\(\int \frac {(a+b \arctan (c x))^3}{(d+e x)^3} \, dx\) [20]

Optimal result

Integrand size = 18, antiderivative size = 936 \[ \int \frac {(a+b \arctan (c x))^3}{(d+e x)^3} \, dx =\text {Too large to display} \] Output:

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 1.54 (sec) , antiderivative size = 921, normalized size of antiderivative = 0.98, 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.

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

\(\Big \downarrow \) 5389

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 = 312.13 (sec) , antiderivative size = 40034, normalized size of antiderivative = 42.77

method	result	size
parts	\(\text {Expression too large to display}\)	\(40034\)
derivativedivides	\(\text {Expression too large to display}\)	\(40283\)
default	\(\text {Expression too large to display}\)	\(40283\)

Input:

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

Output:

result too large to display
 

Fricas [F]

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

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-1)]

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

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

Output:

Timed out
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

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

Output:

(12*atan(c*x)**3*b**3*c**8*d**8*e*x + 6*atan(c*x)**3*b**3*c**8*d**7*e**2*x 
**2 + 18*atan(c*x)**3*b**3*c**6*d**7*e**2 + 20*atan(c*x)**3*b**3*c**6*d**6 
*e**3*x + 10*atan(c*x)**3*b**3*c**6*d**5*e**4*x**2 + 30*atan(c*x)**3*b**3* 
c**4*d**5*e**4 + 4*atan(c*x)**3*b**3*c**4*d**4*e**5*x + 2*atan(c*x)**3*b** 
3*c**4*d**3*e**6*x**2 + 6*atan(c*x)**3*b**3*c**2*d**3*e**6 - 4*atan(c*x)** 
3*b**3*c**2*d**2*e**7*x - 2*atan(c*x)**3*b**3*c**2*d*e**8*x**2 - 6*atan(c* 
x)**3*b**3*d*e**8 - 12*atan(c*x)**2*a*b**2*c**8*d**9 + 12*atan(c*x)**2*a*b 
**2*c**8*d**8*e*x + 6*atan(c*x)**2*a*b**2*c**8*d**7*e**2*x**2 + 18*atan(c* 
x)**2*a*b**2*c**6*d**7*e**2 - 12*atan(c*x)**2*a*b**2*c**6*d**6*e**3*x - 6* 
atan(c*x)**2*a*b**2*c**6*d**5*e**4*x**2 + 54*atan(c*x)**2*a*b**2*c**4*d**5 
*e**4 - 60*atan(c*x)**2*a*b**2*c**4*d**4*e**5*x - 30*atan(c*x)**2*a*b**2*c 
**4*d**3*e**6*x**2 + 6*atan(c*x)**2*a*b**2*c**2*d**3*e**6 - 36*atan(c*x)** 
2*a*b**2*c**2*d**2*e**7*x - 18*atan(c*x)**2*a*b**2*c**2*d*e**8*x**2 - 18*a 
tan(c*x)**2*a*b**2*d*e**8 + 18*atan(c*x)**2*b**3*c**7*d**8*e - 18*atan(c*x 
)**2*b**3*c**7*d**7*e**2*x - 18*atan(c*x)**2*b**3*c**7*d**6*e**3*x**2 + 24 
*atan(c*x)**2*b**3*c**5*d**6*e**3 - 42*atan(c*x)**2*b**3*c**5*d**5*e**4*x 
- 36*atan(c*x)**2*b**3*c**5*d**4*e**5*x**2 - 6*atan(c*x)**2*b**3*c**3*d**4 
*e**5 - 30*atan(c*x)**2*b**3*c**3*d**3*e**6*x - 18*atan(c*x)**2*b**3*c**3* 
d**2*e**7*x**2 - 12*atan(c*x)**2*b**3*c*d**2*e**7 - 6*atan(c*x)**2*b**3*c* 
d*e**8*x + 36*atan(c*x)*a**2*b*c**8*d**8*e*x + 18*atan(c*x)*a**2*b*c**8...