\(\int \frac {a+b \arctan (c x)}{x^2 (d+i c d x)} \, dx\) [48]

Optimal result

Integrand size = 23, antiderivative size = 100 \[ \int \frac {a+b \arctan (c x)}{x^2 (d+i c d x)} \, dx=-\frac {a+b \arctan (c x)}{d x}+\frac {b c \log (x)}{d}-\frac {b c \log \left (1+c^2 x^2\right )}{2 d}-\frac {i c (a+b \arctan (c x)) \log \left (2-\frac {2}{1+i c x}\right )}{d}+\frac {b c \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{2 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {4990, 4946, 272, 36, 29, 31, 4988, 2497} \[ \int \frac {a+b \arctan (c x)}{x^2 (d+i c d x)} \, dx=-\frac {a+b \arctan (c x)}{d x}-\frac {i c \log \left (2-\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{d}-\frac {b c \log \left (c^2 x^2+1\right )}{2 d}+\frac {b c \operatorname {PolyLog}\left (2,\frac {2}{i c x+1}-1\right )}{2 d}+\frac {b c \log (x)}{d} \]

[In]

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

[Out]

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 2497

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/D[u, x])]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rule 4946

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTan[c*x^
n])^p/(m + 1)), x] - Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))),
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] && IntegerQ[m])) && NeQ[m, -1]

Rule 4988

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Simp[(a + b*ArcTan[c*x])
^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Dist[b*c*(p/d), Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[2 - 2/(1 + e*(x/d))
]/(1 + c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]

Rule 4990

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)), x_Symbol] :> Dist[1/d, I
nt[(f*x)^m*(a + b*ArcTan[c*x])^p, x], x] - Dist[e/(d*f), Int[(f*x)^(m + 1)*((a + b*ArcTan[c*x])^p/(d + e*x)),
x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0] && LtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\left ((i c) \int \frac {a+b \arctan (c x)}{x (d+i c d x)} \, dx\right )+\frac {\int \frac {a+b \arctan (c x)}{x^2} \, dx}{d} \\ & = -\frac {a+b \arctan (c x)}{d x}-\frac {i c (a+b \arctan (c x)) \log \left (2-\frac {2}{1+i c x}\right )}{d}+\frac {(b c) \int \frac {1}{x \left (1+c^2 x^2\right )} \, dx}{d}+\frac {\left (i b c^2\right ) \int \frac {\log \left (2-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d} \\ & = -\frac {a+b \arctan (c x)}{d x}-\frac {i c (a+b \arctan (c x)) \log \left (2-\frac {2}{1+i c x}\right )}{d}+\frac {b c \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{2 d}+\frac {(b c) \text {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )}{2 d} \\ & = -\frac {a+b \arctan (c x)}{d x}-\frac {i c (a+b \arctan (c x)) \log \left (2-\frac {2}{1+i c x}\right )}{d}+\frac {b c \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{2 d}+\frac {(b c) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 d}-\frac {\left (b c^3\right ) \text {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right )}{2 d} \\ & = -\frac {a+b \arctan (c x)}{d x}+\frac {b c \log (x)}{d}-\frac {b c \log \left (1+c^2 x^2\right )}{2 d}-\frac {i c (a+b \arctan (c x)) \log \left (2-\frac {2}{1+i c x}\right )}{d}+\frac {b c \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.49 \[ \int \frac {a+b \arctan (c x)}{x^2 (d+i c d x)} \, dx=-\frac {a+b \arctan (c x)}{d x}-\frac {i a c \log (x)}{d}-\frac {i c (a+b \arctan (c x)) \log \left (\frac {2 i}{i-c x}\right )}{d}+\frac {b c \left (2 \log (x)-\log \left (1+c^2 x^2\right )\right )}{2 d}+\frac {b c \operatorname {PolyLog}(2,-i c x)}{2 d}-\frac {b c \operatorname {PolyLog}(2,i c x)}{2 d}+\frac {b c \operatorname {PolyLog}\left (2,-\frac {i+c x}{i-c x}\right )}{2 d} \]

[In]

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

[Out]

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (95 ) = 190\).

Time = 0.86 (sec) , antiderivative size = 200, normalized size of antiderivative = 2.00

method	result	size
parts	\(-\frac {a}{d x}-\frac {i a c \ln \left (x \right )}{d}+\frac {i c a \ln \left (c^{2} x^{2}+1\right )}{2 d}-\frac {c a \arctan \left (c x \right )}{d}+\frac {b c \left (-\frac {\arctan \left (c x \right )}{c x}-i \arctan \left (c x \right ) \ln \left (c x \right )+i \arctan \left (c x \right ) \ln \left (c x -i\right )+\frac {\ln \left (c x \right ) \ln \left (i c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}+\frac {\operatorname {dilog}\left (i c x +1\right )}{2}-\frac {\operatorname {dilog}\left (-i c x +1\right )}{2}+\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{2}+\frac {\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{2}-\frac {\ln \left (c x -i\right )^{2}}{4}-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}+\ln \left (c x \right )\right )}{d}\)	\(200\)
derivativedivides	\(c \left (-\frac {a}{d c x}-\frac {i a \ln \left (c x \right )}{d}+\frac {i a \ln \left (c^{2} x^{2}+1\right )}{2 d}-\frac {a \arctan \left (c x \right )}{d}+\frac {b \left (-\frac {\arctan \left (c x \right )}{c x}-i \arctan \left (c x \right ) \ln \left (c x \right )+i \arctan \left (c x \right ) \ln \left (c x -i\right )+\frac {\ln \left (c x \right ) \ln \left (i c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}+\frac {\operatorname {dilog}\left (i c x +1\right )}{2}-\frac {\operatorname {dilog}\left (-i c x +1\right )}{2}+\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{2}+\frac {\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{2}-\frac {\ln \left (c x -i\right )^{2}}{4}-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}+\ln \left (c x \right )\right )}{d}\right )\)	\(203\)
default	\(c \left (-\frac {a}{d c x}-\frac {i a \ln \left (c x \right )}{d}+\frac {i a \ln \left (c^{2} x^{2}+1\right )}{2 d}-\frac {a \arctan \left (c x \right )}{d}+\frac {b \left (-\frac {\arctan \left (c x \right )}{c x}-i \arctan \left (c x \right ) \ln \left (c x \right )+i \arctan \left (c x \right ) \ln \left (c x -i\right )+\frac {\ln \left (c x \right ) \ln \left (i c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}+\frac {\operatorname {dilog}\left (i c x +1\right )}{2}-\frac {\operatorname {dilog}\left (-i c x +1\right )}{2}+\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{2}+\frac {\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{2}-\frac {\ln \left (c x -i\right )^{2}}{4}-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}+\ln \left (c x \right )\right )}{d}\right )\)	\(203\)
risch	\(\frac {b c \ln \left (i c x +1\right )^{2}}{4 d}+\frac {b c \operatorname {dilog}\left (i c x +1\right )}{2 d}+\frac {b c \ln \left (i c x \right )}{2 d}-\frac {b c \ln \left (i c x +1\right )}{2 d}+\frac {i b \ln \left (i c x +1\right )}{2 d x}-\frac {a}{d x}-\frac {i c \ln \left (-i c x \right ) a}{d}+\frac {i c a \ln \left (c^{2} x^{2}+1\right )}{2 d}-\frac {c a \arctan \left (c x \right )}{d}-\frac {c \operatorname {dilog}\left (-i c x +1\right ) b}{2 d}+\frac {c b \ln \left (-i c x \right )}{2 d}-\frac {c b \ln \left (-i c x +1\right )}{2 d}-\frac {i \ln \left (-i c x +1\right ) b}{2 d x}+\frac {c \ln \left (\frac {1}{2}-\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) b}{2 d}-\frac {c b \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{2 d}+\frac {c b \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{2 d}\)	\(254\)

[In]

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

[Out]

-a/d/x-I/d*a*c*ln(x)+1/2*I*c/d*a*ln(c^2*x^2+1)-c/d*a*arctan(c*x)+1/d*b*c*(-1/c/x*arctan(c*x)-I*arctan(c*x)*ln(
c*x)+I*arctan(c*x)*ln(c*x-I)+1/2*ln(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*x-I)*ln(-1/2*I*(c*x+I))+1/2*dilog(-1/2*I*(c*x+I))-1/4*ln(c*x-I)^2-1/2*ln(c^2*x^2+1)+ln(c*x))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.98 \[ \int \frac {a+b \arctan (c x)}{x^2 (d+i c d x)} \, dx=-\frac {b c x {\rm Li}_2\left (\frac {c x + i}{c x - i} + 1\right ) + 2 \, {\left (i \, a - b\right )} c x \log \left (x\right ) + b c x \log \left (\frac {c x + i}{c}\right ) - {\left (2 i \, a - b\right )} c x \log \left (\frac {c x - i}{c}\right ) + i \, b \log \left (-\frac {c x + i}{c x - i}\right ) + 2 \, a}{2 \, d x} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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