\(\int \frac {a+b \arctan (c x)}{x (d+e x^2)} \, dx\) [1153]

Optimal result

Integrand size = 21, antiderivative size = 353 \[ \int \frac {a+b \arctan (c x)}{x \left (d+e x^2\right )} \, dx=\frac {a \log (x)}{d}+\frac {(a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{d}-\frac {(a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 d}-\frac {(a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 d}+\frac {i b \operatorname {PolyLog}(2,-i c x)}{2 d}-\frac {i b \operatorname {PolyLog}(2,i c x)}{2 d}-\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 d}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 d}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 d} \]

[Out]

a*ln(x)/d+(a+b*arctan(c*x))*ln(2/(1-I*c*x))/d-1/2*(a+b*arctan(c*x))*ln(2*c*((-d)^(1/2)-x*e^(1/2))/(1-I*c*x)/(c
*(-d)^(1/2)-I*e^(1/2)))/d-1/2*(a+b*arctan(c*x))*ln(2*c*((-d)^(1/2)+x*e^(1/2))/(1-I*c*x)/(c*(-d)^(1/2)+I*e^(1/2
)))/d+1/2*I*b*polylog(2,-I*c*x)/d-1/2*I*b*polylog(2,I*c*x)/d-1/2*I*b*polylog(2,1-2/(1-I*c*x))/d+1/4*I*b*polylo
g(2,1-2*c*((-d)^(1/2)-x*e^(1/2))/(1-I*c*x)/(c*(-d)^(1/2)-I*e^(1/2)))/d+1/4*I*b*polylog(2,1-2*c*((-d)^(1/2)+x*e
^(1/2))/(1-I*c*x)/(c*(-d)^(1/2)+I*e^(1/2)))/d

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {5048, 4940, 2438, 5100, 4966, 2449, 2352, 2497} \[ \int \frac {a+b \arctan (c x)}{x \left (d+e x^2\right )} \, dx=-\frac {(a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}-i \sqrt {e}\right )}\right )}{2 d}-\frac {(a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}+i \sqrt {e}\right )}\right )}{2 d}+\frac {\log \left (\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{d}+\frac {a \log (x)}{d}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 d}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right )}{4 d}+\frac {i b \operatorname {PolyLog}(2,-i c x)}{2 d}-\frac {i b \operatorname {PolyLog}(2,i c x)}{2 d}-\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 d} \]

[In]

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

[Out]

(a*Log[x])/d + ((a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/d - ((a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*
x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*d) - ((a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c
*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*d) + ((I/2)*b*PolyLog[2, (-I)*c*x])/d - ((I/2)*b*PolyLog[2, I*c*x])/d
- ((I/2)*b*PolyLog[2, 1 - 2/(1 - I*c*x)])/d + ((I/4)*b*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-
d] - I*Sqrt[e])*(1 - I*c*x))])/d + ((I/4)*b*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[
e])*(1 - I*c*x))])/d

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

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 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

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 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 4966

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

Rule 5048

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*(x_)^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[a
+ b*ArcTan[c*x], x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] &&  !(EqQ[m, 1] && NeQ[a,
0])

Rule 5100

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> With
[{u = ExpandIntegrand[(a + b*ArcTan[c*x])^p, (f*x)^m*(d + e*x^2)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b,
c, d, e, f, m}, x] && IntegerQ[q] && IGtQ[p, 0] && ((EqQ[p, 1] && GtQ[q, 0]) || IntegerQ[m])

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

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 484, normalized size of antiderivative = 1.37 \[ \int \frac {a+b \arctan (c x)}{x \left (d+e x^2\right )} \, dx=\frac {a \log (x)}{d}+\frac {i b \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 d}-\frac {i b \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 d}-\frac {i b \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 d}+\frac {i b \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 d}-\frac {a \log \left (d+e x^2\right )}{2 d}+\frac {i b \operatorname {PolyLog}(2,-i c x)}{2 d}-\frac {i b \operatorname {PolyLog}(2,i c x)}{2 d}-\frac {i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} (1-i c x)}{i c \sqrt {-d}-\sqrt {e}}\right )}{4 d}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1-i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )}{4 d}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} (1+i c x)}{i c \sqrt {-d}-\sqrt {e}}\right )}{4 d}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1+i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )}{4 d} \]

[In]

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

[Out]

(a*Log[x])/d + ((I/4)*b*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/d - ((I/4)*b*
Log[1 - I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/d - ((I/4)*b*Log[1 - I*c*x]*Log[(c*(S
qrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/d + ((I/4)*b*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c
*Sqrt[-d] + I*Sqrt[e])])/d - (a*Log[d + e*x^2])/(2*d) + ((I/2)*b*PolyLog[2, (-I)*c*x])/d - ((I/2)*b*PolyLog[2,
I*c*x])/d - ((I/4)*b*PolyLog[2, -((Sqrt[e]*(1 - I*c*x))/(I*c*Sqrt[-d] - Sqrt[e]))])/d - ((I/4)*b*PolyLog[2, (
Sqrt[e]*(1 - I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/d + ((I/4)*b*PolyLog[2, -((Sqrt[e]*(1 + I*c*x))/(I*c*Sqrt[-d]
- Sqrt[e]))])/d + ((I/4)*b*PolyLog[2, (Sqrt[e]*(1 + I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/d

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.23

method	result	size
risch	\(-\frac {i b \operatorname {dilog}\left (-i c x +1\right )}{2 d}-\frac {i b \ln \left (-i c x +1\right ) \ln \left (\frac {c \sqrt {e d}-\left (-i c x +1\right ) e +e}{c \sqrt {e d}+e}\right )}{4 d}-\frac {i b \ln \left (-i c x +1\right ) \ln \left (\frac {c \sqrt {e d}+\left (-i c x +1\right ) e -e}{c \sqrt {e d}-e}\right )}{4 d}-\frac {i b \operatorname {dilog}\left (\frac {c \sqrt {e d}-\left (-i c x +1\right ) e +e}{c \sqrt {e d}+e}\right )}{4 d}-\frac {i b \operatorname {dilog}\left (\frac {c \sqrt {e d}+\left (-i c x +1\right ) e -e}{c \sqrt {e d}-e}\right )}{4 d}+\frac {a \ln \left (-i c x \right )}{d}-\frac {a \ln \left (\left (-i c x +1\right )^{2} e -c^{2} d -2 \left (-i c x +1\right ) e +e \right )}{2 d}+\frac {i b \operatorname {dilog}\left (i c x +1\right )}{2 d}+\frac {i b \ln \left (i c x +1\right ) \ln \left (\frac {c \sqrt {e d}-\left (i c x +1\right ) e +e}{c \sqrt {e d}+e}\right )}{4 d}+\frac {i b \ln \left (i c x +1\right ) \ln \left (\frac {c \sqrt {e d}+\left (i c x +1\right ) e -e}{c \sqrt {e d}-e}\right )}{4 d}+\frac {i b \operatorname {dilog}\left (\frac {c \sqrt {e d}-\left (i c x +1\right ) e +e}{c \sqrt {e d}+e}\right )}{4 d}+\frac {i b \operatorname {dilog}\left (\frac {c \sqrt {e d}+\left (i c x +1\right ) e -e}{c \sqrt {e d}-e}\right )}{4 d}\)	\(435\)
derivativedivides	\(-\frac {a \ln \left (e \,c^{2} x^{2}+c^{2} d \right )}{2 d}+\frac {a \ln \left (c x \right )}{d}+b \,c^{2} \left (-\frac {\arctan \left (c x \right ) \ln \left (e \,c^{2} x^{2}+c^{2} d \right )}{2 d \,c^{2}}+\frac {\arctan \left (c x \right ) \ln \left (c x \right )}{d \,c^{2}}-\frac {-\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}}{d \,c^{2}}+\frac {-\frac {i \left (\ln \left (c x -i\right ) \ln \left (e \,c^{2} x^{2}+c^{2} d \right )-2 e \left (\frac {\ln \left (c x -i\right ) \left (\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )-c x +i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )}\right )+\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )-c x +i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )-c x +i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )}\right )+\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )-c x +i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )}\right )}{2 e}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (e \,c^{2} x^{2}+c^{2} d \right )-2 e \left (\frac {\ln \left (c x +i\right ) \left (\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )-c x -i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )}\right )+\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )-c x -i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )-c x -i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )}\right )+\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )-c x -i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )}\right )}{2 e}\right )\right )}{2}}{2 d \,c^{2}}\right )\)	\(706\)
default	\(-\frac {a \ln \left (e \,c^{2} x^{2}+c^{2} d \right )}{2 d}+\frac {a \ln \left (c x \right )}{d}+b \,c^{2} \left (-\frac {\arctan \left (c x \right ) \ln \left (e \,c^{2} x^{2}+c^{2} d \right )}{2 d \,c^{2}}+\frac {\arctan \left (c x \right ) \ln \left (c x \right )}{d \,c^{2}}-\frac {-\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}}{d \,c^{2}}+\frac {-\frac {i \left (\ln \left (c x -i\right ) \ln \left (e \,c^{2} x^{2}+c^{2} d \right )-2 e \left (\frac {\ln \left (c x -i\right ) \left (\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )-c x +i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )}\right )+\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )-c x +i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )-c x +i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )}\right )+\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )-c x +i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )}\right )}{2 e}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (e \,c^{2} x^{2}+c^{2} d \right )-2 e \left (\frac {\ln \left (c x +i\right ) \left (\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )-c x -i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )}\right )+\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )-c x -i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )-c x -i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )}\right )+\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )-c x -i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )}\right )}{2 e}\right )\right )}{2}}{2 d \,c^{2}}\right )\)	\(706\)
parts	\(\frac {a \ln \left (x \right )}{d}-\frac {a \ln \left (e \,x^{2}+d \right )}{2 d}+b \left (\frac {\arctan \left (c x \right ) \ln \left (c x \right )}{d}-\frac {\arctan \left (c x \right ) \ln \left (e \,c^{2} x^{2}+c^{2} d \right )}{2 d}-\frac {c^{2} \left (-\frac {i \ln \left (c x \right ) \ln \left (i c x +1\right )}{c^{2} d}+\frac {i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{c^{2} d}-\frac {i \operatorname {dilog}\left (i c x +1\right )}{c^{2} d}+\frac {i \operatorname {dilog}\left (-i c x +1\right )}{c^{2} d}-\frac {-\frac {i \left (\ln \left (c x -i\right ) \ln \left (e \,c^{2} x^{2}+c^{2} d \right )-2 e \left (\frac {\ln \left (c x -i\right ) \left (\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )-c x +i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )}\right )+\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )-c x +i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )-c x +i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )}\right )+\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )-c x +i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )}\right )}{2 e}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (e \,c^{2} x^{2}+c^{2} d \right )-2 e \left (\frac {\ln \left (c x +i\right ) \left (\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )-c x -i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )}\right )+\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )-c x -i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )-c x -i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )}\right )+\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )-c x -i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )}\right )}{2 e}\right )\right )}{2}}{d \,c^{2}}\right )}{2}\right )\)	\(709\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral((b*arctan(c*x) + a)/(e*x^3 + d*x), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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