\(\int \frac {a+b \arctan (c+d x)}{c e+d e x} \, dx\) [4]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {5151, 12, 4940, 2438} \[ \int \frac {a+b \arctan (c+d x)}{c e+d e x} \, dx=\frac {a \log (c+d x)}{d e}+\frac {i b \operatorname {PolyLog}(2,-i (c+d x))}{2 d e}-\frac {i b \operatorname {PolyLog}(2,i (c+d x))}{2 d e} \]

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2438

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 4940

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

Rule 5151

Int[((a_.) + ArcTan[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[I
nt[(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]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a+b \arctan (x)}{e x} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {a+b \arctan (x)}{x} \, dx,x,c+d x\right )}{d e} \\ & = \frac {a \log (c+d x)}{d e}+\frac {(i b) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,c+d x\right )}{2 d e}-\frac {(i b) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,c+d x\right )}{2 d e} \\ & = \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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83 \[ \int \frac {a+b \arctan (c+d x)}{c e+d e x} \, dx=\frac {a \log (c+d x)+\frac {1}{2} i b \operatorname {PolyLog}(2,-i (c+d x))-\frac {1}{2} i b \operatorname {PolyLog}(2,i (c+d x))}{d e} \]

[In]

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

[Out]

(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)

Maple [A] (verified)

Time = 0.35 (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	\(\frac {a \ln \left (d x +c \right )}{d e}+\frac {b \left (\ln \left (d x +c \right ) \arctan \left (d x +c \right )+\frac {i \ln \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )\right )}{2}-\frac {i \ln \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )\right )}{2}+\frac {i \operatorname {dilog}\left (1+i \left (d x +c \right )\right )}{2}-\frac {i \operatorname {dilog}\left (1-i \left (d x +c \right )\right )}{2}\right )}{e d}\)	\(106\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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