\(\int \frac {a+b \arctan (c x)}{x^3 (d+i c d x)^2} \, dx\) [57]

Optimal result

Integrand size = 23, antiderivative size = 244 \[ \int \frac {a+b \arctan (c x)}{x^3 (d+i c d x)^2} \, dx=-\frac {b c}{2 d^2 x}-\frac {b c^2}{2 d^2 (i-c x)}-\frac {a+b \arctan (c x)}{2 d^2 x^2}+\frac {2 i c (a+b \arctan (c x))}{d^2 x}-\frac {i c^2 (a+b \arctan (c x))}{d^2 (i-c x)}-\frac {3 a c^2 \log (x)}{d^2}-\frac {2 i b c^2 \log (x)}{d^2}-\frac {3 c^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{d^2}+\frac {i b c^2 \log \left (1+c^2 x^2\right )}{d^2}-\frac {3 i b c^2 \operatorname {PolyLog}(2,-i c x)}{2 d^2}+\frac {3 i b c^2 \operatorname {PolyLog}(2,i c x)}{2 d^2}-\frac {3 i b c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 d^2} \] Output:

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.23 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.91 \[ \int \frac {a+b \arctan (c x)}{x^3 (d+i c d x)^2} \, dx=-\frac {-b c^2 \left (\frac {1}{-i+c x}+\arctan (c x)\right )+\frac {a+b \arctan (c x)}{x^2}-\frac {4 i c (a+b \arctan (c x))}{x}-\frac {2 i c^2 (a+b \arctan (c x))}{-i+c x}+\frac {b c \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-c^2 x^2\right )}{x}+6 a c^2 \log (x)+6 c^2 (a+b \arctan (c x)) \log \left (\frac {2 i}{i-c x}\right )+2 i b c^2 \left (2 \log (x)-\log \left (1+c^2 x^2\right )\right )+3 i b c^2 \operatorname {PolyLog}(2,-i c x)-3 i b c^2 \operatorname {PolyLog}(2,i c x)+3 i b c^2 \operatorname {PolyLog}\left (2,\frac {i+c x}{-i+c x}\right )}{2 d^2} \] Input:

Integrate[(a + b*ArcTan[c*x])/(x^3*(d + I*c*d*x)^2),x]
 

Output:

-1/2*(-(b*c^2*((-I + c*x)^(-1) + ArcTan[c*x])) + (a + b*ArcTan[c*x])/x^2 - 
 ((4*I)*c*(a + b*ArcTan[c*x]))/x - ((2*I)*c^2*(a + b*ArcTan[c*x]))/(-I + c 
*x) + (b*c*Hypergeometric2F1[-1/2, 1, 1/2, -(c^2*x^2)])/x + 6*a*c^2*Log[x] 
 + 6*c^2*(a + b*ArcTan[c*x])*Log[(2*I)/(I - c*x)] + (2*I)*b*c^2*(2*Log[x] 
- Log[1 + c^2*x^2]) + (3*I)*b*c^2*PolyLog[2, (-I)*c*x] - (3*I)*b*c^2*PolyL 
og[2, I*c*x] + (3*I)*b*c^2*PolyLog[2, (I + c*x)/(-I + c*x)])/d^2
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {5411, 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.

\(\displaystyle \int \frac {a+b \arctan (c x)}{x^3 (d+i c d x)^2} \, dx\)

\(\Big \downarrow \) 5411

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

\(\Big \downarrow \) 2009

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

Input:

Int[(a + b*ArcTan[c*x])/(x^3*(d + I*c*d*x)^2),x]
 

Output:

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

Defintions of rubi rules used

rule 2009

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

rule 5411

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_ 
.)*(x_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcTan[c*x])^p, (f* 
x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] & 
& IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])
 

Maple [A] (verified)

Time = 0.45 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.17

method	result	size
derivativedivides	\(c^{2} \left (-\frac {a}{2 d^{2} c^{2} x^{2}}+\frac {2 i a}{d^{2} c x}-\frac {3 a \ln \left (c x \right )}{d^{2}}+\frac {i a}{d^{2} \left (c x -i\right )}+\frac {3 a \ln \left (c^{2} x^{2}+1\right )}{2 d^{2}}+\frac {3 i a \arctan \left (c x \right )}{d^{2}}+\frac {b \left (-\frac {\arctan \left (c x \right )}{2 c^{2} x^{2}}+\frac {2 i \arctan \left (c x \right )}{c x}-3 \ln \left (c x \right ) \arctan \left (c x \right )+\frac {i \arctan \left (c x \right )}{c x -i}+3 \arctan \left (c x \right ) \ln \left (c x -i\right )-\frac {3 i \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}+\frac {3 i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-\frac {3 i \operatorname {dilog}\left (i c x +1\right )}{2}+\frac {3 i \operatorname {dilog}\left (-i c x +1\right )}{2}-\frac {3 i \left (\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )+\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}+\frac {3 i \ln \left (c x -i\right )^{2}}{4}-\frac {1}{2 c x}-2 i \ln \left (c x \right )+i \ln \left (c^{2} x^{2}+1\right )+\frac {1}{2 c x -2 i}\right )}{d^{2}}\right )\)	\(286\)
default	\(c^{2} \left (-\frac {a}{2 d^{2} c^{2} x^{2}}+\frac {2 i a}{d^{2} c x}-\frac {3 a \ln \left (c x \right )}{d^{2}}+\frac {i a}{d^{2} \left (c x -i\right )}+\frac {3 a \ln \left (c^{2} x^{2}+1\right )}{2 d^{2}}+\frac {3 i a \arctan \left (c x \right )}{d^{2}}+\frac {b \left (-\frac {\arctan \left (c x \right )}{2 c^{2} x^{2}}+\frac {2 i \arctan \left (c x \right )}{c x}-3 \ln \left (c x \right ) \arctan \left (c x \right )+\frac {i \arctan \left (c x \right )}{c x -i}+3 \arctan \left (c x \right ) \ln \left (c x -i\right )-\frac {3 i \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}+\frac {3 i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-\frac {3 i \operatorname {dilog}\left (i c x +1\right )}{2}+\frac {3 i \operatorname {dilog}\left (-i c x +1\right )}{2}-\frac {3 i \left (\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )+\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}+\frac {3 i \ln \left (c x -i\right )^{2}}{4}-\frac {1}{2 c x}-2 i \ln \left (c x \right )+i \ln \left (c^{2} x^{2}+1\right )+\frac {1}{2 c x -2 i}\right )}{d^{2}}\right )\)	\(286\)
parts	\(-\frac {i a \,c^{2}}{d^{2} \left (-c x +i\right )}+\frac {3 c^{2} a \ln \left (c^{2} x^{2}+1\right )}{2 d^{2}}+\frac {3 i c^{2} a \arctan \left (c x \right )}{d^{2}}-\frac {a}{2 x^{2} d^{2}}+\frac {2 i c a}{d^{2} x}-\frac {3 a \,c^{2} \ln \left (x \right )}{d^{2}}+\frac {b \,c^{2} \left (-\frac {\arctan \left (c x \right )}{2 c^{2} x^{2}}+\frac {2 i \arctan \left (c x \right )}{c x}-3 \ln \left (c x \right ) \arctan \left (c x \right )+\frac {i \arctan \left (c x \right )}{c x -i}+3 \arctan \left (c x \right ) \ln \left (c x -i\right )-\frac {3 i \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}+\frac {3 i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-\frac {3 i \operatorname {dilog}\left (i c x +1\right )}{2}+\frac {3 i \operatorname {dilog}\left (-i c x +1\right )}{2}-\frac {3 i \left (\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )+\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}+\frac {3 i \ln \left (c x -i\right )^{2}}{4}-\frac {1}{2 c x}-2 i \ln \left (c x \right )+i \ln \left (c^{2} x^{2}+1\right )+\frac {1}{2 c x -2 i}\right )}{d^{2}}\)	\(291\)
risch	\(\frac {c^{2} a}{d^{2} \left (-i c x -1\right )}-\frac {3 c^{2} a \ln \left (-i c x \right )}{d^{2}}-\frac {a}{2 x^{2} d^{2}}+\frac {3 c^{2} a \ln \left (c^{2} x^{2}+1\right )}{2 d^{2}}+\frac {3 i c^{2} b \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{2 d^{2}}+\frac {i c^{2} b \ln \left (-i c x +1\right )}{4 d^{2} \left (-i c x -1\right )}-\frac {3 i c^{2} b \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i c x}{2}\right )}{2 d^{2}}+\frac {i b \,c^{2} \ln \left (i c x +1\right )}{2 d^{2} \left (i c x +1\right )}+\frac {c^{3} b \ln \left (-i c x +1\right ) x}{4 d^{2} \left (-i c x -1\right )}+\frac {b c \ln \left (i c x +1\right )}{d^{2} x}-\frac {5 i b \,c^{2} \ln \left (i c x \right )}{4 d^{2}}+\frac {5 i b \,c^{2} \ln \left (i c x +1\right )}{4 d^{2}}+\frac {i b \ln \left (i c x +1\right )}{4 d^{2} x^{2}}+\frac {i b \,c^{2}}{2 d^{2} \left (i c x +1\right )}-\frac {3 i b \,c^{2} \ln \left (i c x +1\right )^{2}}{4 d^{2}}-\frac {3 i b \,c^{2} \operatorname {dilog}\left (i c x +1\right )}{2 d^{2}}-\frac {c b \ln \left (-i c x +1\right )}{d^{2} x}-\frac {i b \,c^{2} \ln \left (c^{2} x^{2}+1\right )}{8 d^{2}}-\frac {3 i c^{2} b \ln \left (-i c x \right )}{4 d^{2}}-\frac {i b \ln \left (-i c x +1\right )}{4 d^{2} x^{2}}-\frac {3 i c^{2} b \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{2 d^{2}}+\frac {3 i c^{2} b \operatorname {dilog}\left (-i c x +1\right )}{2 d^{2}}+\frac {3 i c^{2} \ln \left (-i c x +1\right ) b}{4 d^{2}}+\frac {3 i c^{2} a \arctan \left (c x \right )}{d^{2}}+\frac {2 i c a}{d^{2} x}+\frac {b \,c^{2} \arctan \left (c x \right )}{4 d^{2}}-\frac {b c}{2 d^{2} x}\)	\(495\)

Input:

int((a+b*arctan(c*x))/x^3/(d+I*c*d*x)^2,x,method=_RETURNVERBOSE)
 

Output:

c^2*(-1/2*a/d^2/c^2/x^2+2*I*a/d^2/c/x-3*a/d^2*ln(c*x)+I*a/d^2/(c*x-I)+3/2* 
a/d^2*ln(c^2*x^2+1)+3*I*a/d^2*arctan(c*x)+b/d^2*(-1/2/c^2/x^2*arctan(c*x)+ 
2*I*arctan(c*x)/c/x-3*ln(c*x)*arctan(c*x)+I*arctan(c*x)/(c*x-I)+3*arctan(c 
*x)*ln(c*x-I)-3/2*I*ln(c*x)*ln(1+I*c*x)+3/2*I*ln(c*x)*ln(1-I*c*x)-3/2*I*di 
log(1+I*c*x)+3/2*I*dilog(1-I*c*x)-3/2*I*(dilog(-1/2*I*(c*x+I))+ln(c*x-I)*l 
n(-1/2*I*(c*x+I)))+3/4*I*ln(c*x-I)^2-1/2/c/x-2*I*ln(c*x)+I*ln(c^2*x^2+1)+1 
/2/(c*x-I)))
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.91 \[ \int \frac {a+b \arctan (c x)}{x^3 (d+i c d x)^2} \, dx=\frac {12 i \, a c^{2} x^{2} + 2 \, {\left (3 \, a + i \, b\right )} c x - 6 \, {\left (-i \, b c^{3} x^{3} - b c^{2} x^{2}\right )} {\rm Li}_2\left (\frac {c x + i}{c x - i} + 1\right ) - 4 \, {\left ({\left (3 \, a + 2 i \, b\right )} c^{3} x^{3} + {\left (-3 i \, a + 2 \, b\right )} c^{2} x^{2}\right )} \log \left (x\right ) - {\left (6 \, b c^{2} x^{2} - 3 i \, b c x + b\right )} \log \left (-\frac {c x + i}{c x - i}\right ) - 4 \, {\left (-i \, b c^{3} x^{3} - b c^{2} x^{2}\right )} \log \left (\frac {c x + i}{c}\right ) + 4 \, {\left ({\left (3 \, a + i \, b\right )} c^{3} x^{3} - {\left (3 i \, a - b\right )} c^{2} x^{2}\right )} \log \left (\frac {c x - i}{c}\right ) + 2 i \, a}{4 \, {\left (c d^{2} x^{3} - i \, d^{2} x^{2}\right )}} \] Input:

integrate((a+b*arctan(c*x))/x^3/(d+I*c*d*x)^2,x, algorithm="fricas")
 

Output:

1/4*(12*I*a*c^2*x^2 + 2*(3*a + I*b)*c*x - 6*(-I*b*c^3*x^3 - b*c^2*x^2)*dil 
og((c*x + I)/(c*x - I) + 1) - 4*((3*a + 2*I*b)*c^3*x^3 + (-3*I*a + 2*b)*c^ 
2*x^2)*log(x) - (6*b*c^2*x^2 - 3*I*b*c*x + b)*log(-(c*x + I)/(c*x - I)) - 
4*(-I*b*c^3*x^3 - b*c^2*x^2)*log((c*x + I)/c) + 4*((3*a + I*b)*c^3*x^3 - ( 
3*I*a - b)*c^2*x^2)*log((c*x - I)/c) + 2*I*a)/(c*d^2*x^3 - I*d^2*x^2)
 

Sympy [F]

\[ \int \frac {a+b \arctan (c x)}{x^3 (d+i c d x)^2} \, dx=- \frac {\int \frac {a}{c^{2} x^{5} - 2 i c x^{4} - x^{3}}\, dx + \int \frac {b \operatorname {atan}{\left (c x \right )}}{c^{2} x^{5} - 2 i c x^{4} - x^{3}}\, dx}{d^{2}} \] Input:

integrate((a+b*atan(c*x))/x**3/(d+I*c*d*x)**2,x)
 

Output:

-(Integral(a/(c**2*x**5 - 2*I*c*x**4 - x**3), x) + Integral(b*atan(c*x)/(c 
**2*x**5 - 2*I*c*x**4 - x**3), x))/d**2
 

Maxima [F]

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

integrate((a+b*arctan(c*x))/x^3/(d+I*c*d*x)^2,x, algorithm="maxima")
 

Output:

(-2*I*c*integrate(arctan(c*x)/(c^4*d^2*x^6 + 2*c^2*d^2*x^4 + d^2*x^2), x) 
- integrate((c^2*x^2 - 1)*arctan(c*x)/(c^4*d^2*x^7 + 2*c^2*d^2*x^5 + d^2*x 
^3), x))*b - 1/2*a*(-2*I*c^2/(c*d^2*x - I*d^2) - 6*c^2*log(c*x - I)/d^2 + 
6*c^2*log(x)/d^2 - (4*I*c*x - 1)/(d^2*x^2))
 

Giac [F]

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

integrate((a+b*arctan(c*x))/x^3/(d+I*c*d*x)^2,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \arctan (c x)}{x^3 (d+i c d x)^2} \, dx=\int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{x^3\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^2} \,d x \] Input:

int((a + b*atan(c*x))/(x^3*(d + c*d*x*1i)^2),x)
 

Output:

int((a + b*atan(c*x))/(x^3*(d + c*d*x*1i)^2), x)
 

Reduce [F]

\[ \int \frac {a+b \arctan (c x)}{x^3 (d+i c d x)^2} \, dx=\frac {-\left (\int \frac {\mathit {atan} \left (c x \right )}{c^{2} x^{5}-2 c i \,x^{4}-x^{3}}d x \right ) b -\left (\int \frac {1}{c^{2} x^{5}-2 c i \,x^{4}-x^{3}}d x \right ) a}{d^{2}} \] Input:

int((a+b*atan(c*x))/x^3/(d+I*c*d*x)^2,x)
 

Output:

( - (int(atan(c*x)/(c**2*x**5 - 2*c*i*x**4 - x**3),x)*b + int(1/(c**2*x**5 
 - 2*c*i*x**4 - x**3),x)*a))/d**2