\(\int \frac {a+b \arctan (c x)}{x^3 (d+i c d x)} \, dx\) [49]

Optimal result

Integrand size = 23, antiderivative size = 161 \[ \int \frac {a+b \arctan (c x)}{x^3 (d+i c d x)} \, dx=-\frac {b c}{2 d x}-\frac {b c^2 \arctan (c x)}{2 d}-\frac {a+b \arctan (c x)}{2 d x^2}+\frac {i c (a+b \arctan (c x))}{d x}-\frac {i b c^2 \log (x)}{d}+\frac {i b c^2 \log \left (1+c^2 x^2\right )}{2 d}-\frac {c^2 (a+b \arctan (c x)) \log \left (2-\frac {2}{1+i c x}\right )}{d}-\frac {i b c^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{2 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {4990, 4946, 331, 209, 272, 36, 29, 31, 4988, 2497} \[ \int \frac {a+b \arctan (c x)}{x^3 (d+i c d x)} \, dx=-\frac {c^2 \log \left (2-\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{d}-\frac {a+b \arctan (c x)}{2 d x^2}+\frac {i c (a+b \arctan (c x))}{d x}-\frac {b c^2 \arctan (c x)}{2 d}-\frac {i b c^2 \operatorname {PolyLog}\left (2,\frac {2}{i c x+1}-1\right )}{2 d}+\frac {i b c^2 \log \left (c^2 x^2+1\right )}{2 d}-\frac {i b c^2 \log (x)}{d}-\frac {b c}{2 d x} \]

[In]

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

[Out]

-1/2*(b*c)/(d*x) - (b*c^2*ArcTan[c*x])/(2*d) - (a + b*ArcTan[c*x])/(2*d*x^2) + (I*c*(a + b*ArcTan[c*x]))/(d*x)
- (I*b*c^2*Log[x])/d + ((I/2)*b*c^2*Log[1 + c^2*x^2])/d - (c^2*(a + b*ArcTan[c*x])*Log[2 - 2/(1 + I*c*x)])/d
- ((I/2)*b*c^2*PolyLog[2, -1 + 2/(1 + I*c*x)])/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 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 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 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

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^2 (d+i c d x)} \, dx\right )+\frac {\int \frac {a+b \arctan (c x)}{x^3} \, dx}{d} \\ & = -\frac {a+b \arctan (c x)}{2 d x^2}-c^2 \int \frac {a+b \arctan (c x)}{x (d+i c d x)} \, dx-\frac {(i c) \int \frac {a+b \arctan (c x)}{x^2} \, dx}{d}+\frac {(b c) \int \frac {1}{x^2 \left (1+c^2 x^2\right )} \, dx}{2 d} \\ & = -\frac {b c}{2 d x}-\frac {a+b \arctan (c x)}{2 d x^2}+\frac {i c (a+b \arctan (c x))}{d x}-\frac {c^2 (a+b \arctan (c x)) \log \left (2-\frac {2}{1+i c x}\right )}{d}-\frac {\left (i b c^2\right ) \int \frac {1}{x \left (1+c^2 x^2\right )} \, dx}{d}-\frac {\left (b c^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{2 d}+\frac {\left (b c^3\right ) \int \frac {\log \left (2-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d} \\ & = -\frac {b c}{2 d x}-\frac {b c^2 \arctan (c x)}{2 d}-\frac {a+b \arctan (c x)}{2 d x^2}+\frac {i c (a+b \arctan (c x))}{d x}-\frac {c^2 (a+b \arctan (c x)) \log \left (2-\frac {2}{1+i c x}\right )}{d}-\frac {i b c^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{2 d}-\frac {\left (i b c^2\right ) \text {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )}{2 d} \\ & = -\frac {b c}{2 d x}-\frac {b c^2 \arctan (c x)}{2 d}-\frac {a+b \arctan (c x)}{2 d x^2}+\frac {i c (a+b \arctan (c x))}{d x}-\frac {c^2 (a+b \arctan (c x)) \log \left (2-\frac {2}{1+i c x}\right )}{d}-\frac {i b c^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{2 d}-\frac {\left (i b c^2\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 d}+\frac {\left (i b c^4\right ) \text {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right )}{2 d} \\ & = -\frac {b c}{2 d x}-\frac {b c^2 \arctan (c x)}{2 d}-\frac {a+b \arctan (c x)}{2 d x^2}+\frac {i c (a+b \arctan (c x))}{d x}-\frac {i b c^2 \log (x)}{d}+\frac {i b c^2 \log \left (1+c^2 x^2\right )}{2 d}-\frac {c^2 (a+b \arctan (c x)) \log \left (2-\frac {2}{1+i c x}\right )}{d}-\frac {i b c^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{2 d} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 0.14 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.11 \[ \int \frac {a+b \arctan (c x)}{x^3 (d+i c d x)} \, dx=-\frac {\frac {a+b \arctan (c x)}{x^2}-\frac {2 i c (a+b \arctan (c x))}{x}+\frac {b c \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-c^2 x^2\right )}{x}+2 a c^2 \log (x)+2 c^2 (a+b \arctan (c x)) \log \left (\frac {2 i}{i-c x}\right )+i b c^2 \left (2 \log (x)-\log \left (1+c^2 x^2\right )\right )+i b c^2 \operatorname {PolyLog}(2,-i c x)-i b c^2 \operatorname {PolyLog}(2,i c x)+i b c^2 \operatorname {PolyLog}\left (2,\frac {i+c x}{-i+c x}\right )}{2 d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.03 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.56

method	result	size
derivativedivides	\(c^{2} \left (-\frac {a}{2 d \,c^{2} x^{2}}+\frac {i a}{d c x}-\frac {a \ln \left (c x \right )}{d}+\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d}+\frac {i a \arctan \left (c x \right )}{d}+\frac {b \left (-\frac {\arctan \left (c x \right )}{2 c^{2} x^{2}}+\frac {i \arctan \left (c x \right )}{c x}-\arctan \left (c x \right ) \ln \left (c x \right )+\arctan \left (c x \right ) \ln \left (c x -i\right )-i \ln \left (c x \right )-\frac {1}{2 c x}+\frac {i \ln \left (c^{2} x^{2}+1\right )}{2}-\frac {\arctan \left (c x \right )}{2}-\frac {i \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}+\frac {i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-\frac {i \operatorname {dilog}\left (i c x +1\right )}{2}+\frac {i \operatorname {dilog}\left (-i c x +1\right )}{2}-\frac {i \left (\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )+\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}+\frac {i \ln \left (c x -i\right )^{2}}{4}\right )}{d}\right )\)	\(251\)
default	\(c^{2} \left (-\frac {a}{2 d \,c^{2} x^{2}}+\frac {i a}{d c x}-\frac {a \ln \left (c x \right )}{d}+\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d}+\frac {i a \arctan \left (c x \right )}{d}+\frac {b \left (-\frac {\arctan \left (c x \right )}{2 c^{2} x^{2}}+\frac {i \arctan \left (c x \right )}{c x}-\arctan \left (c x \right ) \ln \left (c x \right )+\arctan \left (c x \right ) \ln \left (c x -i\right )-i \ln \left (c x \right )-\frac {1}{2 c x}+\frac {i \ln \left (c^{2} x^{2}+1\right )}{2}-\frac {\arctan \left (c x \right )}{2}-\frac {i \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}+\frac {i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-\frac {i \operatorname {dilog}\left (i c x +1\right )}{2}+\frac {i \operatorname {dilog}\left (-i c x +1\right )}{2}-\frac {i \left (\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )+\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}+\frac {i \ln \left (c x -i\right )^{2}}{4}\right )}{d}\right )\)	\(251\)
parts	\(-\frac {a}{2 d \,x^{2}}+\frac {i a c}{d x}-\frac {a \,c^{2} \ln \left (x \right )}{d}+\frac {c^{2} a \ln \left (c^{2} x^{2}+1\right )}{2 d}+\frac {i c^{2} a \arctan \left (c x \right )}{d}+\frac {b \,c^{2} \left (-\frac {\arctan \left (c x \right )}{2 c^{2} x^{2}}+\frac {i \arctan \left (c x \right )}{c x}-\arctan \left (c x \right ) \ln \left (c x \right )+\arctan \left (c x \right ) \ln \left (c x -i\right )-i \ln \left (c x \right )-\frac {1}{2 c x}+\frac {i \ln \left (c^{2} x^{2}+1\right )}{2}-\frac {\arctan \left (c x \right )}{2}-\frac {i \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}+\frac {i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-\frac {i \operatorname {dilog}\left (i c x +1\right )}{2}+\frac {i \operatorname {dilog}\left (-i c x +1\right )}{2}-\frac {i \left (\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )+\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}+\frac {i \ln \left (c x -i\right )^{2}}{4}\right )}{d}\)	\(252\)
risch	\(-\frac {i c^{2} b \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{2 d}-\frac {i b \ln \left (-i c x +1\right )}{4 d \,x^{2}}-\frac {i c^{2} b \ln \left (-i c x \right )}{4 d}+\frac {i c^{2} b \ln \left (-i c x +1\right )}{4 d}-\frac {3 b \,c^{2} \arctan \left (c x \right )}{4 d}+\frac {b c \ln \left (i c x +1\right )}{2 d x}-\frac {b c}{2 d x}+\frac {i c^{2} a \arctan \left (c x \right )}{d}-\frac {a}{2 d \,x^{2}}+\frac {i c^{2} \operatorname {dilog}\left (-i c x +1\right ) b}{2 d}-\frac {c^{2} \ln \left (-i c x \right ) a}{d}+\frac {c^{2} a \ln \left (c^{2} x^{2}+1\right )}{2 d}+\frac {i c^{2} b \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{2 d}-\frac {i b \,c^{2} \operatorname {dilog}\left (i c x +1\right )}{2 d}+\frac {i a c}{d x}+\frac {i b \ln \left (i c x +1\right )}{4 d \,x^{2}}-\frac {c b \ln \left (-i c x +1\right )}{2 d x}-\frac {i c^{2} \ln \left (\frac {1}{2}-\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) b}{2 d}-\frac {3 i b \,c^{2} \ln \left (i c x \right )}{4 d}-\frac {i b \,c^{2} \ln \left (i c x +1\right )^{2}}{4 d}+\frac {3 i b \,c^{2} \ln \left (c^{2} x^{2}+1\right )}{8 d}\)	\(361\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.81 \[ \int \frac {a+b \arctan (c x)}{x^3 (d+i c d x)} \, dx=\frac {2 i \, b c^{2} x^{2} {\rm Li}_2\left (\frac {c x + i}{c x - i} + 1\right ) - 4 \, {\left (a + i \, b\right )} c^{2} x^{2} \log \left (x\right ) + i \, b c^{2} x^{2} \log \left (\frac {c x + i}{c}\right ) + {\left (4 \, a + 3 i \, b\right )} c^{2} x^{2} \log \left (\frac {c x - i}{c}\right ) - 2 \, {\left (-2 i \, a + b\right )} c x - {\left (2 \, b c x + i \, b\right )} \log \left (-\frac {c x + i}{c x - i}\right ) - 2 \, a}{4 \, d x^{2}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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