\(\int \frac {(a+b \arctan (c x)) (d+e \log (f+g x^2))}{x^2} \, dx\) [1300]

Optimal result

Integrand size = 24, antiderivative size = 672 \[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{x^2} \, dx=\frac {2 a e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}-\frac {i b e \sqrt {g} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}-i \sqrt {g}}\right )}{2 \sqrt {-f}}+\frac {i b e \sqrt {g} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}+i \sqrt {g}}\right )}{2 \sqrt {-f}}-\frac {i b e \sqrt {g} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}-i \sqrt {g}}\right )}{2 \sqrt {-f}}+\frac {i b e \sqrt {g} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}+i \sqrt {g}}\right )}{2 \sqrt {-f}}-\frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{x}+\frac {1}{2} b c \log \left (-\frac {g x^2}{f}\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c \log \left (-\frac {g \left (1+c^2 x^2\right )}{c^2 f-g}\right ) \left (d+e \log \left (f+g x^2\right )\right )+\frac {i b e \sqrt {g} \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (i-c x)}{c \sqrt {-f}+i \sqrt {g}}\right )}{2 \sqrt {-f}}-\frac {i b e \sqrt {g} \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (1-i c x)}{i c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {-f}}-\frac {i b e \sqrt {g} \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (1+i c x)}{i c \sqrt {-f}+\sqrt {g}}\right )}{2 \sqrt {-f}}+\frac {i b e \sqrt {g} \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (i+c x)}{c \sqrt {-f}+i \sqrt {g}}\right )}{2 \sqrt {-f}}-\frac {1}{2} b c e \operatorname {PolyLog}\left (2,\frac {c^2 \left (f+g x^2\right )}{c^2 f-g}\right )+\frac {1}{2} b c e \operatorname {PolyLog}\left (2,1+\frac {g x^2}{f}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 672, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5137, 2525, 36, 29, 31, 2463, 2441, 2352, 2440, 2438, 5030, 211, 5028, 2456} \[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{x^2} \, dx=-\frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{x}+\frac {2 a e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}-\frac {1}{2} b c \log \left (-\frac {g \left (c^2 x^2+1\right )}{c^2 f-g}\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c e \operatorname {PolyLog}\left (2,\frac {c^2 \left (g x^2+f\right )}{c^2 f-g}\right )+\frac {1}{2} b c \log \left (-\frac {g x^2}{f}\right ) \left (d+e \log \left (f+g x^2\right )\right )+\frac {1}{2} b c e \operatorname {PolyLog}\left (2,\frac {g x^2}{f}+1\right )+\frac {i b e \sqrt {g} \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (i-c x)}{\sqrt {-f} c+i \sqrt {g}}\right )}{2 \sqrt {-f}}-\frac {i b e \sqrt {g} \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (1-i c x)}{i \sqrt {-f} c+\sqrt {g}}\right )}{2 \sqrt {-f}}-\frac {i b e \sqrt {g} \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (i c x+1)}{i \sqrt {-f} c+\sqrt {g}}\right )}{2 \sqrt {-f}}+\frac {i b e \sqrt {g} \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (c x+i)}{\sqrt {-f} c+i \sqrt {g}}\right )}{2 \sqrt {-f}}-\frac {i b e \sqrt {g} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}-i \sqrt {g}}\right )}{2 \sqrt {-f}}+\frac {i b e \sqrt {g} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}+i \sqrt {g}}\right )}{2 \sqrt {-f}}-\frac {i b e \sqrt {g} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}-i \sqrt {g}}\right )}{2 \sqrt {-f}}+\frac {i b e \sqrt {g} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}+i \sqrt {g}}\right )}{2 \sqrt {-f}} \]

[In]

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

[Out]

(2*a*e*Sqrt[g]*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/Sqrt[f] - ((I/2)*b*e*Sqrt[g]*Log[1 + I*c*x]*Log[(c*(Sqrt[-f] - Sqr
t[g]*x))/(c*Sqrt[-f] - I*Sqrt[g])])/Sqrt[-f] + ((I/2)*b*e*Sqrt[g]*Log[1 - I*c*x]*Log[(c*(Sqrt[-f] - Sqrt[g]*x)
)/(c*Sqrt[-f] + I*Sqrt[g])])/Sqrt[-f] - ((I/2)*b*e*Sqrt[g]*Log[1 - I*c*x]*Log[(c*(Sqrt[-f] + Sqrt[g]*x))/(c*Sq
rt[-f] - I*Sqrt[g])])/Sqrt[-f] + ((I/2)*b*e*Sqrt[g]*Log[1 + I*c*x]*Log[(c*(Sqrt[-f] + Sqrt[g]*x))/(c*Sqrt[-f]
+ I*Sqrt[g])])/Sqrt[-f] - ((a + b*ArcTan[c*x])*(d + e*Log[f + g*x^2]))/x + (b*c*Log[-((g*x^2)/f)]*(d + e*Log[f
+ g*x^2]))/2 - (b*c*Log[-((g*(1 + c^2*x^2))/(c^2*f - g))]*(d + e*Log[f + g*x^2]))/2 + ((I/2)*b*e*Sqrt[g]*Poly
Log[2, (Sqrt[g]*(I - c*x))/(c*Sqrt[-f] + I*Sqrt[g])])/Sqrt[-f] - ((I/2)*b*e*Sqrt[g]*PolyLog[2, (Sqrt[g]*(1 - I
*c*x))/(I*c*Sqrt[-f] + Sqrt[g])])/Sqrt[-f] - ((I/2)*b*e*Sqrt[g]*PolyLog[2, (Sqrt[g]*(1 + I*c*x))/(I*c*Sqrt[-f]
+ Sqrt[g])])/Sqrt[-f] + ((I/2)*b*e*Sqrt[g]*PolyLog[2, (Sqrt[g]*(I + c*x))/(c*Sqrt[-f] + I*Sqrt[g])])/Sqrt[-f]
- (b*c*e*PolyLog[2, (c^2*(f + g*x^2))/(c^2*f - g)])/2 + (b*c*e*PolyLog[2, 1 + (g*x^2)/f])/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 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 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 2463

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2525

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(f + g*x^(s/n))^r*(a + b*Log[c*(d + e*x)^p])^q,
x], x, x^n], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && IntegerQ[r] && IntegerQ[s/n] && Intege
rQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0])

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 5137

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

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

Mathematica [A] (verified)

Time = 0.81 (sec) , antiderivative size = 552, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{x^2} \, dx=\frac {1}{2} \left (-\frac {2 (a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{x}+\frac {e \sqrt {g} \left (4 a \sqrt {-f} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )+i b \sqrt {f} \left (\log (1+i c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}+i \sqrt {g}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {g} (i-c x)}{c \sqrt {-f}+i \sqrt {g}}\right )\right )-i b \sqrt {f} \left (\log (1-i c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}-i \sqrt {g}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {g} (1-i c x)}{i c \sqrt {-f}+\sqrt {g}}\right )\right )-i b \sqrt {f} \left (\log (1+i c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}-i \sqrt {g}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {g} (1+i c x)}{i c \sqrt {-f}+\sqrt {g}}\right )\right )+i b \sqrt {f} \left (\log (1-i c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}+i \sqrt {g}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {g} (i+c x)}{c \sqrt {-f}+i \sqrt {g}}\right )\right )\right )}{\sqrt {-f^2}}+b c \left (\left (\log \left (-\frac {g x^2}{f}\right )-\log \left (-\frac {g \left (1+c^2 x^2\right )}{c^2 f-g}\right )\right ) \left (d+e \log \left (f+g x^2\right )\right )-e \operatorname {PolyLog}\left (2,\frac {c^2 \left (f+g x^2\right )}{c^2 f-g}\right )+e \operatorname {PolyLog}\left (2,1+\frac {g x^2}{f}\right )\right )\right ) \]

[In]

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

[Out]

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

Maple [F]

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

[In]

int((a+b*arctan(c*x))*(d+e*ln(g*x^2+f))/x^2,x)

[Out]

int((a+b*arctan(c*x))*(d+e*ln(g*x^2+f))/x^2,x)

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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