\(\int (d+e x)^3 (a+b \arctan (c x))^3 \, dx\) [15]

Optimal result

Integrand size = 18, antiderivative size = 652 \[ \int (d+e x)^3 (a+b \arctan (c x))^3 \, dx=\frac {3 a b^2 d e^2 x}{c^2}-\frac {b^3 e^3 x}{4 c^3}+\frac {b^3 e^3 \arctan (c x)}{4 c^4}+\frac {3 b^3 d e^2 x \arctan (c x)}{c^2}+\frac {b^2 e^3 x^2 (a+b \arctan (c x))}{4 c^2}-\frac {3 b d e^2 (a+b \arctan (c x))^2}{2 c^3}+\frac {i b e^3 (a+b \arctan (c x))^2}{4 c^4}-\frac {3 i b e \left (6 c^2 d^2-e^2\right ) (a+b \arctan (c x))^2}{4 c^4}-\frac {3 b e \left (6 c^2 d^2-e^2\right ) x (a+b \arctan (c x))^2}{4 c^3}-\frac {3 b d e^2 x^2 (a+b \arctan (c x))^2}{2 c}-\frac {b e^3 x^3 (a+b \arctan (c x))^2}{4 c}+\frac {i d (c d-e) (c d+e) (a+b \arctan (c x))^3}{c^3}-\frac {\left (c^4 d^4-6 c^2 d^2 e^2+e^4\right ) (a+b \arctan (c x))^3}{4 c^4 e}+\frac {(d+e x)^4 (a+b \arctan (c x))^3}{4 e}+\frac {b^2 e^3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{2 c^4}-\frac {3 b^2 e \left (6 c^2 d^2-e^2\right ) (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{2 c^4}+\frac {3 b d (c d-e) (c d+e) (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^3}-\frac {3 b^3 d e^2 \log \left (1+c^2 x^2\right )}{2 c^3}+\frac {i b^3 e^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{4 c^4}-\frac {3 i b^3 e \left (6 c^2 d^2-e^2\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{4 c^4}+\frac {3 i b^2 d (c d-e) (c d+e) (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^3}+\frac {3 b^3 d (c d-e) (c d+e) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^3} \] Output:

3*a*b^2*d*e^2*x/c^2-1/4*b^3*e^3*x/c^3+1/4*b^3*e^3*arctan(c*x)/c^4+3*b^3*d* 
e^2*x*arctan(c*x)/c^2+1/4*b^2*e^3*x^2*(a+b*arctan(c*x))/c^2-3/2*b*d*e^2*(a 
+b*arctan(c*x))^2/c^3-3/4*I*b^3*e*(6*c^2*d^2-e^2)*polylog(2,1-2/(1+I*c*x)) 
/c^4-3/4*I*b*e*(6*c^2*d^2-e^2)*(a+b*arctan(c*x))^2/c^4-3/4*b*e*(6*c^2*d^2- 
e^2)*x*(a+b*arctan(c*x))^2/c^3-3/2*b*d*e^2*x^2*(a+b*arctan(c*x))^2/c-1/4*b 
*e^3*x^3*(a+b*arctan(c*x))^2/c+3*I*b^2*d*(c*d-e)*(c*d+e)*(a+b*arctan(c*x)) 
*polylog(2,1-2/(1+I*c*x))/c^3-1/4*(c^4*d^4-6*c^2*d^2*e^2+e^4)*(a+b*arctan( 
c*x))^3/c^4/e+1/4*(e*x+d)^4*(a+b*arctan(c*x))^3/e+1/2*b^2*e^3*(a+b*arctan( 
c*x))*ln(2/(1+I*c*x))/c^4-3/2*b^2*e*(6*c^2*d^2-e^2)*(a+b*arctan(c*x))*ln(2 
/(1+I*c*x))/c^4+3*b*d*(c*d-e)*(c*d+e)*(a+b*arctan(c*x))^2*ln(2/(1+I*c*x))/ 
c^3-3/2*b^3*d*e^2*ln(c^2*x^2+1)/c^3+1/4*I*b*e^3*(a+b*arctan(c*x))^2/c^4+1/ 
4*I*b^3*e^3*polylog(2,1-2/(1+I*c*x))/c^4+I*d*(c*d-e)*(c*d+e)*(a+b*arctan(c 
*x))^3/c^3+3/2*b^3*d*(c*d-e)*(c*d+e)*polylog(3,1-2/(1+I*c*x))/c^3
 

Mathematica [A] (verified)

Time = 1.15 (sec) , antiderivative size = 855, normalized size of antiderivative = 1.31 \[ \int (d+e x)^3 (a+b \arctan (c x))^3 \, dx =\text {Too large to display} \] Input:

Integrate[(d + e*x)^3*(a + b*ArcTan[c*x])^3,x]
 

Output:

(a^2*c*(4*a*c^3*d^3 + 3*b*e*(-6*c^2*d^2 + e^2))*x + 6*a^2*c^3*d*e*(a*c*d - 
 b*e)*x^2 + a^2*c^3*e^2*(4*a*c*d - b*e)*x^3 + a^3*c^4*e^3*x^4 + 3*a^2*b*(6 
*c^2*d^2*e - e^3)*ArcTan[c*x] + 3*a^2*b*c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2 
*x^2 + e^3*x^3)*ArcTan[c*x] + a*b^2*e^3*(1 + c^2*x^2 + (6*c*x - 2*c^3*x^3) 
*ArcTan[c*x] + 3*(-1 + c^4*x^4)*ArcTan[c*x]^2 - 4*Log[1 + c^2*x^2]) - 6*a^ 
2*b*c*d*(c^2*d^2 - e^2)*Log[1 + c^2*x^2] + 18*a*b^2*c^2*d^2*e*(-2*c*x*ArcT 
an[c*x] + (1 + c^2*x^2)*ArcTan[c*x]^2 + Log[1 + c^2*x^2]) + 12*a*b^2*c^3*d 
^3*(ArcTan[c*x]*((-I + c*x)*ArcTan[c*x] + 2*Log[1 + E^((2*I)*ArcTan[c*x])] 
) - I*PolyLog[2, -E^((2*I)*ArcTan[c*x])]) + 12*a*b^2*c*d*e^2*(c*x + (I + c 
^3*x^3)*ArcTan[c*x]^2 - ArcTan[c*x]*(1 + c^2*x^2 + 2*Log[1 + E^((2*I)*ArcT 
an[c*x])]) + I*PolyLog[2, -E^((2*I)*ArcTan[c*x])]) + 6*b^3*c^2*d^2*e*(ArcT 
an[c*x]*((3*I - 3*c*x)*ArcTan[c*x] + (1 + c^2*x^2)*ArcTan[c*x]^2 - 6*Log[1 
 + E^((2*I)*ArcTan[c*x])]) + (3*I)*PolyLog[2, -E^((2*I)*ArcTan[c*x])]) + b 
^3*e^3*(-(c*x) - (4*I - 3*c*x + c^3*x^3)*ArcTan[c*x]^2 + (-1 + c^4*x^4)*Ar 
cTan[c*x]^3 + ArcTan[c*x]*(1 + c^2*x^2 + 8*Log[1 + E^((2*I)*ArcTan[c*x])]) 
 - (4*I)*PolyLog[2, -E^((2*I)*ArcTan[c*x])]) + 2*b^3*c*d*e^2*(6*c*x*ArcTan 
[c*x] - 3*ArcTan[c*x]^2 - 3*c^2*x^2*ArcTan[c*x]^2 + (2*I)*ArcTan[c*x]^3 + 
2*c^3*x^3*ArcTan[c*x]^3 - 6*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] - 
 3*Log[1 + c^2*x^2] + (6*I)*ArcTan[c*x]*PolyLog[2, -E^((2*I)*ArcTan[c*x])] 
 - 3*PolyLog[3, -E^((2*I)*ArcTan[c*x])]) + 2*b^3*c^3*d^3*(2*ArcTan[c*x]...
 

Rubi [A] (verified)

Time = 1.51 (sec) , antiderivative size = 645, normalized size of antiderivative = 0.99, 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.

\(\displaystyle \int (d+e x)^3 (a+b \arctan (c x))^3 \, dx\)

\(\Big \downarrow \) 5389

\(\displaystyle \frac {(d+e x)^4 (a+b \arctan (c x))^3}{4 e}-\frac {3 b c \int \left (\frac {x^2 (a+b \arctan (c x))^2 e^4}{c^2}+\frac {4 d x (a+b \arctan (c x))^2 e^3}{c^2}+\frac {\left (6 c^2 d^2-e^2\right ) (a+b \arctan (c x))^2 e^2}{c^4}+\frac {\left (c^4 d^4-6 c^2 e^2 d^2+4 c^2 (c d-e) e (c d+e) x d+e^4\right ) (a+b \arctan (c x))^2}{c^4 \left (c^2 x^2+1\right )}\right )dx}{4 e}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(d+e x)^4 (a+b \arctan (c x))^3}{4 e}-\frac {3 b c \left (-\frac {i e^4 (a+b \arctan (c x))^2}{3 c^5}-\frac {2 b e^4 \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{3 c^5}+\frac {2 d e^3 (a+b \arctan (c x))^2}{c^4}-\frac {4 i b d e (c d-e) (c d+e) \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{c^4}-\frac {4 i d e (c d-e) (c d+e) (a+b \arctan (c x))^3}{3 b c^4}-\frac {4 d e (c d-e) (c d+e) \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c^4}-\frac {b e^4 x^2 (a+b \arctan (c x))}{3 c^3}+\frac {2 d e^3 x^2 (a+b \arctan (c x))^2}{c^2}+\frac {e^4 x^3 (a+b \arctan (c x))^2}{3 c^2}+\frac {i e^2 \left (6 c^2 d^2-e^2\right ) (a+b \arctan (c x))^2}{c^5}+\frac {2 b e^2 \left (6 c^2 d^2-e^2\right ) \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c^5}+\frac {e^2 x \left (6 c^2 d^2-e^2\right ) (a+b \arctan (c x))^2}{c^4}+\frac {\left (c^4 d^4-6 c^2 d^2 e^2+e^4\right ) (a+b \arctan (c x))^3}{3 b c^5}-\frac {4 a b d e^3 x}{c^3}-\frac {b^2 e^4 \arctan (c x)}{3 c^5}-\frac {4 b^2 d e^3 x \arctan (c x)}{c^3}-\frac {i b^2 e^4 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{3 c^5}-\frac {2 b^2 d e (c d-e) (c d+e) \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right )}{c^4}+\frac {b^2 e^4 x}{3 c^4}+\frac {i b^2 e^2 \left (6 c^2 d^2-e^2\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{c^5}+\frac {2 b^2 d e^3 \log \left (c^2 x^2+1\right )}{c^4}\right )}{4 e}\)

Input:

Int[(d + e*x)^3*(a + b*ArcTan[c*x])^3,x]
 

Output:

((d + e*x)^4*(a + b*ArcTan[c*x])^3)/(4*e) - (3*b*c*((-4*a*b*d*e^3*x)/c^3 + 
 (b^2*e^4*x)/(3*c^4) - (b^2*e^4*ArcTan[c*x])/(3*c^5) - (4*b^2*d*e^3*x*ArcT 
an[c*x])/c^3 - (b*e^4*x^2*(a + b*ArcTan[c*x]))/(3*c^3) + (2*d*e^3*(a + b*A 
rcTan[c*x])^2)/c^4 - ((I/3)*e^4*(a + b*ArcTan[c*x])^2)/c^5 + (I*e^2*(6*c^2 
*d^2 - e^2)*(a + b*ArcTan[c*x])^2)/c^5 + (e^2*(6*c^2*d^2 - e^2)*x*(a + b*A 
rcTan[c*x])^2)/c^4 + (2*d*e^3*x^2*(a + b*ArcTan[c*x])^2)/c^2 + (e^4*x^3*(a 
 + b*ArcTan[c*x])^2)/(3*c^2) - (((4*I)/3)*d*(c*d - e)*e*(c*d + e)*(a + b*A 
rcTan[c*x])^3)/(b*c^4) + ((c^4*d^4 - 6*c^2*d^2*e^2 + e^4)*(a + b*ArcTan[c* 
x])^3)/(3*b*c^5) - (2*b*e^4*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(3*c^5 
) + (2*b*e^2*(6*c^2*d^2 - e^2)*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c^5 
 - (4*d*(c*d - e)*e*(c*d + e)*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/c^ 
4 + (2*b^2*d*e^3*Log[1 + c^2*x^2])/c^4 - ((I/3)*b^2*e^4*PolyLog[2, 1 - 2/( 
1 + I*c*x)])/c^5 + (I*b^2*e^2*(6*c^2*d^2 - e^2)*PolyLog[2, 1 - 2/(1 + I*c* 
x)])/c^5 - ((4*I)*b*d*(c*d - e)*e*(c*d + e)*(a + b*ArcTan[c*x])*PolyLog[2, 
 1 - 2/(1 + I*c*x)])/c^4 - (2*b^2*d*(c*d - e)*e*(c*d + e)*PolyLog[3, 1 - 2 
/(1 + I*c*x)])/c^4))/(4*e)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5389

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]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 17.92 (sec) , antiderivative size = 3122, normalized size of antiderivative = 4.79

method	result	size
parts	\(\text {Expression too large to display}\)	\(3122\)
derivativedivides	\(\text {Expression too large to display}\)	\(3153\)
default	\(\text {Expression too large to display}\)	\(3153\)

Input:

int((e*x+d)^3*(a+b*arctan(c*x))^3,x,method=_RETURNVERBOSE)
 

Output:

1/4*a^3*(e*x+d)^4/e+b^3/c*(1/4*c*e^3*arctan(c*x)^3*x^4+c*e^2*arctan(c*x)^3 
*x^3*d+3/2*c*e*arctan(c*x)^3*x^2*d^2+arctan(c*x)^3*c*x*d^3+1/4*c/e*arctan( 
c*x)^3*d^4-3/4/c^3/e*(-2/3*e^4*arctan(c*x)*(c*x-I)*(c*x+I)+2/3*I*e^4*arcta 
n(c*x)*(c*x-I)-8/3*e^4*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-8/3 
*e^4*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+2*I*e*c^3*d^3*Pi*csgn 
(I*(1+(1+I*c*x)^2/(c^2*x^2+1)))*csgn(I*(1+(1+I*c*x)^2/(c^2*x^2+1))^2)^2*ar 
ctan(c*x)^2-I*e^3*c*d*Pi*csgn(I*(1+I*c*x)/(c^2*x^2+1)^(1/2))^2*csgn(I*(1+I 
*c*x)^2/(c^2*x^2+1))*arctan(c*x)^2-I*e*c^3*d^3*Pi*csgn(I*(1+I*c*x)^2/(c^2* 
x^2+1))*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/(1+(1+I*c*x)^2/(c^2*x^2+1))^2)^2*ar 
ctan(c*x)^2-2*I*e*c^3*d^3*Pi*csgn(I*(1+I*c*x)/(c^2*x^2+1)^(1/2))*csgn(I*(1 
+I*c*x)^2/(c^2*x^2+1))^2*arctan(c*x)^2+2*I*e^3*c*d*Pi*csgn(I*(1+I*c*x)/(c^ 
2*x^2+1)^(1/2))*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))^2*arctan(c*x)^2+I*e^3*c*d* 
Pi*csgn(I/(1+(1+I*c*x)^2/(c^2*x^2+1))^2)*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/(1 
+(1+I*c*x)^2/(c^2*x^2+1))^2)^2*arctan(c*x)^2+I*e^3*c*d*Pi*csgn(I*(1+I*c*x) 
^2/(c^2*x^2+1))*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/(1+(1+I*c*x)^2/(c^2*x^2+1)) 
^2)^2*arctan(c*x)^2-2*I*e^3*c*d*Pi*csgn(I*(1+(1+I*c*x)^2/(c^2*x^2+1)))*csg 
n(I*(1+(1+I*c*x)^2/(c^2*x^2+1))^2)^2*arctan(c*x)^2-I*e*c^3*d^3*Pi*csgn(I*( 
1+(1+I*c*x)^2/(c^2*x^2+1)))^2*csgn(I*(1+(1+I*c*x)^2/(c^2*x^2+1))^2)*arctan 
(c*x)^2+I*e*c^3*d^3*Pi*csgn(I*(1+I*c*x)/(c^2*x^2+1)^(1/2))^2*csgn(I*(1+I*c 
*x)^2/(c^2*x^2+1))*arctan(c*x)^2-I*e*c^3*d^3*Pi*csgn(I/(1+(1+I*c*x)^2/(...
 

Fricas [F]

\[ \int (d+e x)^3 (a+b \arctan (c x))^3 \, dx=\int { {\left (e x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{3} \,d x } \] Input:

integrate((e*x+d)^3*(a+b*arctan(c*x))^3,x, algorithm="fricas")
 

Output:

integral(a^3*e^3*x^3 + 3*a^3*d*e^2*x^2 + 3*a^3*d^2*e*x + a^3*d^3 + (b^3*e^ 
3*x^3 + 3*b^3*d*e^2*x^2 + 3*b^3*d^2*e*x + b^3*d^3)*arctan(c*x)^3 + 3*(a*b^ 
2*e^3*x^3 + 3*a*b^2*d*e^2*x^2 + 3*a*b^2*d^2*e*x + a*b^2*d^3)*arctan(c*x)^2 
 + 3*(a^2*b*e^3*x^3 + 3*a^2*b*d*e^2*x^2 + 3*a^2*b*d^2*e*x + a^2*b*d^3)*arc 
tan(c*x), x)
 

Sympy [F]

\[ \int (d+e x)^3 (a+b \arctan (c x))^3 \, dx=\int \left (a + b \operatorname {atan}{\left (c x \right )}\right )^{3} \left (d + e x\right )^{3}\, dx \] Input:

integrate((e*x+d)**3*(a+b*atan(c*x))**3,x)
 

Output:

Integral((a + b*atan(c*x))**3*(d + e*x)**3, x)
 

Maxima [F]

\[ \int (d+e x)^3 (a+b \arctan (c x))^3 \, dx=\int { {\left (e x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{3} \,d x } \] Input:

integrate((e*x+d)^3*(a+b*arctan(c*x))^3,x, algorithm="maxima")
 

Output:

1/4*a^3*e^3*x^4 + a^3*d*e^2*x^3 + 7/32*b^3*d^3*arctan(c*x)^4/c + 112*b^3*c 
^2*e^3*integrate(1/128*x^5*arctan(c*x)^3/(c^2*x^2 + 1), x) + 12*b^3*c^2*e^ 
3*integrate(1/128*x^5*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 3 
84*a*b^2*c^2*e^3*integrate(1/128*x^5*arctan(c*x)^2/(c^2*x^2 + 1), x) + 336 
*b^3*c^2*d*e^2*integrate(1/128*x^4*arctan(c*x)^3/(c^2*x^2 + 1), x) + 12*b^ 
3*c^2*e^3*integrate(1/128*x^5*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^2 + 1), 
x) + 36*b^3*c^2*d*e^2*integrate(1/128*x^4*arctan(c*x)*log(c^2*x^2 + 1)^2/( 
c^2*x^2 + 1), x) + 1152*a*b^2*c^2*d*e^2*integrate(1/128*x^4*arctan(c*x)^2/ 
(c^2*x^2 + 1), x) + 336*b^3*c^2*d^2*e*integrate(1/128*x^3*arctan(c*x)^3/(c 
^2*x^2 + 1), x) + 48*b^3*c^2*d*e^2*integrate(1/128*x^4*arctan(c*x)*log(c^2 
*x^2 + 1)/(c^2*x^2 + 1), x) + 36*b^3*c^2*d^2*e*integrate(1/128*x^3*arctan( 
c*x)*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 1152*a*b^2*c^2*d^2*e*integrate 
(1/128*x^3*arctan(c*x)^2/(c^2*x^2 + 1), x) + 112*b^3*c^2*d^3*integrate(1/1 
28*x^2*arctan(c*x)^3/(c^2*x^2 + 1), x) + 72*b^3*c^2*d^2*e*integrate(1/128* 
x^3*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) + 12*b^3*c^2*d^3*integr 
ate(1/128*x^2*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 384*a*b^2 
*c^2*d^3*integrate(1/128*x^2*arctan(c*x)^2/(c^2*x^2 + 1), x) + 48*b^3*c^2* 
d^3*integrate(1/128*x^2*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) + 3 
/2*a^3*d^2*e*x^2 + a*b^2*d^3*arctan(c*x)^3/c - 12*b^3*c*e^3*integrate(1/12 
8*x^4*arctan(c*x)^2/(c^2*x^2 + 1), x) + 3*b^3*c*e^3*integrate(1/128*x^4...
 

Giac [F]

\[ \int (d+e x)^3 (a+b \arctan (c x))^3 \, dx=\int { {\left (e x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{3} \,d x } \] Input:

integrate((e*x+d)^3*(a+b*arctan(c*x))^3,x, algorithm="giac")
 

Output:

integrate((e*x + d)^3*(b*arctan(c*x) + a)^3, x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int (d+e x)^3 (a+b \arctan (c x))^3 \, dx=\int {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^3\,{\left (d+e\,x\right )}^3 \,d x \] Input:

int((a + b*atan(c*x))^3*(d + e*x)^3,x)
 

Output:

int((a + b*atan(c*x))^3*(d + e*x)^3, x)
 

Reduce [F]

\[ \int (d+e x)^3 (a+b \arctan (c x))^3 \, dx =\text {Too large to display} \] Input:

int((e*x+d)^3*(a+b*atan(c*x))^3,x)
 

Output:

(4*atan(c*x)**3*b**3*c**4*d**3*x + 6*atan(c*x)**3*b**3*c**4*d**2*e*x**2 + 
4*atan(c*x)**3*b**3*c**4*d*e**2*x**3 + atan(c*x)**3*b**3*c**4*e**3*x**4 + 
6*atan(c*x)**3*b**3*c**2*d**2*e - atan(c*x)**3*b**3*e**3 + 12*atan(c*x)**2 
*a*b**2*c**4*d**3*x + 18*atan(c*x)**2*a*b**2*c**4*d**2*e*x**2 + 12*atan(c* 
x)**2*a*b**2*c**4*d*e**2*x**3 + 3*atan(c*x)**2*a*b**2*c**4*e**3*x**4 + 18* 
atan(c*x)**2*a*b**2*c**2*d**2*e - 3*atan(c*x)**2*a*b**2*e**3 - 18*atan(c*x 
)**2*b**3*c**3*d**2*e*x - 6*atan(c*x)**2*b**3*c**3*d*e**2*x**2 - atan(c*x) 
**2*b**3*c**3*e**3*x**3 - 6*atan(c*x)**2*b**3*c*d*e**2 + 3*atan(c*x)**2*b* 
*3*c*e**3*x + 12*atan(c*x)*a**2*b*c**4*d**3*x + 18*atan(c*x)*a**2*b*c**4*d 
**2*e*x**2 + 12*atan(c*x)*a**2*b*c**4*d*e**2*x**3 + 3*atan(c*x)*a**2*b*c** 
4*e**3*x**4 + 18*atan(c*x)*a**2*b*c**2*d**2*e - 3*atan(c*x)*a**2*b*e**3 - 
36*atan(c*x)*a*b**2*c**3*d**2*e*x - 12*atan(c*x)*a*b**2*c**3*d*e**2*x**2 - 
 2*atan(c*x)*a*b**2*c**3*e**3*x**3 - 12*atan(c*x)*a*b**2*c*d*e**2 + 6*atan 
(c*x)*a*b**2*c*e**3*x + 12*atan(c*x)*b**3*c**2*d*e**2*x + atan(c*x)*b**3*c 
**2*e**3*x**2 + atan(c*x)*b**3*e**3 - 24*int((atan(c*x)*x)/(c**2*x**2 + 1) 
,x)*a*b**2*c**5*d**3 + 24*int((atan(c*x)*x)/(c**2*x**2 + 1),x)*a*b**2*c**3 
*d*e**2 + 36*int((atan(c*x)*x)/(c**2*x**2 + 1),x)*b**3*c**4*d**2*e - 8*int 
((atan(c*x)*x)/(c**2*x**2 + 1),x)*b**3*c**2*e**3 - 12*int((atan(c*x)**2*x) 
/(c**2*x**2 + 1),x)*b**3*c**5*d**3 + 12*int((atan(c*x)**2*x)/(c**2*x**2 + 
1),x)*b**3*c**3*d*e**2 - 6*log(c**2*x**2 + 1)*a**2*b*c**3*d**3 + 6*log(...