\(\int \frac {(a+b \arctan (c x))^3}{d+e x} \, dx\) [18]

Optimal result

Integrand size = 18, antiderivative size = 320 \[ \int \frac {(a+b \arctan (c x))^3}{d+e x} \, dx=-\frac {(a+b \arctan (c x))^3 \log \left (\frac {2}{1-i c x}\right )}{e}+\frac {(a+b \arctan (c x))^3 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e}+\frac {3 i b (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 e}-\frac {3 i b (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e}-\frac {3 b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 e}+\frac {3 b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e}-\frac {3 i b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1-i c x}\right )}{4 e}+\frac {3 i b^3 \operatorname {PolyLog}\left (4,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{4 e} \] Output:

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {5385}

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.

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

\(\Big \downarrow \) 5385

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

Input:

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

Output:

-(((a + b*ArcTan[c*x])^3*Log[2/(1 - I*c*x)])/e) + ((a + b*ArcTan[c*x])^3*L 
og[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e + (((3*I)/2)*b*(a + b*Arc 
Tan[c*x])^2*PolyLog[2, 1 - 2/(1 - I*c*x)])/e - (((3*I)/2)*b*(a + b*ArcTan[ 
c*x])^2*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e - (3* 
b^2*(a + b*ArcTan[c*x])*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*e) + (3*b^2*(a + 
 b*ArcTan[c*x])*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))]) 
/(2*e) - (((3*I)/4)*b^3*PolyLog[4, 1 - 2/(1 - I*c*x)])/e + (((3*I)/4)*b^3* 
PolyLog[4, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e
 

Defintions of rubi rules used

rule 5385

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^3/((d_) + (e_.)*(x_)), x_Symbol] :> 
Simp[(-(a + b*ArcTan[c*x])^3)*(Log[2/(1 - I*c*x)]/e), x] + (Simp[(a + b*Arc 
Tan[c*x])^3*(Log[2*c*((d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/e), x] + Simp[3 
*I*b*(a + b*ArcTan[c*x])^2*(PolyLog[2, 1 - 2/(1 - I*c*x)]/(2*e)), x] - Simp 
[3*I*b*(a + b*ArcTan[c*x])^2*(PolyLog[2, 1 - 2*c*((d + e*x)/((c*d + I*e)*(1 
 - I*c*x)))]/(2*e)), x] - Simp[3*b^2*(a + b*ArcTan[c*x])*(PolyLog[3, 1 - 2/ 
(1 - I*c*x)]/(2*e)), x] + Simp[3*b^2*(a + b*ArcTan[c*x])*(PolyLog[3, 1 - 2* 
c*((d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/(2*e)), x] - Simp[3*I*b^3*(PolyLog 
[4, 1 - 2/(1 - I*c*x)]/(4*e)), x] + Simp[3*I*b^3*(PolyLog[4, 1 - 2*c*((d + 
e*x)/((c*d + I*e)*(1 - I*c*x)))]/(4*e)), x]) /; FreeQ[{a, b, c, d, e}, x] & 
& NeQ[c^2*d^2 + e^2, 0]
 

Maple [C] (warning: unable to verify)

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

Time = 4.46 (sec) , antiderivative size = 2398, normalized size of antiderivative = 7.49

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2398\)
default	\(\text {Expression too large to display}\)	\(2398\)
parts	\(\text {Expression too large to display}\)	\(2406\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

a^3*log(e*x + d)/e + integrate(1/32*(28*b^3*arctan(c*x)^3 + 3*b^3*arctan(c 
*x)*log(c^2*x^2 + 1)^2 + 96*a*b^2*arctan(c*x)^2 + 96*a^2*b*arctan(c*x))/(e 
*x + d), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

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

Output:

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