*(a+b*arctan(c*x))/c-1/6*b*e^3*x^3*(a+b*arctan(c*x))/c+I*d*(c*d-e)*(c*d+ e)*(a+b*arctan(c*x))^2/c^3-1/4*(c^4*d^4-6*c^2*d^2*e^2+e^4)*(a+b*arctan(c*x ))^2/c^4/e+1/4*(e*x+d)^4*(a+b*arctan(c*x))^2/e+2*b*d*(c*d-e)*(c*d+e)*(a+b* arctan(c*x))*ln(2/(1+I*c*x))/c^3-1/12*b^2*e^3*ln(c^2*x^2+1)/c^4+1/4*b^2*e* (6*c^2*d^2-e^2)*ln(c^2*x^2+1)/c^4+I*b^2*d*(c*d-e)*(c*d+e)*polylog(2,1-2/(1 +I*c*x))/c^3
Time = 0.53 (sec) , antiderivative size = 472, normalized size of antiderivative = 1.26 \[ \int (d+e x)^3 (a+b \arctan (c x))^2 \, dx=\frac {b^2 e^3+12 a^2 c^4 d^3 x-36 a b c^3 d^2 e x+12 b^2 c^2 d e^2 x+6 a b c e^3 x+18 a^2 c^4 d^2 e x^2-12 a b c^3 d e^2 x^2+b^2 c^2 e^3 x^2+12 a^2 c^4 d e^2 x^3-2 a b c^3 e^3 x^3+3 a^2 c^4 e^3 x^4+3 b^2 \left (-4 i c^3 d^3+6 c^2 d^2 e+4 i c d e^2-e^3+c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )\right ) \arctan (c x)^2+2 b \arctan (c x) \left (-b c e \left (18 c^2 d^2 x+e^2 x \left (-3+c^2 x^2\right )+6 d \left (e+c^2 e x^2\right )\right )+3 a \left (6 c^2 d^2 e-e^3+c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )\right )+12 b c d \left (c^2 d^2-e^2\right ) \log \left (1+e^{2 i \arctan (c x)}\right )\right )-12 a b c^3 d^3 \log \left (1+c^2 x^2\right )+18 b^2 c^2 d^2 e \log \left (1+c^2 x^2\right )+12 a b c d e^2 \log \left (1+c^2 x^2\right )-4 b^2 e^3 \log \left (1+c^2 x^2\right )-12 i b^2 c d \left (c^2 d^2-e^2\right ) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )}{12 c^4} \] Input:
Integrate[(d + e*x)^3*(a + b*ArcTan[c*x])^2,x]
Output:
(b^2*e^3 + 12*a^2*c^4*d^3*x - 36*a*b*c^3*d^2*e*x + 12*b^2*c^2*d*e^2*x + 6* a*b*c*e^3*x + 18*a^2*c^4*d^2*e*x^2 - 12*a*b*c^3*d*e^2*x^2 + b^2*c^2*e^3*x^ 2 + 12*a^2*c^4*d*e^2*x^3 - 2*a*b*c^3*e^3*x^3 + 3*a^2*c^4*e^3*x^4 + 3*b^2*( (-4*I)*c^3*d^3 + 6*c^2*d^2*e + (4*I)*c*d*e^2 - e^3 + c^4*x*(4*d^3 + 6*d^2* e*x + 4*d*e^2*x^2 + e^3*x^3))*ArcTan[c*x]^2 + 2*b*ArcTan[c*x]*(-(b*c*e*(18 *c^2*d^2*x + e^2*x*(-3 + c^2*x^2) + 6*d*(e + c^2*e*x^2))) + 3*a*(6*c^2*d^2 *e - e^3 + c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3)) + 12*b*c*d*( c^2*d^2 - e^2)*Log[1 + E^((2*I)*ArcTan[c*x])]) - 12*a*b*c^3*d^3*Log[1 + c^ 2*x^2] + 18*b^2*c^2*d^2*e*Log[1 + c^2*x^2] + 12*a*b*c*d*e^2*Log[1 + c^2*x^ 2] - 4*b^2*e^3*Log[1 + c^2*x^2] - (12*I)*b^2*c*d*(c^2*d^2 - e^2)*PolyLog[2 , -E^((2*I)*ArcTan[c*x])])/(12*c^4)
Time = 0.81 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, 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.
| \(\displaystyle \int (d+e x)^3 (a+b \arctan (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5389 |
| \(\displaystyle \frac {(d+e x)^4 (a+b \arctan (c x))^2}{4 e}-\frac {b c \int \left (\frac {x^2 (a+b \arctan (c x)) e^4}{c^2}+\frac {4 d x (a+b \arctan (c x)) e^3}{c^2}+\frac {\left (6 c^2 d^2-e^2\right ) (a+b \arctan (c x)) e^2}{c^4}+\frac {\left (c^4 d^4-6 c^2 e^2 d^2+4 c^2 (c d-e) e (c d+e) x d+e^4\right ) (a+b \arctan (c x))}{c^4 \left (c^2 x^2+1\right )}\right )dx}{2 e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(d+e x)^4 (a+b \arctan (c x))^2}{4 e}-\frac {b c \left (-\frac {2 i d e (c d-e) (c d+e) (a+b \arctan (c x))^2}{b c^4}-\frac {4 d e (c d-e) (c d+e) \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c^4}+\frac {2 d e^3 x^2 (a+b \arctan (c x))}{c^2}+\frac {e^4 x^3 (a+b \arctan (c x))}{3 c^2}+\frac {\left (c^4 d^4-6 c^2 d^2 e^2+e^4\right ) (a+b \arctan (c x))^2}{2 b c^5}+\frac {a e^2 x \left (6 c^2 d^2-e^2\right )}{c^4}+\frac {2 b d e^3 \arctan (c x)}{c^4}+\frac {b e^2 x \arctan (c x) \left (6 c^2 d^2-e^2\right )}{c^4}-\frac {2 i b d e (c d-e) (c d+e) \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{c^4}-\frac {2 b d e^3 x}{c^3}-\frac {b e^4 x^2}{6 c^3}-\frac {b e^2 \left (6 c^2 d^2-e^2\right ) \log \left (c^2 x^2+1\right )}{2 c^5}+\frac {b e^4 \log \left (c^2 x^2+1\right )}{6 c^5}\right )}{2 e}\) |
Input:
Int[(d + e*x)^3*(a + b*ArcTan[c*x])^2,x]
Output:
((d + e*x)^4*(a + b*ArcTan[c*x])^2)/(4*e) - (b*c*((-2*b*d*e^3*x)/c^3 + (a* e^2*(6*c^2*d^2 - e^2)*x)/c^4 - (b*e^4*x^2)/(6*c^3) + (2*b*d*e^3*ArcTan[c*x ])/c^4 + (b*e^2*(6*c^2*d^2 - e^2)*x*ArcTan[c*x])/c^4 + (2*d*e^3*x^2*(a + b *ArcTan[c*x]))/c^2 + (e^4*x^3*(a + b*ArcTan[c*x]))/(3*c^2) - ((2*I)*d*(c*d - e)*e*(c*d + e)*(a + b*ArcTan[c*x])^2)/(b*c^4) + ((c^4*d^4 - 6*c^2*d^2*e ^2 + e^4)*(a + b*ArcTan[c*x])^2)/(2*b*c^5) - (4*d*(c*d - e)*e*(c*d + e)*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c^4 + (b*e^4*Log[1 + c^2*x^2])/(6*c^ 5) - (b*e^2*(6*c^2*d^2 - e^2)*Log[1 + c^2*x^2])/(2*c^5) - ((2*I)*b*d*(c*d - e)*e*(c*d + e)*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^4))/(2*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]
Time = 0.82 (sec) , antiderivative size = 667, normalized size of antiderivative = 1.77
| method | result | size |
| parts | \(\frac {a^{2} \left (e x +d \right )^{4}}{4 e}+\frac {b^{2} \left (\frac {c \,e^{3} \arctan \left (c x \right )^{2} x^{4}}{4}+c \,e^{2} \arctan \left (c x \right )^{2} x^{3} d +\frac {3 c e \arctan \left (c x \right )^{2} x^{2} d^{2}}{2}+\arctan \left (c x \right )^{2} c x \,d^{3}+\frac {c \arctan \left (c x \right )^{2} d^{4}}{4 e}-\frac {6 \arctan \left (c x \right ) c^{3} d^{2} e^{2} x +2 \arctan \left (c x \right ) c^{3} d \,e^{3} x^{2}+\frac {\arctan \left (c x \right ) e^{4} c^{3} x^{3}}{3}-\arctan \left (c x \right ) e^{4} c x +2 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{3} d^{3} e -2 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c d \,e^{3}+\arctan \left (c x \right )^{2} c^{4} d^{4}-6 \arctan \left (c x \right )^{2} c^{2} d^{2} e^{2}+\arctan \left (c x \right )^{2} e^{4}-\frac {\left (6 c^{4} d^{4}-36 c^{2} d^{2} e^{2}+6 e^{4}\right ) \arctan \left (c x \right )^{2}}{12}-\frac {e^{2} \left (6 c^{2} d e x +\frac {e^{2} c^{2} x^{2}}{2}+\frac {\left (18 c^{2} d^{2}-4 e^{2}\right ) \ln \left (c^{2} x^{2}+1\right )}{2}-6 c d e \arctan \left (c x \right )\right )}{3}-2 c d e \left (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 c^{3} e}\right )}{c}+\frac {2 a b \left (\frac {c \,e^{3} \arctan \left (c x \right ) x^{4}}{4}+c \,e^{2} \arctan \left (c x \right ) x^{3} d +\frac {3 c e \arctan \left (c x \right ) x^{2} d^{2}}{2}+\arctan \left (c x \right ) c x \,d^{3}+\frac {c \arctan \left (c x \right ) d^{4}}{4 e}-\frac {6 c^{3} d^{2} e^{2} x +2 c^{3} d \,e^{3} x^{2}+\frac {e^{4} c^{3} x^{3}}{3}-e^{4} c x +\frac {\left (4 c^{3} d^{3} e -4 c d \,e^{3}\right ) \ln \left (c^{2} x^{2}+1\right )}{2}+\left (c^{4} d^{4}-6 c^{2} d^{2} e^{2}+e^{4}\right ) \arctan \left (c x \right )}{4 c^{3} e}\right )}{c}\) | \(667\) |
| derivativedivides | \(\frac {\frac {a^{2} \left (c e x +c d \right )^{4}}{4 c^{3} e}+\frac {b^{2} \left (\frac {\arctan \left (c x \right )^{2} c^{4} d^{4}}{4 e}+\arctan \left (c x \right )^{2} c^{4} d^{3} x +\frac {3 e \arctan \left (c x \right )^{2} c^{4} d^{2} x^{2}}{2}+e^{2} \arctan \left (c x \right )^{2} c^{4} d \,x^{3}+\frac {e^{3} \arctan \left (c x \right )^{2} c^{4} x^{4}}{4}-\frac {6 \arctan \left (c x \right ) c^{3} d^{2} e^{2} x +2 \arctan \left (c x \right ) c^{3} d \,e^{3} x^{2}+\frac {\arctan \left (c x \right ) e^{4} c^{3} x^{3}}{3}-\arctan \left (c x \right ) e^{4} c x +2 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{3} d^{3} e -2 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c d \,e^{3}+\arctan \left (c x \right )^{2} c^{4} d^{4}-6 \arctan \left (c x \right )^{2} c^{2} d^{2} e^{2}+\arctan \left (c x \right )^{2} e^{4}-\frac {\left (6 c^{4} d^{4}-36 c^{2} d^{2} e^{2}+6 e^{4}\right ) \arctan \left (c x \right )^{2}}{12}-\frac {e^{2} \left (6 c^{2} d e x +\frac {e^{2} c^{2} x^{2}}{2}+\frac {\left (18 c^{2} d^{2}-4 e^{2}\right ) \ln \left (c^{2} x^{2}+1\right )}{2}-6 c d e \arctan \left (c x \right )\right )}{3}-2 c d e \left (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 e}\right )}{c^{3}}+\frac {2 a b \left (\frac {\arctan \left (c x \right ) c^{4} d^{4}}{4 e}+\arctan \left (c x \right ) c^{4} d^{3} x +\frac {3 e \arctan \left (c x \right ) c^{4} d^{2} x^{2}}{2}+e^{2} \arctan \left (c x \right ) c^{4} d \,x^{3}+\frac {e^{3} \arctan \left (c x \right ) c^{4} x^{4}}{4}-\frac {6 c^{3} d^{2} e^{2} x +2 c^{3} d \,e^{3} x^{2}+\frac {e^{4} c^{3} x^{3}}{3}-e^{4} c x +\frac {\left (4 c^{3} d^{3} e -4 c d \,e^{3}\right ) \ln \left (c^{2} x^{2}+1\right )}{2}+\left (c^{4} d^{4}-6 c^{2} d^{2} e^{2}+e^{4}\right ) \arctan \left (c x \right )}{4 e}\right )}{c^{3}}}{c}\) | \(691\) |
| default | \(\frac {\frac {a^{2} \left (c e x +c d \right )^{4}}{4 c^{3} e}+\frac {b^{2} \left (\frac {\arctan \left (c x \right )^{2} c^{4} d^{4}}{4 e}+\arctan \left (c x \right )^{2} c^{4} d^{3} x +\frac {3 e \arctan \left (c x \right )^{2} c^{4} d^{2} x^{2}}{2}+e^{2} \arctan \left (c x \right )^{2} c^{4} d \,x^{3}+\frac {e^{3} \arctan \left (c x \right )^{2} c^{4} x^{4}}{4}-\frac {6 \arctan \left (c x \right ) c^{3} d^{2} e^{2} x +2 \arctan \left (c x \right ) c^{3} d \,e^{3} x^{2}+\frac {\arctan \left (c x \right ) e^{4} c^{3} x^{3}}{3}-\arctan \left (c x \right ) e^{4} c x +2 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{3} d^{3} e -2 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c d \,e^{3}+\arctan \left (c x \right )^{2} c^{4} d^{4}-6 \arctan \left (c x \right )^{2} c^{2} d^{2} e^{2}+\arctan \left (c x \right )^{2} e^{4}-\frac {\left (6 c^{4} d^{4}-36 c^{2} d^{2} e^{2}+6 e^{4}\right ) \arctan \left (c x \right )^{2}}{12}-\frac {e^{2} \left (6 c^{2} d e x +\frac {e^{2} c^{2} x^{2}}{2}+\frac {\left (18 c^{2} d^{2}-4 e^{2}\right ) \ln \left (c^{2} x^{2}+1\right )}{2}-6 c d e \arctan \left (c x \right )\right )}{3}-2 c d e \left (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 e}\right )}{c^{3}}+\frac {2 a b \left (\frac {\arctan \left (c x \right ) c^{4} d^{4}}{4 e}+\arctan \left (c x \right ) c^{4} d^{3} x +\frac {3 e \arctan \left (c x \right ) c^{4} d^{2} x^{2}}{2}+e^{2} \arctan \left (c x \right ) c^{4} d \,x^{3}+\frac {e^{3} \arctan \left (c x \right ) c^{4} x^{4}}{4}-\frac {6 c^{3} d^{2} e^{2} x +2 c^{3} d \,e^{3} x^{2}+\frac {e^{4} c^{3} x^{3}}{3}-e^{4} c x +\frac {\left (4 c^{3} d^{3} e -4 c d \,e^{3}\right ) \ln \left (c^{2} x^{2}+1\right )}{2}+\left (c^{4} d^{4}-6 c^{2} d^{2} e^{2}+e^{4}\right ) \arctan \left (c x \right )}{4 e}\right )}{c^{3}}}{c}\) | \(691\) |
| risch | \(\text {Expression too large to display}\) | \(1348\) |
Input:
int((e*x+d)^3*(a+b*arctan(c*x))^2,x,method=_RETURNVERBOSE)
Output:
1/4*a^2*(e*x+d)^4/e+b^2/c*(1/4*c*e^3*arctan(c*x)^2*x^4+c*e^2*arctan(c*x)^2 *x^3*d+3/2*c*e*arctan(c*x)^2*x^2*d^2+arctan(c*x)^2*c*x*d^3+1/4*c/e*arctan( c*x)^2*d^4-1/2/c^3/e*(6*arctan(c*x)*c^3*d^2*e^2*x+2*arctan(c*x)*c^3*d*e^3* x^2+1/3*arctan(c*x)*e^4*c^3*x^3-arctan(c*x)*e^4*c*x+2*arctan(c*x)*ln(c^2*x ^2+1)*c^3*d^3*e-2*arctan(c*x)*ln(c^2*x^2+1)*c*d*e^3+arctan(c*x)^2*c^4*d^4- 6*arctan(c*x)^2*c^2*d^2*e^2+arctan(c*x)^2*e^4-1/12*(6*c^4*d^4-36*c^2*d^2*e ^2+6*e^4)*arctan(c*x)^2-1/3*e^2*(6*c^2*d*e*x+1/2*e^2*c^2*x^2+1/2*(18*c^2*d ^2-4*e^2)*ln(c^2*x^2+1)-6*c*d*e*arctan(c*x))-2*c*d*e*(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))))))+2*a*b/c*(1/4*c*e^3*arctan(c* x)*x^4+c*e^2*arctan(c*x)*x^3*d+3/2*c*e*arctan(c*x)*x^2*d^2+arctan(c*x)*c*x *d^3+1/4*c/e*arctan(c*x)*d^4-1/4/c^3/e*(6*c^3*d^2*e^2*x+2*c^3*d*e^3*x^2+1/ 3*e^4*c^3*x^3-e^4*c*x+1/2*(4*c^3*d^3*e-4*c*d*e^3)*ln(c^2*x^2+1)+(c^4*d^4-6 *c^2*d^2*e^2+e^4)*arctan(c*x)))
\[ \int (d+e x)^3 (a+b \arctan (c x))^2 \, dx=\int { {\left (e x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2} \,d x } \] Input:
integrate((e*x+d)^3*(a+b*arctan(c*x))^2,x, algorithm="fricas")
Output:
integral(a^2*e^3*x^3 + 3*a^2*d*e^2*x^2 + 3*a^2*d^2*e*x + a^2*d^3 + (b^2*e^ 3*x^3 + 3*b^2*d*e^2*x^2 + 3*b^2*d^2*e*x + b^2*d^3)*arctan(c*x)^2 + 2*(a*b* e^3*x^3 + 3*a*b*d*e^2*x^2 + 3*a*b*d^2*e*x + a*b*d^3)*arctan(c*x), x)
\[ \int (d+e x)^3 (a+b \arctan (c x))^2 \, dx=\int \left (a + b \operatorname {atan}{\left (c x \right )}\right )^{2} \left (d + e x\right )^{3}\, dx \] Input:
integrate((e*x+d)**3*(a+b*atan(c*x))**2,x)
Output:
Integral((a + b*atan(c*x))**2*(d + e*x)**3, x)
\[ \int (d+e x)^3 (a+b \arctan (c x))^2 \, dx=\int { {\left (e x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2} \,d x } \] Input:
integrate((e*x+d)^3*(a+b*arctan(c*x))^2,x, algorithm="maxima")
Output:
1/4*a^2*e^3*x^4 + a^2*d*e^2*x^3 + 12*b^2*c^2*e^3*integrate(1/16*x^5*arctan (c*x)^2/(c^2*x^2 + 1), x) + b^2*c^2*e^3*integrate(1/16*x^5*log(c^2*x^2 + 1 )^2/(c^2*x^2 + 1), x) + 36*b^2*c^2*d*e^2*integrate(1/16*x^4*arctan(c*x)^2/ (c^2*x^2 + 1), x) + b^2*c^2*e^3*integrate(1/16*x^5*log(c^2*x^2 + 1)/(c^2*x ^2 + 1), x) + 3*b^2*c^2*d*e^2*integrate(1/16*x^4*log(c^2*x^2 + 1)^2/(c^2*x ^2 + 1), x) + 36*b^2*c^2*d^2*e*integrate(1/16*x^3*arctan(c*x)^2/(c^2*x^2 + 1), x) + 4*b^2*c^2*d*e^2*integrate(1/16*x^4*log(c^2*x^2 + 1)/(c^2*x^2 + 1 ), x) + 3*b^2*c^2*d^2*e*integrate(1/16*x^3*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1 ), x) + 12*b^2*c^2*d^3*integrate(1/16*x^2*arctan(c*x)^2/(c^2*x^2 + 1), x) + 6*b^2*c^2*d^2*e*integrate(1/16*x^3*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) + b^2*c^2*d^3*integrate(1/16*x^2*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 4*b^ 2*c^2*d^3*integrate(1/16*x^2*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) + 3/2*a^2* d^2*e*x^2 + 1/4*b^2*d^3*arctan(c*x)^3/c - 2*b^2*c*e^3*integrate(1/16*x^4*a rctan(c*x)/(c^2*x^2 + 1), x) - 8*b^2*c*d*e^2*integrate(1/16*x^3*arctan(c*x )/(c^2*x^2 + 1), x) - 12*b^2*c*d^2*e*integrate(1/16*x^2*arctan(c*x)/(c^2*x ^2 + 1), x) - 8*b^2*c*d^3*integrate(1/16*x*arctan(c*x)/(c^2*x^2 + 1), x) + 3*(x^2*arctan(c*x) - c*(x/c^2 - arctan(c*x)/c^3))*a*b*d^2*e + (2*x^3*arct an(c*x) - c*(x^2/c^2 - log(c^2*x^2 + 1)/c^4))*a*b*d*e^2 + 1/6*(3*x^4*arcta n(c*x) - c*((c^2*x^3 - 3*x)/c^4 + 3*arctan(c*x)/c^5))*a*b*e^3 + a^2*d^3*x + 12*b^2*e^3*integrate(1/16*x^3*arctan(c*x)^2/(c^2*x^2 + 1), x) + b^2*e...
\[ \int (d+e x)^3 (a+b \arctan (c x))^2 \, dx=\int { {\left (e x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2} \,d x } \] Input:
integrate((e*x+d)^3*(a+b*arctan(c*x))^2,x, algorithm="giac")
Output:
integrate((e*x + d)^3*(b*arctan(c*x) + a)^2, x)
Timed out. \[ \int (d+e x)^3 (a+b \arctan (c x))^2 \, dx=\int {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (d+e\,x\right )}^3 \,d x \] Input:
int((a + b*atan(c*x))^2*(d + e*x)^3,x)
Output:
int((a + b*atan(c*x))^2*(d + e*x)^3, x)
\[ \int (d+e x)^3 (a+b \arctan (c x))^2 \, dx=\frac {-3 \mathit {atan} \left (c x \right )^{2} b^{2} e^{3}-4 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) b^{2} e^{3}-12 \mathit {atan} \left (c x \right ) b^{2} c^{3} d \,e^{2} x^{2}+12 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) a b c d \,e^{2}-36 a b \,c^{3} d^{2} e x -12 a b \,c^{3} d \,e^{2} x^{2}+18 \mathit {atan} \left (c x \right )^{2} b^{2} c^{4} d^{2} e \,x^{2}+12 \mathit {atan} \left (c x \right )^{2} b^{2} c^{4} d \,e^{2} x^{3}+24 \mathit {atan} \left (c x \right ) a b \,c^{4} d^{3} x +6 \mathit {atan} \left (c x \right ) a b \,c^{4} e^{3} x^{4}+36 \mathit {atan} \left (c x \right ) a b \,c^{2} d^{2} e -36 \mathit {atan} \left (c x \right ) b^{2} c^{3} d^{2} e x +b^{2} c^{2} e^{3} x^{2}-6 \mathit {atan} \left (c x \right ) a b \,e^{3}+12 a^{2} c^{4} d^{3} x +3 a^{2} c^{4} e^{3} x^{4}-24 \left (\int \frac {\mathit {atan} \left (c x \right ) x}{c^{2} x^{2}+1}d x \right ) b^{2} c^{5} d^{3}+24 \mathit {atan} \left (c x \right ) a b \,c^{4} d \,e^{2} x^{3}+12 \mathit {atan} \left (c x \right )^{2} b^{2} c^{4} d^{3} x +3 \mathit {atan} \left (c x \right )^{2} b^{2} c^{4} e^{3} x^{4}+18 \mathit {atan} \left (c x \right )^{2} b^{2} c^{2} d^{2} e -2 \mathit {atan} \left (c x \right ) b^{2} c^{3} e^{3} x^{3}-12 \mathit {atan} \left (c x \right ) b^{2} c d \,e^{2}+6 \mathit {atan} \left (c x \right ) b^{2} c \,e^{3} x -12 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) a b \,c^{3} d^{3}+18 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) b^{2} c^{2} d^{2} e +18 a^{2} c^{4} d^{2} e \,x^{2}+12 a^{2} c^{4} d \,e^{2} x^{3}-2 a b \,c^{3} e^{3} x^{3}+6 a b c \,e^{3} x +12 b^{2} c^{2} d \,e^{2} x +24 \left (\int \frac {\mathit {atan} \left (c x \right ) x}{c^{2} x^{2}+1}d x \right ) b^{2} c^{3} d \,e^{2}+36 \mathit {atan} \left (c x \right ) a b \,c^{4} d^{2} e \,x^{2}}{12 c^{4}} \] Input:
int((e*x+d)^3*(a+b*atan(c*x))^2,x)
Output:
(12*atan(c*x)**2*b**2*c**4*d**3*x + 18*atan(c*x)**2*b**2*c**4*d**2*e*x**2 + 12*atan(c*x)**2*b**2*c**4*d*e**2*x**3 + 3*atan(c*x)**2*b**2*c**4*e**3*x* *4 + 18*atan(c*x)**2*b**2*c**2*d**2*e - 3*atan(c*x)**2*b**2*e**3 + 24*atan (c*x)*a*b*c**4*d**3*x + 36*atan(c*x)*a*b*c**4*d**2*e*x**2 + 24*atan(c*x)*a *b*c**4*d*e**2*x**3 + 6*atan(c*x)*a*b*c**4*e**3*x**4 + 36*atan(c*x)*a*b*c* *2*d**2*e - 6*atan(c*x)*a*b*e**3 - 36*atan(c*x)*b**2*c**3*d**2*e*x - 12*at an(c*x)*b**2*c**3*d*e**2*x**2 - 2*atan(c*x)*b**2*c**3*e**3*x**3 - 12*atan( c*x)*b**2*c*d*e**2 + 6*atan(c*x)*b**2*c*e**3*x - 24*int((atan(c*x)*x)/(c** 2*x**2 + 1),x)*b**2*c**5*d**3 + 24*int((atan(c*x)*x)/(c**2*x**2 + 1),x)*b* *2*c**3*d*e**2 - 12*log(c**2*x**2 + 1)*a*b*c**3*d**3 + 12*log(c**2*x**2 + 1)*a*b*c*d*e**2 + 18*log(c**2*x**2 + 1)*b**2*c**2*d**2*e - 4*log(c**2*x**2 + 1)*b**2*e**3 + 12*a**2*c**4*d**3*x + 18*a**2*c**4*d**2*e*x**2 + 12*a**2 *c**4*d*e**2*x**3 + 3*a**2*c**4*e**3*x**4 - 36*a*b*c**3*d**2*e*x - 12*a*b* c**3*d*e**2*x**2 - 2*a*b*c**3*e**3*x**3 + 6*a*b*c*e**3*x + 12*b**2*c**2*d* e**2*x + b**2*c**2*e**3*x**2)/(12*c**4)