\(\int (d+e x) (a+b \arctan (c x))^3 \, dx\) [17]

Optimal result

Integrand size = 16, antiderivative size = 264 \[ \int (d+e x) (a+b \arctan (c x))^3 \, dx=-\frac {3 i b e (a+b \arctan (c x))^2}{2 c^2}-\frac {3 b e x (a+b \arctan (c x))^2}{2 c}+\frac {i d (a+b \arctan (c x))^3}{c}-\frac {\left (d^2-\frac {e^2}{c^2}\right ) (a+b \arctan (c x))^3}{2 e}+\frac {(d+e x)^2 (a+b \arctan (c x))^3}{2 e}-\frac {3 b^2 e (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^2}+\frac {3 b d (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c}-\frac {3 i b^3 e \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c^2}+\frac {3 i b^2 d (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c}+\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c} \] Output:

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

Mathematica [A] (verified)

Time = 0.56 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.30 \[ \int (d+e x) (a+b \arctan (c x))^3 \, dx=\frac {a^2 c (2 a c d-3 b e) x+a^3 c^2 e x^2+3 a^2 b e \arctan (c x)+3 a^2 b c^2 x (2 d+e x) \arctan (c x)-3 a^2 b c d \log \left (1+c^2 x^2\right )+3 a b^2 e \left (-2 c x \arctan (c x)+\left (1+c^2 x^2\right ) \arctan (c x)^2+\log \left (1+c^2 x^2\right )\right )+6 a b^2 c d \left (\arctan (c x) \left ((-i+c x) \arctan (c x)+2 \log \left (1+e^{2 i \arctan (c x)}\right )\right )-i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )+b^3 e \left (\arctan (c x) \left ((3 i-3 c x) \arctan (c x)+\left (1+c^2 x^2\right ) \arctan (c x)^2-6 \log \left (1+e^{2 i \arctan (c x)}\right )\right )+3 i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )+b^3 c d \left (2 \arctan (c x)^2 \left ((-i+c x) \arctan (c x)+3 \log \left (1+e^{2 i \arctan (c x)}\right )\right )-6 i \arctan (c x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )+3 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )\right )}{2 c^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.05, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, 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.

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

\(\Big \downarrow \) 5389

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

((d + e*x)^2*(a + b*ArcTan[c*x])^3)/(2*e) - (3*b*c*((I*e^2*(a + b*ArcTan[c 
*x])^2)/c^3 + (e^2*x*(a + b*ArcTan[c*x])^2)/c^2 - (((2*I)/3)*d*e*(a + b*Ar 
cTan[c*x])^3)/(b*c^2) + ((c*d - e)*(c*d + e)*(a + b*ArcTan[c*x])^3)/(3*b*c 
^3) + (2*b*e^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c^3 - (2*d*e*(a + b 
*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/c^2 + (I*b^2*e^2*PolyLog[2, 1 - 2/(1 + 
 I*c*x)])/c^3 - ((2*I)*b*d*e*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c 
*x)])/c^2 - (b^2*d*e*PolyLog[3, 1 - 2/(1 + I*c*x)])/c^2))/(2*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 = 2.64 (sec) , antiderivative size = 3886, normalized size of antiderivative = 14.72

method	result	size
parts	\(\text {Expression too large to display}\)	\(3886\)
derivativedivides	\(\text {Expression too large to display}\)	\(3901\)
default	\(\text {Expression too large to display}\)	\(3901\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

7/32*b^3*d*arctan(c*x)^4/c + 56*b^3*c^2*e*integrate(1/64*x^3*arctan(c*x)^3 
/(c^2*x^2 + 1), x) + 6*b^3*c^2*e*integrate(1/64*x^3*arctan(c*x)*log(c^2*x^ 
2 + 1)^2/(c^2*x^2 + 1), x) + 192*a*b^2*c^2*e*integrate(1/64*x^3*arctan(c*x 
)^2/(c^2*x^2 + 1), x) + 56*b^3*c^2*d*integrate(1/64*x^2*arctan(c*x)^3/(c^2 
*x^2 + 1), x) + 12*b^3*c^2*e*integrate(1/64*x^3*arctan(c*x)*log(c^2*x^2 + 
1)/(c^2*x^2 + 1), x) + 6*b^3*c^2*d*integrate(1/64*x^2*arctan(c*x)*log(c^2* 
x^2 + 1)^2/(c^2*x^2 + 1), x) + 192*a*b^2*c^2*d*integrate(1/64*x^2*arctan(c 
*x)^2/(c^2*x^2 + 1), x) + 24*b^3*c^2*d*integrate(1/64*x^2*arctan(c*x)*log( 
c^2*x^2 + 1)/(c^2*x^2 + 1), x) + 1/2*a^3*e*x^2 + a*b^2*d*arctan(c*x)^3/c - 
 12*b^3*c*e*integrate(1/64*x^2*arctan(c*x)^2/(c^2*x^2 + 1), x) + 3*b^3*c*e 
*integrate(1/64*x^2*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) - 24*b^3*c*d*inte 
grate(1/64*x*arctan(c*x)^2/(c^2*x^2 + 1), x) + 6*b^3*c*d*integrate(1/64*x* 
log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 3/2*(x^2*arctan(c*x) - c*(x/c^2 - a 
rctan(c*x)/c^3))*a^2*b*e + a^3*d*x + 56*b^3*e*integrate(1/64*x*arctan(c*x) 
^3/(c^2*x^2 + 1), x) + 6*b^3*e*integrate(1/64*x*arctan(c*x)*log(c^2*x^2 + 
1)^2/(c^2*x^2 + 1), x) + 192*a*b^2*e*integrate(1/64*x*arctan(c*x)^2/(c^2*x 
^2 + 1), x) + 6*b^3*d*integrate(1/64*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^2*x 
^2 + 1), x) + 3/2*(2*c*x*arctan(c*x) - log(c^2*x^2 + 1))*a^2*b*d/c + 1/16* 
(b^3*e*x^2 + 2*b^3*d*x)*arctan(c*x)^3 - 3/64*(b^3*e*x^2 + 2*b^3*d*x)*arcta 
n(c*x)*log(c^2*x^2 + 1)^2
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (d+e x) (a+b \arctan (c x))^3 \, dx=\frac {2 \mathit {atan} \left (c x \right )^{3} b^{3} c^{2} d x +\mathit {atan} \left (c x \right )^{3} b^{3} c^{2} e \,x^{2}+\mathit {atan} \left (c x \right )^{3} b^{3} e +6 \mathit {atan} \left (c x \right )^{2} a \,b^{2} c^{2} d x +3 \mathit {atan} \left (c x \right )^{2} a \,b^{2} c^{2} e \,x^{2}+3 \mathit {atan} \left (c x \right )^{2} a \,b^{2} e -3 \mathit {atan} \left (c x \right )^{2} b^{3} c e x +6 \mathit {atan} \left (c x \right ) a^{2} b \,c^{2} d x +3 \mathit {atan} \left (c x \right ) a^{2} b \,c^{2} e \,x^{2}+3 \mathit {atan} \left (c x \right ) a^{2} b e -6 \mathit {atan} \left (c x \right ) a \,b^{2} c e x -12 \left (\int \frac {\mathit {atan} \left (c x \right ) x}{c^{2} x^{2}+1}d x \right ) a \,b^{2} c^{3} d +6 \left (\int \frac {\mathit {atan} \left (c x \right ) x}{c^{2} x^{2}+1}d x \right ) b^{3} c^{2} e -6 \left (\int \frac {\mathit {atan} \left (c x \right )^{2} x}{c^{2} x^{2}+1}d x \right ) b^{3} c^{3} d -3 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) a^{2} b c d +3 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) a \,b^{2} e +2 a^{3} c^{2} d x +a^{3} c^{2} e \,x^{2}-3 a^{2} b c e x}{2 c^{2}} \] Input:

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

Output:

(2*atan(c*x)**3*b**3*c**2*d*x + atan(c*x)**3*b**3*c**2*e*x**2 + atan(c*x)* 
*3*b**3*e + 6*atan(c*x)**2*a*b**2*c**2*d*x + 3*atan(c*x)**2*a*b**2*c**2*e* 
x**2 + 3*atan(c*x)**2*a*b**2*e - 3*atan(c*x)**2*b**3*c*e*x + 6*atan(c*x)*a 
**2*b*c**2*d*x + 3*atan(c*x)*a**2*b*c**2*e*x**2 + 3*atan(c*x)*a**2*b*e - 6 
*atan(c*x)*a*b**2*c*e*x - 12*int((atan(c*x)*x)/(c**2*x**2 + 1),x)*a*b**2*c 
**3*d + 6*int((atan(c*x)*x)/(c**2*x**2 + 1),x)*b**3*c**2*e - 6*int((atan(c 
*x)**2*x)/(c**2*x**2 + 1),x)*b**3*c**3*d - 3*log(c**2*x**2 + 1)*a**2*b*c*d 
 + 3*log(c**2*x**2 + 1)*a*b**2*e + 2*a**3*c**2*d*x + a**3*c**2*e*x**2 - 3* 
a**2*b*c*e*x)/(2*c**2)