\(\int \frac {(a+b \arctan (c x))^2}{d+e x} \, dx\) [12]

Optimal result

Integrand size = 18, antiderivative size = 223 \[ \int \frac {(a+b \arctan (c x))^2}{d+e x} \, dx=-\frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{1-i c x}\right )}{e}+\frac {(a+b \arctan (c x))^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e}+\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{e}-\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 e}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e} \] Output:

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

Mathematica [F]

\[ \int \frac {(a+b \arctan (c x))^2}{d+e x} \, dx=\int \frac {(a+b \arctan (c x))^2}{d+e x} \, dx \] Input:

Integrate[(a + b*ArcTan[c*x])^2/(d + e*x),x]
 

Output:

Integrate[(a + b*ArcTan[c*x])^2/(d + e*x), x]
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 223, 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 = {5383}

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.

\(\displaystyle \int \frac {(a+b \arctan (c x))^2}{d+e x} \, dx\)

\(\Big \downarrow \) 5383

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

Input:

Int[(a + b*ArcTan[c*x])^2/(d + e*x),x]
 

Output:

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

Defintions of rubi rules used

rule 5383

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^2/((d_) + (e_.)*(x_)), x_Symbol] :> 
Simp[(-(a + b*ArcTan[c*x])^2)*(Log[2/(1 - I*c*x)]/e), x] + (Simp[(a + b*Arc 
Tan[c*x])^2*(Log[2*c*((d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/e), x] + Simp[I 
*b*(a + b*ArcTan[c*x])*(PolyLog[2, 1 - 2/(1 - I*c*x)]/e), x] - Simp[I*b*(a 
+ b*ArcTan[c*x])*(PolyLog[2, 1 - 2*c*((d + e*x)/((c*d + I*e)*(1 - I*c*x)))] 
/e), x] - Simp[b^2*(PolyLog[3, 1 - 2/(1 - I*c*x)]/(2*e)), x] + Simp[b^2*(Po 
lyLog[3, 1 - 2*c*((d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/(2*e)), x]) /; Free 
Q[{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 = 4.97 (sec) , antiderivative size = 1199, normalized size of antiderivative = 5.38

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1199\)
default	\(\text {Expression too large to display}\)	\(1199\)
parts	\(\text {Expression too large to display}\)	\(1203\)

Input:

int((a+b*arctan(c*x))^2/(e*x+d),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F]

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

integrate((a+b*arctan(c*x))^2/(e*x+d),x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate((a+b*atan(c*x))**2/(e*x+d),x)
 

Output:

Integral((a + b*atan(c*x))**2/(d + e*x), x)
                                                                                    
                                                                                    
 

Maxima [F]

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

integrate((a+b*arctan(c*x))^2/(e*x+d),x, algorithm="maxima")
 

Output:

a^2*log(e*x + d)/e + integrate(1/16*(12*b^2*arctan(c*x)^2 + b^2*log(c^2*x^ 
2 + 1)^2 + 32*a*b*arctan(c*x))/(e*x + d), x)
 

Giac [F]

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

integrate((a+b*arctan(c*x))^2/(e*x+d),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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