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

Optimal result

Integrand size = 21, antiderivative size = 63 \[ \int \frac {a+b \arctan (c+d x)}{c e+d e x} \, dx=\frac {a \log (c+d x)}{d e}+\frac {i b \operatorname {PolyLog}(2,-i (c+d x))}{2 d e}-\frac {i b \operatorname {PolyLog}(2,i (c+d x))}{2 d e} \] Output:

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83 \[ \int \frac {a+b \arctan (c+d x)}{c e+d e x} \, dx=\frac {a \log (c+d x)+\frac {1}{2} i b \operatorname {PolyLog}(2,-i (c+d x))-\frac {1}{2} i b \operatorname {PolyLog}(2,i (c+d x))}{d e} \] Input:

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

Output:

(a*Log[c + d*x] + (I/2)*b*PolyLog[2, (-I)*(c + d*x)] - (I/2)*b*PolyLog[2, 
I*(c + d*x)])/(d*e)
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {5566, 27, 5355, 2838}

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.

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

\(\Big \downarrow \) 5566

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5355

\(\displaystyle \frac {\frac {1}{2} i b \int \frac {\log (1-i (c+d x))}{c+d x}d(c+d x)-\frac {1}{2} i b \int \frac {\log (i (c+d x)+1)}{c+d x}d(c+d x)+a \log (c+d x)}{d e}\)

\(\Big \downarrow \) 2838

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

Input:

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

Output:

(a*Log[c + d*x] + (I/2)*b*PolyLog[2, (-I)*(c + d*x)] - (I/2)*b*PolyLog[2, 
I*(c + d*x)])/(d*e)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 5355

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] 
 + (Simp[I*(b/2)   Int[Log[1 - I*c*x]/x, x], x] - Simp[I*(b/2)   Int[Log[1 
+ I*c*x]/x, x], x]) /; FreeQ[{a, b, c}, x]
 

rule 5566

Int[((a_.) + ArcTan[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m 
_.), x_Symbol] :> Simp[1/d   Subst[Int[(f*(x/d))^m*(a + b*ArcTan[x])^p, x], 
 x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[d*e - c*f, 0] && 
 IGtQ[p, 0]
 

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.03

method	result	size
risch	\(-\frac {i b \operatorname {dilog}\left (-i d x -i c +1\right )}{2 e d}+\frac {a \ln \left (-i d x -i c \right )}{e d}+\frac {i b \operatorname {dilog}\left (i d x +i c +1\right )}{2 e d}\)	\(65\)
derivativedivides	\(\frac {\frac {a \ln \left (d x +c \right )}{e}+\frac {b \left (\ln \left (d x +c \right ) \arctan \left (d x +c \right )+\frac {i \ln \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )\right )}{2}-\frac {i \ln \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )\right )}{2}+\frac {i \operatorname {dilog}\left (1+i \left (d x +c \right )\right )}{2}-\frac {i \operatorname {dilog}\left (1-i \left (d x +c \right )\right )}{2}\right )}{e}}{d}\)	\(104\)
default	\(\frac {\frac {a \ln \left (d x +c \right )}{e}+\frac {b \left (\ln \left (d x +c \right ) \arctan \left (d x +c \right )+\frac {i \ln \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )\right )}{2}-\frac {i \ln \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )\right )}{2}+\frac {i \operatorname {dilog}\left (1+i \left (d x +c \right )\right )}{2}-\frac {i \operatorname {dilog}\left (1-i \left (d x +c \right )\right )}{2}\right )}{e}}{d}\)	\(104\)
parts	\(\frac {a \ln \left (d x +c \right )}{d e}+\frac {b \left (\ln \left (d x +c \right ) \arctan \left (d x +c \right )+\frac {i \ln \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )\right )}{2}-\frac {i \ln \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )\right )}{2}+\frac {i \operatorname {dilog}\left (1+i \left (d x +c \right )\right )}{2}-\frac {i \operatorname {dilog}\left (1-i \left (d x +c \right )\right )}{2}\right )}{e d}\)	\(106\)

Input:

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

Output:

-1/2*I/e/d*b*dilog(1-I*c-I*d*x)+1/e/d*a*ln(-I*d*x-I*c)+1/2*I*b/e/d*dilog(1 
+I*c+I*d*x)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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