\(\int \frac {a+b \arctan (c x)}{x^3 (d+e x^2)} \, dx\) [1154]

Optimal result

Integrand size = 21, antiderivative size = 409 \[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )} \, dx=-\frac {b c}{2 d x}-\frac {b c^2 \arctan (c x)}{2 d}-\frac {a+b \arctan (c x)}{2 d x^2}-\frac {a e \log (x)}{d^2}-\frac {e (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{d^2}+\frac {e (a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 d^2}+\frac {e (a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 d^2}-\frac {i b e \operatorname {PolyLog}(2,-i c x)}{2 d^2}+\frac {i b e \operatorname {PolyLog}(2,i c x)}{2 d^2}+\frac {i b e \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 d^2}-\frac {i b e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 d^2}-\frac {i b e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 d^2} \]

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5038, 4946, 331, 209, 5048, 4940, 2438, 5100, 4966, 2449, 2352, 2497} \[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )} \, dx=-\frac {e \log \left (\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{d^2}+\frac {e (a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}-i \sqrt {e}\right )}\right )}{2 d^2}+\frac {e (a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}+i \sqrt {e}\right )}\right )}{2 d^2}-\frac {a+b \arctan (c x)}{2 d x^2}-\frac {a e \log (x)}{d^2}-\frac {b c^2 \arctan (c x)}{2 d}-\frac {i b e \operatorname {PolyLog}(2,-i c x)}{2 d^2}+\frac {i b e \operatorname {PolyLog}(2,i c x)}{2 d^2}+\frac {i b e \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 d^2}-\frac {i b e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 d^2}-\frac {i b e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right )}{4 d^2}-\frac {b c}{2 d x} \]

[In]

Int[(a + b*ArcTan[c*x])/(x^3*(d + e*x^2)),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) - (a*e*Log[x])/d^2 - (e*(a + b*Ar
cTan[c*x])*Log[2/(1 - I*c*x)])/d^2 + (e*(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^2) + (e*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqr
t[e])*(1 - I*c*x))])/(2*d^2) - ((I/2)*b*e*PolyLog[2, (-I)*c*x])/d^2 + ((I/2)*b*e*PolyLog[2, I*c*x])/d^2 + ((I/
2)*b*e*PolyLog[2, 1 - 2/(1 - I*c*x)])/d^2 - ((I/4)*b*e*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d
] - I*Sqrt[e])*(1 - I*c*x))])/d^2 - ((I/4)*b*e*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sq
rt[e])*(1 - I*c*x))])/d^2

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 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 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 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 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 5038

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

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

Mathematica [C] (verified)

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

Time = 0.23 (sec) , antiderivative size = 514, normalized size of antiderivative = 1.26 \[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )} \, dx=-\frac {2 a d+2 b d \arctan (c x)+2 b c d x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-c^2 x^2\right )+4 a e x^2 \log (x)+i b e x^2 \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )-i b e x^2 \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )-i b e x^2 \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )+i b e x^2 \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )-2 a e x^2 \log \left (d+e x^2\right )+2 i b e x^2 \operatorname {PolyLog}(2,-i c x)-2 i b e x^2 \operatorname {PolyLog}(2,i c x)+i b e x^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i-c x)}{c \sqrt {-d}+i \sqrt {e}}\right )-i b e x^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1-i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )+i b e x^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1+i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )-i b e x^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i+c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 d^2 x^2} \]

[In]

Integrate[(a + b*ArcTan[c*x])/(x^3*(d + e*x^2)),x]

[Out]

-1/4*(2*a*d + 2*b*d*ArcTan[c*x] + 2*b*c*d*x*Hypergeometric2F1[-1/2, 1, 1/2, -(c^2*x^2)] + 4*a*e*x^2*Log[x] + I
*b*e*x^2*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])] - I*b*e*x^2*Log[1 - I*c*x]*Lo
g[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])] - I*b*e*x^2*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x
))/(c*Sqrt[-d] - I*Sqrt[e])] + I*b*e*x^2*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e]
)] - 2*a*e*x^2*Log[d + e*x^2] + (2*I)*b*e*x^2*PolyLog[2, (-I)*c*x] - (2*I)*b*e*x^2*PolyLog[2, I*c*x] + I*b*e*x
^2*PolyLog[2, (Sqrt[e]*(I - c*x))/(c*Sqrt[-d] + I*Sqrt[e])] - I*b*e*x^2*PolyLog[2, (Sqrt[e]*(1 - I*c*x))/(I*c*
Sqrt[-d] + Sqrt[e])] + I*b*e*x^2*PolyLog[2, (Sqrt[e]*(1 + I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])] - I*b*e*x^2*PolyLo
g[2, (Sqrt[e]*(I + c*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(d^2*x^2)

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 571, normalized size of antiderivative = 1.40

method	result	size
risch	\(-\frac {i b \ln \left (-i c x +1\right )}{4 d \,x^{2}}-\frac {b c}{2 d x}+\frac {i b e \ln \left (-i c x +1\right ) \ln \left (\frac {c \sqrt {e d}+\left (-i c x +1\right ) e -e}{c \sqrt {e d}-e}\right )}{4 d^{2}}+\frac {i c^{2} b \ln \left (-i c x \right )}{4 d}-\frac {i c^{2} b \ln \left (i c x \right )}{4 d}-\frac {i c^{2} b \ln \left (-i c x +1\right )}{4 d}+\frac {i b e \ln \left (-i c x +1\right ) \ln \left (\frac {c \sqrt {e d}-\left (-i c x +1\right ) e +e}{c \sqrt {e d}+e}\right )}{4 d^{2}}+\frac {i b e \operatorname {dilog}\left (\frac {c \sqrt {e d}+\left (-i c x +1\right ) e -e}{c \sqrt {e d}-e}\right )}{4 d^{2}}-\frac {i b e \operatorname {dilog}\left (\frac {c \sqrt {e d}-\left (i c x +1\right ) e +e}{c \sqrt {e d}+e}\right )}{4 d^{2}}-\frac {a}{2 d \,x^{2}}-\frac {a e \ln \left (-i c x \right )}{d^{2}}+\frac {a e \ln \left (\left (-i c x +1\right )^{2} e -c^{2} d -2 \left (-i c x +1\right ) e +e \right )}{2 d^{2}}+\frac {i b e \operatorname {dilog}\left (-i c x +1\right )}{2 d^{2}}-\frac {i b e \operatorname {dilog}\left (i c x +1\right )}{2 d^{2}}-\frac {i b e \ln \left (i c x +1\right ) \ln \left (\frac {c \sqrt {e d}-\left (i c x +1\right ) e +e}{c \sqrt {e d}+e}\right )}{4 d^{2}}+\frac {i c^{2} b \ln \left (i c x +1\right )}{4 d}+\frac {i b e \operatorname {dilog}\left (\frac {c \sqrt {e d}-\left (-i c x +1\right ) e +e}{c \sqrt {e d}+e}\right )}{4 d^{2}}-\frac {i b e \ln \left (i c x +1\right ) \ln \left (\frac {c \sqrt {e d}+\left (i c x +1\right ) e -e}{c \sqrt {e d}-e}\right )}{4 d^{2}}+\frac {i b \ln \left (i c x +1\right )}{4 d \,x^{2}}-\frac {i b e \operatorname {dilog}\left (\frac {c \sqrt {e d}+\left (i c x +1\right ) e -e}{c \sqrt {e d}-e}\right )}{4 d^{2}}\)	\(571\)
parts	\(a \left (-\frac {1}{2 d \,x^{2}}-\frac {e \ln \left (x \right )}{d^{2}}+\frac {e \ln \left (e \,x^{2}+d \right )}{2 d^{2}}\right )+b \,c^{2} \left (-\frac {\arctan \left (c x \right )}{2 d \,c^{2} x^{2}}-\frac {\arctan \left (c x \right ) e \ln \left (c x \right )}{c^{2} d^{2}}+\frac {\arctan \left (c x \right ) e \ln \left (e \,c^{2} x^{2}+c^{2} d \right )}{2 c^{2} d^{2}}-\frac {c^{2} \left (\frac {e \left (-\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}\right )}{d^{2} c^{4}}-\frac {-\frac {1}{c x}-\arctan \left (c x \right )}{d \,c^{2}}-\frac {2 e \left (-\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}\right )}{d^{2} c^{4}}\right )}{2}\right )\)	\(757\)
derivativedivides	\(c^{2} \left (\frac {a e \ln \left (e \,c^{2} x^{2}+c^{2} d \right )}{2 c^{2} d^{2}}-\frac {a}{2 d \,c^{2} x^{2}}-\frac {a e \ln \left (c x \right )}{c^{2} d^{2}}+b \,c^{2} \left (\frac {\arctan \left (c x \right ) e \ln \left (e \,c^{2} x^{2}+c^{2} d \right )}{2 d^{2} c^{4}}-\frac {\arctan \left (c x \right )}{2 d \,c^{4} x^{2}}-\frac {\arctan \left (c x \right ) e \ln \left (c x \right )}{d^{2} c^{4}}-\frac {e \left (-\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}\right )}{2 d^{2} c^{4}}+\frac {-\frac {1}{c x}-\arctan \left (c x \right )}{2 d \,c^{2}}+\frac {e \left (-\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}\right )}{d^{2} c^{4}}\right )\right )\)	\(773\)
default	\(c^{2} \left (\frac {a e \ln \left (e \,c^{2} x^{2}+c^{2} d \right )}{2 c^{2} d^{2}}-\frac {a}{2 d \,c^{2} x^{2}}-\frac {a e \ln \left (c x \right )}{c^{2} d^{2}}+b \,c^{2} \left (\frac {\arctan \left (c x \right ) e \ln \left (e \,c^{2} x^{2}+c^{2} d \right )}{2 d^{2} c^{4}}-\frac {\arctan \left (c x \right )}{2 d \,c^{4} x^{2}}-\frac {\arctan \left (c x \right ) e \ln \left (c x \right )}{d^{2} c^{4}}-\frac {e \left (-\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}\right )}{2 d^{2} c^{4}}+\frac {-\frac {1}{c x}-\arctan \left (c x \right )}{2 d \,c^{2}}+\frac {e \left (-\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}\right )}{d^{2} c^{4}}\right )\right )\)	\(773\)

[In]

int((a+b*arctan(c*x))/x^3/(e*x^2+d),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F]

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

[In]

integrate((a+b*arctan(c*x))/x^3/(e*x^2+d),x, algorithm="fricas")

[Out]

integral((b*arctan(c*x) + a)/(e*x^5 + d*x^3), x)

Sympy [F]

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

[In]

integrate((a+b*atan(c*x))/x**3/(e*x**2+d),x)

[Out]

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

Maxima [F]

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

[In]

integrate((a+b*arctan(c*x))/x^3/(e*x^2+d),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate((a+b*arctan(c*x))/x^3/(e*x^2+d),x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )} \, dx=\int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{x^3\,\left (e\,x^2+d\right )} \,d x \]

[In]

int((a + b*atan(c*x))/(x^3*(d + e*x^2)),x)

[Out]

int((a + b*atan(c*x))/(x^3*(d + e*x^2)), x)