\(\int \frac {(a+b \arctan (c x)) (d+e \log (f+g x^2))}{x^2} \, dx\) [1298]

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.70 (sec) , antiderivative size = 552, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{x^2} \, dx=\frac {1}{2} \left (-\frac {2 (a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{x}+\frac {e \sqrt {g} \left (4 a \sqrt {-f} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )+i b \sqrt {f} \left (\log (1+i c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}+i \sqrt {g}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {g} (i-c x)}{c \sqrt {-f}+i \sqrt {g}}\right )\right )-i b \sqrt {f} \left (\log (1-i c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}-i \sqrt {g}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {g} (1-i c x)}{i c \sqrt {-f}+\sqrt {g}}\right )\right )-i b \sqrt {f} \left (\log (1+i c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}-i \sqrt {g}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {g} (1+i c x)}{i c \sqrt {-f}+\sqrt {g}}\right )\right )+i b \sqrt {f} \left (\log (1-i c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}+i \sqrt {g}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {g} (i+c x)}{c \sqrt {-f}+i \sqrt {g}}\right )\right )\right )}{\sqrt {-f^2}}+b c \left (\left (\log \left (-\frac {g x^2}{f}\right )-\log \left (-\frac {g \left (1+c^2 x^2\right )}{c^2 f-g}\right )\right ) \left (d+e \log \left (f+g x^2\right )\right )-e \operatorname {PolyLog}\left (2,\frac {c^2 \left (f+g x^2\right )}{c^2 f-g}\right )+e \operatorname {PolyLog}\left (2,1+\frac {g x^2}{f}\right )\right )\right ) \] Input:

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

Output:

((-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*(Sqr 
t[-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)] - L 
og[-((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
 

Rubi [A] (verified)

Time = 1.44 (sec) , antiderivative size = 649, normalized size of antiderivative = 0.97, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5552, 2925, 2863, 2009, 5445, 218, 5443, 2856, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 5552

\(\displaystyle 2 e g \int \frac {a+b \arctan (c x)}{g x^2+f}dx+b c \int \frac {d+e \log \left (g x^2+f\right )}{x \left (c^2 x^2+1\right )}dx-\frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{x}\)

\(\Big \downarrow \) 2925

\(\displaystyle 2 e g \int \frac {a+b \arctan (c x)}{g x^2+f}dx+\frac {1}{2} b c \int \frac {d+e \log \left (g x^2+f\right )}{x^2 \left (c^2 x^2+1\right )}dx^2-\frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{x}\)

\(\Big \downarrow \) 2863

\(\displaystyle 2 e g \int \frac {a+b \arctan (c x)}{g x^2+f}dx+\frac {1}{2} b c \int \left (\frac {d+e \log \left (g x^2+f\right )}{x^2}-\frac {c^2 \left (d+e \log \left (g x^2+f\right )\right )}{c^2 x^2+1}\right )dx^2-\frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{x}\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 e g \int \frac {a+b \arctan (c x)}{g x^2+f}dx-\frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{x}+\frac {1}{2} b c \left (-\log \left (-\frac {g \left (c^2 x^2+1\right )}{c^2 f-g}\right ) \left (d+e \log \left (f+g x^2\right )\right )-e \operatorname {PolyLog}\left (2,\frac {c^2 \left (g x^2+f\right )}{c^2 f-g}\right )+\log \left (-\frac {g x^2}{f}\right ) \left (d+e \log \left (f+g x^2\right )\right )+e \operatorname {PolyLog}\left (2,\frac {g x^2}{f}+1\right )\right )\)

\(\Big \downarrow \) 5445

\(\displaystyle 2 e g \left (a \int \frac {1}{g x^2+f}dx+b \int \frac {\arctan (c x)}{g x^2+f}dx\right )-\frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{x}+\frac {1}{2} b c \left (-\log \left (-\frac {g \left (c^2 x^2+1\right )}{c^2 f-g}\right ) \left (d+e \log \left (f+g x^2\right )\right )-e \operatorname {PolyLog}\left (2,\frac {c^2 \left (g x^2+f\right )}{c^2 f-g}\right )+\log \left (-\frac {g x^2}{f}\right ) \left (d+e \log \left (f+g x^2\right )\right )+e \operatorname {PolyLog}\left (2,\frac {g x^2}{f}+1\right )\right )\)

\(\Big \downarrow \) 218

\(\displaystyle 2 e g \left (b \int \frac {\arctan (c x)}{g x^2+f}dx+\frac {a \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}\right )-\frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{x}+\frac {1}{2} b c \left (-\log \left (-\frac {g \left (c^2 x^2+1\right )}{c^2 f-g}\right ) \left (d+e \log \left (f+g x^2\right )\right )-e \operatorname {PolyLog}\left (2,\frac {c^2 \left (g x^2+f\right )}{c^2 f-g}\right )+\log \left (-\frac {g x^2}{f}\right ) \left (d+e \log \left (f+g x^2\right )\right )+e \operatorname {PolyLog}\left (2,\frac {g x^2}{f}+1\right )\right )\)

\(\Big \downarrow \) 5443

\(\displaystyle 2 e g \left (\frac {a \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}+b \left (\frac {1}{2} i \int \frac {\log (1-i c x)}{g x^2+f}dx-\frac {1}{2} i \int \frac {\log (i c x+1)}{g x^2+f}dx\right )\right )-\frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{x}+\frac {1}{2} b c \left (-\log \left (-\frac {g \left (c^2 x^2+1\right )}{c^2 f-g}\right ) \left (d+e \log \left (f+g x^2\right )\right )-e \operatorname {PolyLog}\left (2,\frac {c^2 \left (g x^2+f\right )}{c^2 f-g}\right )+\log \left (-\frac {g x^2}{f}\right ) \left (d+e \log \left (f+g x^2\right )\right )+e \operatorname {PolyLog}\left (2,\frac {g x^2}{f}+1\right )\right )\)

\(\Big \downarrow \) 2856

\(\displaystyle 2 e g \left (\frac {a \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}+b \left (\frac {1}{2} i \int \left (\frac {\sqrt {-f} \log (1-i c x)}{2 f \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {\sqrt {-f} \log (1-i c x)}{2 f \left (\sqrt {g} x+\sqrt {-f}\right )}\right )dx-\frac {1}{2} i \int \left (\frac {\sqrt {-f} \log (i c x+1)}{2 f \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {\sqrt {-f} \log (i c x+1)}{2 f \left (\sqrt {g} x+\sqrt {-f}\right )}\right )dx\right )\right )-\frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{x}+\frac {1}{2} b c \left (-\log \left (-\frac {g \left (c^2 x^2+1\right )}{c^2 f-g}\right ) \left (d+e \log \left (f+g x^2\right )\right )-e \operatorname {PolyLog}\left (2,\frac {c^2 \left (g x^2+f\right )}{c^2 f-g}\right )+\log \left (-\frac {g x^2}{f}\right ) \left (d+e \log \left (f+g x^2\right )\right )+e \operatorname {PolyLog}\left (2,\frac {g x^2}{f}+1\right )\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{x}+2 e g \left (\frac {a \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}}+b \left (\frac {1}{2} i \left (-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {g} (1-i c x)}{i \sqrt {-f} c+\sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {g} (c x+i)}{\sqrt {-f} c+i \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\log (1-i c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}+i \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {\log (1-i c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}-i \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}\right )-\frac {1}{2} i \left (-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {g} (i-c x)}{\sqrt {-f} c+i \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {g} (i c x+1)}{i \sqrt {-f} c+\sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\log (1+i c x) \log \left (\frac {c \left (\sqrt {-f}-\sqrt {g} x\right )}{c \sqrt {-f}-i \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {\log (1+i c x) \log \left (\frac {c \left (\sqrt {-f}+\sqrt {g} x\right )}{c \sqrt {-f}+i \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}\right )\right )\right )+\frac {1}{2} b c \left (-\log \left (-\frac {g \left (c^2 x^2+1\right )}{c^2 f-g}\right ) \left (d+e \log \left (f+g x^2\right )\right )-e \operatorname {PolyLog}\left (2,\frac {c^2 \left (g x^2+f\right )}{c^2 f-g}\right )+\log \left (-\frac {g x^2}{f}\right ) \left (d+e \log \left (f+g x^2\right )\right )+e \operatorname {PolyLog}\left (2,\frac {g x^2}{f}+1\right )\right )\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 218

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2856

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_. 
)*(x_)^(r_))^(q_.), x_Symbol] :> Int[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] && I 
GtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))
 

rule 2863

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 2925

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m 
_.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Si 
mplify[(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] && Integer 
Q[r] && IntegerQ[s/n] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0 
] || IGtQ[q, 0])
 

rule 5443

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

rule 5445

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

rule 5552

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] + (-Simp[b*(c/(m + 1))   Int[x^(m + 1)*((d + e* 
Log[f + g*x^2])/(1 + c^2*x^2)), x], x] - Simp[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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-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]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{x^2} \, dx=\frac {4 \sqrt {g}\, \sqrt {f}\, \mathit {atan} \left (\frac {g x}{\sqrt {g}\, \sqrt {f}}\right ) a e x -2 \mathit {atan} \left (c x \right ) b d f +2 \left (\int \frac {\mathit {atan} \left (c x \right ) \mathrm {log}\left (g \,x^{2}+f \right )}{x^{2}}d x \right ) b e f x -\mathrm {log}\left (c^{2} x^{2}+1\right ) b c d f x -2 \,\mathrm {log}\left (g \,x^{2}+f \right ) a e f +2 \,\mathrm {log}\left (x \right ) b c d f x -2 a d f}{2 f x} \] Input:

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

Output:

(4*sqrt(g)*sqrt(f)*atan((g*x)/(sqrt(g)*sqrt(f)))*a*e*x - 2*atan(c*x)*b*d*f 
 + 2*int((atan(c*x)*log(f + g*x**2))/x**2,x)*b*e*f*x - log(c**2*x**2 + 1)* 
b*c*d*f*x - 2*log(f + g*x**2)*a*e*f + 2*log(x)*b*c*d*f*x - 2*a*d*f)/(2*f*x 
)