\(\int \frac {(d+i c d x) (a+b \arctan (c x))}{x^2} \, dx\) [6]

Optimal result

Integrand size = 21, antiderivative size = 77 \[ \int \frac {(d+i c d x) (a+b \arctan (c x))}{x^2} \, dx=-\frac {d (a+b \arctan (c x))}{x}+i a c d \log (x)+b c d \log (x)-\frac {1}{2} b c d \log \left (1+c^2 x^2\right )-\frac {1}{2} b c d \operatorname {PolyLog}(2,-i c x)+\frac {1}{2} b c d \operatorname {PolyLog}(2,i c x) \] Output:

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.97 \[ \int \frac {(d+i c d x) (a+b \arctan (c x))}{x^2} \, dx=\frac {d \left (-2 a-2 b \arctan (c x)+2 i a c x \log (x)+2 b c x \log (x)-b c x \log \left (1+c^2 x^2\right )-b c x \operatorname {PolyLog}(2,-i c x)+b c x \operatorname {PolyLog}(2,i c x)\right )}{2 x} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, 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.

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

\(\Big \downarrow \) 5411

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-((d*(a + b*ArcTan[c*x]))/x) + I*a*c*d*Log[x] + b*c*d*Log[x] - (b*c*d*Log[ 
1 + c^2*x^2])/2 - (b*c*d*PolyLog[2, (-I)*c*x])/2 + (b*c*d*PolyLog[2, I*c*x 
])/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 = 0.34 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.42

method	result	size
parts	\(d a \left (-\frac {1}{x}+i c \ln \left (x \right )\right )+d b c \left (-\frac {\arctan \left (c x \right )}{c x}+i \arctan \left (c x \right ) \ln \left (c x \right )-\frac {\ln \left (c x \right ) \ln \left (i c x +1\right )}{2}+\frac {\ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-\frac {\operatorname {dilog}\left (i c x +1\right )}{2}+\frac {\operatorname {dilog}\left (-i c x +1\right )}{2}-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}+\ln \left (c x \right )\right )\)	\(109\)
derivativedivides	\(c \left (d a \left (-\frac {1}{c x}+i \ln \left (c x \right )\right )+d b \left (-\frac {\arctan \left (c x \right )}{c x}+i \arctan \left (c x \right ) \ln \left (c x \right )-\frac {\ln \left (c x \right ) \ln \left (i c x +1\right )}{2}+\frac {\ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-\frac {\operatorname {dilog}\left (i c x +1\right )}{2}+\frac {\operatorname {dilog}\left (-i c x +1\right )}{2}-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}+\ln \left (c x \right )\right )\right )\)	\(114\)
default	\(c \left (d a \left (-\frac {1}{c x}+i \ln \left (c x \right )\right )+d b \left (-\frac {\arctan \left (c x \right )}{c x}+i \arctan \left (c x \right ) \ln \left (c x \right )-\frac {\ln \left (c x \right ) \ln \left (i c x +1\right )}{2}+\frac {\ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-\frac {\operatorname {dilog}\left (i c x +1\right )}{2}+\frac {\operatorname {dilog}\left (-i c x +1\right )}{2}-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}+\ln \left (c x \right )\right )\right )\)	\(114\)
risch	\(\frac {c d b \ln \left (i c x \right )}{2}-\frac {c d b \ln \left (i c x +1\right )}{2}+\frac {i d b \ln \left (i c x +1\right )}{2 x}-\frac {c d b \operatorname {dilog}\left (i c x +1\right )}{2}-\frac {d a}{x}+i c d a \ln \left (-i c x \right )+\frac {c d b \ln \left (-i c x \right )}{2}-\frac {\ln \left (-i c x +1\right ) b c d}{2}-\frac {i d b \ln \left (-i c x +1\right )}{2 x}+\frac {c d b \operatorname {dilog}\left (-i c x +1\right )}{2}\)	\(127\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(1/2*(2*I*a*c*d*x + 2*a*d - (b*c*d*x - I*b*d)*log(-(c*x + I)/(c*x 
- I)))/x^2, x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

I*b*c*d*integrate(arctan(c*x)/x, x) + I*a*c*d*log(x) - 1/2*(c*(log(c^2*x^2 
 + 1) - log(x^2)) + 2*arctan(c*x)/x)*b*d - a*d/x
 

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 1.18 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.21 \[ \int \frac {(d+i c d x) (a+b \arctan (c x))}{x^2} \, dx=\left \{\begin {array}{cl} -\frac {a\,d}{x} & \text {\ if\ \ }c=0\\ \frac {b\,d\,\left (c^2\,\ln \left (x\right )-\frac {c^2\,\ln \left (c^2\,x^2+1\right )}{2}\right )}{c}+\frac {b\,c\,d\,\left ({\mathrm {Li}}_{\mathrm {2}}\left (1-c\,x\,1{}\mathrm {i}\right )-{\mathrm {Li}}_{\mathrm {2}}\left (1+c\,x\,1{}\mathrm {i}\right )\right )}{2}+\frac {a\,d\,\left (-1+c\,x\,\ln \left (x\right )\,1{}\mathrm {i}\right )}{x}-\frac {b\,d\,\mathrm {atan}\left (c\,x\right )}{x} & \text {\ if\ \ }c\neq 0 \end {array}\right . \] Input:

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

Output:

piecewise(c == 0, -(a*d)/x, c ~= 0, (b*d*(c^2*log(x) - (c^2*log(c^2*x^2 + 
1))/2))/c + (b*c*d*(dilog(- c*x*1i + 1) - dilog(c*x*1i + 1)))/2 + (a*d*(c* 
x*log(x)*1i - 1))/x - (b*d*atan(c*x))/x)
 

Reduce [F]

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

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

Output:

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