\(\int \frac {a+b \arctan (c x)}{x^2 (d+e x^2)} \, dx\) [1157]

Optimal result

Integrand size = 21, antiderivative size = 561 \[ \int \frac {a+b \arctan (c x)}{x^2 \left (d+e x^2\right )} \, dx=-\frac {a+b \arctan (c x)}{d x}-\frac {a \sqrt {e} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{3/2}}+\frac {b c \log (x)}{d}-\frac {i b \sqrt {e} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 (-d)^{3/2}}+\frac {i b \sqrt {e} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{3/2}}-\frac {i b \sqrt {e} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 (-d)^{3/2}}+\frac {i b \sqrt {e} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{3/2}}-\frac {b c \log \left (1+c^2 x^2\right )}{2 d}+\frac {i b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i-c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{3/2}}-\frac {i b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1-i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )}{4 (-d)^{3/2}}-\frac {i b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1+i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )}{4 (-d)^{3/2}}+\frac {i b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i+c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{3/2}} \]

[Out]

(-a-b*arctan(c*x))/d/x+b*c*ln(x)/d-1/2*b*c*ln(c^2*x^2+1)/d-a*arctan(x*e^(1/2)/d^(1/2))*e^(1/2)/d^(3/2)-1/4*I*b
*ln(1+I*c*x)*ln(c*((-d)^(1/2)-x*e^(1/2))/(c*(-d)^(1/2)-I*e^(1/2)))*e^(1/2)/(-d)^(3/2)+1/4*I*b*ln(1-I*c*x)*ln(c
*((-d)^(1/2)-x*e^(1/2))/(c*(-d)^(1/2)+I*e^(1/2)))*e^(1/2)/(-d)^(3/2)-1/4*I*b*ln(1-I*c*x)*ln(c*((-d)^(1/2)+x*e^
(1/2))/(c*(-d)^(1/2)-I*e^(1/2)))*e^(1/2)/(-d)^(3/2)+1/4*I*b*ln(1+I*c*x)*ln(c*((-d)^(1/2)+x*e^(1/2))/(c*(-d)^(1
/2)+I*e^(1/2)))*e^(1/2)/(-d)^(3/2)+1/4*I*b*polylog(2,(I-c*x)*e^(1/2)/(c*(-d)^(1/2)+I*e^(1/2)))*e^(1/2)/(-d)^(3
/2)+1/4*I*b*polylog(2,(I+c*x)*e^(1/2)/(c*(-d)^(1/2)+I*e^(1/2)))*e^(1/2)/(-d)^(3/2)-1/4*I*b*polylog(2,(1-I*c*x)
*e^(1/2)/(I*c*(-d)^(1/2)+e^(1/2)))*e^(1/2)/(-d)^(3/2)-1/4*I*b*polylog(2,(1+I*c*x)*e^(1/2)/(I*c*(-d)^(1/2)+e^(1
/2)))*e^(1/2)/(-d)^(3/2)

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {5038, 4946, 272, 36, 29, 31, 5030, 211, 5028, 2456, 2441, 2440, 2438} \[ \int \frac {a+b \arctan (c x)}{x^2 \left (d+e x^2\right )} \, dx=-\frac {a+b \arctan (c x)}{d x}-\frac {a \sqrt {e} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{3/2}}-\frac {b c \log \left (c^2 x^2+1\right )}{2 d}+\frac {i b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i-c x)}{\sqrt {-d} c+i \sqrt {e}}\right )}{4 (-d)^{3/2}}-\frac {i b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1-i c x)}{i \sqrt {-d} c+\sqrt {e}}\right )}{4 (-d)^{3/2}}-\frac {i b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i c x+1)}{i \sqrt {-d} c+\sqrt {e}}\right )}{4 (-d)^{3/2}}+\frac {i b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (c x+i)}{\sqrt {-d} c+i \sqrt {e}}\right )}{4 (-d)^{3/2}}-\frac {i b \sqrt {e} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 (-d)^{3/2}}+\frac {i b \sqrt {e} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{3/2}}-\frac {i b \sqrt {e} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 (-d)^{3/2}}+\frac {i b \sqrt {e} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{3/2}}+\frac {b c \log (x)}{d} \]

[In]

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

[Out]

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

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 211

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

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 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 2440

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

Rule 2441

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[e*((f + g
*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x)^n])/g), x] - Dist[b*e*(n/g), Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2456

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_)^(r_))^(q_.), x_Symbol] :> In
t[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x]
&& IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))

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 5028

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

Rule 5030

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

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]

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

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 467, normalized size of antiderivative = 0.83 \[ \int \frac {a+b \arctan (c x)}{x^2 \left (d+e x^2\right )} \, dx=\frac {-\frac {a+b \arctan (c x)}{x}+b c \left (\log (x)-\frac {1}{2} \log \left (1+c^2 x^2\right )\right )-\frac {\sqrt {e} \left (4 a \sqrt {-d} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )+i b \sqrt {d} \left (\log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i-c x)}{c \sqrt {-d}+i \sqrt {e}}\right )\right )-i b \sqrt {d} \left (\log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1-i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )\right )-i b \sqrt {d} \left (\log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1+i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )\right )+i b \sqrt {d} \left (\log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i+c x)}{c \sqrt {-d}+i \sqrt {e}}\right )\right )\right )}{4 \sqrt {-d^2}}}{d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.89 (sec) , antiderivative size = 537, normalized size of antiderivative = 0.96

method	result	size
risch	\(-\frac {b e \ln \left (-i c x +1\right ) \ln \left (\frac {c \sqrt {e d}-\left (-i c x +1\right ) e +e}{c \sqrt {e d}+e}\right )}{4 d \sqrt {e d}}+\frac {b e \ln \left (-i c x +1\right ) \ln \left (\frac {c \sqrt {e d}+\left (-i c x +1\right ) e -e}{c \sqrt {e d}-e}\right )}{4 d \sqrt {e d}}-\frac {b e \operatorname {dilog}\left (\frac {c \sqrt {e d}-\left (-i c x +1\right ) e +e}{c \sqrt {e d}+e}\right )}{4 d \sqrt {e d}}+\frac {b e \operatorname {dilog}\left (\frac {c \sqrt {e d}+\left (-i c x +1\right ) e -e}{c \sqrt {e d}-e}\right )}{4 d \sqrt {e d}}+\frac {c b \ln \left (-i c x \right )}{2 d}-\frac {c b \ln \left (-i c x +1\right )}{2 d}+\frac {i b \ln \left (i c x +1\right )}{2 d x}-\frac {i b \ln \left (-i c x +1\right )}{2 d x}-\frac {a}{d x}-\frac {b e \ln \left (i c x +1\right ) \ln \left (\frac {c \sqrt {e d}-\left (i c x +1\right ) e +e}{c \sqrt {e d}+e}\right )}{4 d \sqrt {e d}}+\frac {b e \ln \left (i c x +1\right ) \ln \left (\frac {c \sqrt {e d}+\left (i c x +1\right ) e -e}{c \sqrt {e d}-e}\right )}{4 d \sqrt {e d}}-\frac {b e \operatorname {dilog}\left (\frac {c \sqrt {e d}-\left (i c x +1\right ) e +e}{c \sqrt {e d}+e}\right )}{4 d \sqrt {e d}}+\frac {b e \operatorname {dilog}\left (\frac {c \sqrt {e d}+\left (i c x +1\right ) e -e}{c \sqrt {e d}-e}\right )}{4 d \sqrt {e d}}+\frac {b c \ln \left (i c x \right )}{2 d}-\frac {b c \ln \left (i c x +1\right )}{2 d}-\frac {i a e \,\operatorname {arctanh}\left (\frac {2 \left (-i c x +1\right ) e -2 e}{2 c \sqrt {e d}}\right )}{d \sqrt {e d}}\)	\(537\)
parts	\(\text {Expression too large to display}\)	\(2411\)
derivativedivides	\(\text {Expression too large to display}\)	\(2432\)
default	\(\text {Expression too large to display}\)	\(2432\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral((b*arctan(c*x) + a)/(e*x^4 + d*x^2), x)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \arctan (c x)}{x^2 \left (d+e x^2\right )} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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