3.13.73 \(\int \frac {(a+b \arctan (c x))^2}{x^2 (d+e x^2)^2} \, dx\) [1273]

3.13.73.1 Optimal result

Integrand size = 23, antiderivative size = 1141 \[ \int \frac {(a+b \arctan (c x))^2}{x^2 \left (d+e x^2\right )^2} \, dx=-\frac {i c (a+b \arctan (c x))^2}{d^2}-\frac {i c e (a+b \arctan (c x))^2}{2 d^2 \left (c^2 d-e\right )}-\frac {(a+b \arctan (c x))^2}{d^2 x}+\frac {\sqrt {e} (a+b \arctan (c x))^2}{4 d^2 \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\sqrt {e} (a+b \arctan (c x))^2}{4 d^2 \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c e (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{d^2 \left (c^2 d-e\right )}-\frac {b c e (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{d^2 \left (c^2 d-e\right )}-\frac {b c 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 \left (c^2 d-e\right )}-\frac {3 \sqrt {e} (a+b \arctan (c x))^2 \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 )}{4 (-d)^{5/2}}-\frac {b c 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 \left (c^2 d-e\right )}+\frac {3 \sqrt {e} (a+b \arctan (c x))^2 \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 )}{4 (-d)^{5/2}}+\frac {2 b c (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )}{d^2}-\frac {i b^2 c e \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 d^2 \left (c^2 d-e\right )}-\frac {i b^2 c \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )}{d^2}-\frac {i b^2 c e \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 d^2 \left (c^2 d-e\right )}+\frac {i b^2 c 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 \left (c^2 d-e\right )}+\frac {3 i b \sqrt {e} (a+b \arctan (c x)) \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)^{5/2}}+\frac {i b^2 c 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 \left (c^2 d-e\right )}-\frac {3 i b \sqrt {e} (a+b \arctan (c x)) \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)^{5/2}}-\frac {3 b^2 \sqrt {e} \operatorname {PolyLog}\left (3,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{8 (-d)^{5/2}}+\frac {3 b^2 \sqrt {e} \operatorname {PolyLog}\left (3,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{8 (-d)^{5/2}} \]

output

-1/2*I*b^2*c*e*polylog(2,1-2/(1-I*c*x))/d^2/(c^2*d-e)-1/2*I*b^2*c*e*polylo 
g(2,1-2/(1+I*c*x))/d^2/(c^2*d-e)-(a+b*arctan(c*x))^2/d^2/x+b*c*e*(a+b*arct 
an(c*x))*ln(2/(1-I*c*x))/d^2/(c^2*d-e)-b*c*e*(a+b*arctan(c*x))*ln(2/(1+I*c 
*x))/d^2/(c^2*d-e)+2*b*c*(a+b*arctan(c*x))*ln(2-2/(1-I*c*x))/d^2-1/2*b*c*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/(c^2*d-e)-1/2*b*c*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/(c^2*d-e)+3/4*I*b*(a+b*ar 
ctan(c*x))*polylog(2,1-2*c*((-d)^(1/2)-x*e^(1/2))/(1-I*c*x)/(c*(-d)^(1/2)- 
I*e^(1/2)))*e^(1/2)/(-d)^(5/2)-I*b^2*c*polylog(2,-1+2/(1-I*c*x))/d^2-3/4*I 
*b*(a+b*arctan(c*x))*polylog(2,1-2*c*((-d)^(1/2)+x*e^(1/2))/(1-I*c*x)/(c*( 
-d)^(1/2)+I*e^(1/2)))*e^(1/2)/(-d)^(5/2)-I*c*(a+b*arctan(c*x))^2/d^2+1/4*I 
*b^2*c*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/(c^2*d-e)-3/4*(a+b*arctan(c*x))^2*ln(2*c*((-d)^(1/2)-x*e^(1/ 
2))/(1-I*c*x)/(c*(-d)^(1/2)-I*e^(1/2)))*e^(1/2)/(-d)^(5/2)+3/4*(a+b*arctan 
(c*x))^2*ln(2*c*((-d)^(1/2)+x*e^(1/2))/(1-I*c*x)/(c*(-d)^(1/2)+I*e^(1/2))) 
*e^(1/2)/(-d)^(5/2)+1/4*I*b^2*c*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/(c^2*d-e)-1/2*I*c*e*(a+b*arctan(c*x 
))^2/d^2/(c^2*d-e)-3/8*b^2*polylog(3,1-2*c*((-d)^(1/2)-x*e^(1/2))/(1-I*c*x 
)/(c*(-d)^(1/2)-I*e^(1/2)))*e^(1/2)/(-d)^(5/2)+3/8*b^2*polylog(3,1-2*c*((- 
d)^(1/2)+x*e^(1/2))/(1-I*c*x)/(c*(-d)^(1/2)+I*e^(1/2)))*e^(1/2)/(-d)^(5...
 

3.13.73.2 Mathematica [F(-1)]

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

input

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

output

$Aborted
 

3.13.73.3 Rubi [A] (verified)

Time = 2.06 (sec) , antiderivative size = 1141, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {5515, 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.

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

\(\Big \downarrow \) 5515

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {i c e \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right ) b^2}{2 d^2 \left (c^2 d-e\right )}-\frac {i c \operatorname {PolyLog}\left (2,\frac {2}{1-i c x}-1\right ) b^2}{d^2}-\frac {i c e \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) b^2}{2 d^2 \left (c^2 d-e\right )}+\frac {i c 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 ) b^2}{4 d^2 \left (c^2 d-e\right )}+\frac {i c 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 ) b^2}{4 d^2 \left (c^2 d-e\right )}-\frac {3 \sqrt {e} \operatorname {PolyLog}\left (3,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right ) b^2}{8 (-d)^{5/2}}+\frac {3 \sqrt {e} \operatorname {PolyLog}\left (3,1-\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right ) b^2}{8 (-d)^{5/2}}+\frac {c e (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right ) b}{d^2 \left (c^2 d-e\right )}-\frac {c e (a+b \arctan (c x)) \log \left (\frac {2}{i c x+1}\right ) b}{d^2 \left (c^2 d-e\right )}-\frac {c 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 ) b}{2 d^2 \left (c^2 d-e\right )}-\frac {c e (a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right ) b}{2 d^2 \left (c^2 d-e\right )}+\frac {2 c (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right ) b}{d^2}+\frac {3 i \sqrt {e} (a+b \arctan (c x)) \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 ) b}{4 (-d)^{5/2}}-\frac {3 i \sqrt {e} (a+b \arctan (c x)) \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 ) b}{4 (-d)^{5/2}}-\frac {i c e (a+b \arctan (c x))^2}{2 d^2 \left (c^2 d-e\right )}-\frac {(a+b \arctan (c x))^2}{d^2 x}+\frac {\sqrt {e} (a+b \arctan (c x))^2}{4 d^2 \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\sqrt {e} (a+b \arctan (c x))^2}{4 d^2 \left (\sqrt {e} x+\sqrt {-d}\right )}-\frac {i c (a+b \arctan (c x))^2}{d^2}-\frac {3 \sqrt {e} (a+b \arctan (c x))^2 \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 )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} (a+b \arctan (c x))^2 \log \left (\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)^{5/2}}\)

input

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

output

((-I)*c*(a + b*ArcTan[c*x])^2)/d^2 - ((I/2)*c*e*(a + b*ArcTan[c*x])^2)/(d^ 
2*(c^2*d - e)) - (a + b*ArcTan[c*x])^2/(d^2*x) + (Sqrt[e]*(a + b*ArcTan[c* 
x])^2)/(4*d^2*(Sqrt[-d] - Sqrt[e]*x)) - (Sqrt[e]*(a + b*ArcTan[c*x])^2)/(4 
*d^2*(Sqrt[-d] + Sqrt[e]*x)) + (b*c*e*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x 
)])/(d^2*(c^2*d - e)) - (b*c*e*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(d^ 
2*(c^2*d - e)) - (b*c*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*(c^2*d - e)) - (3*Sqrt[ 
e]*(a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I 
*Sqrt[e])*(1 - I*c*x))])/(4*(-d)^(5/2)) - (b*c*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*(c^2*d - e)) + (3*Sqrt[e]*(a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] + Sqr 
t[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*(-d)^(5/2)) + (2*b*c* 
(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)])/d^2 - ((I/2)*b^2*c*e*PolyLog[2 
, 1 - 2/(1 - I*c*x)])/(d^2*(c^2*d - e)) - (I*b^2*c*PolyLog[2, -1 + 2/(1 - 
I*c*x)])/d^2 - ((I/2)*b^2*c*e*PolyLog[2, 1 - 2/(1 + I*c*x)])/(d^2*(c^2*d - 
 e)) + ((I/4)*b^2*c*e*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt 
[-d] - I*Sqrt[e])*(1 - I*c*x))])/(d^2*(c^2*d - e)) + (((3*I)/4)*b*Sqrt[e]* 
(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[- 
d] - I*Sqrt[e])*(1 - I*c*x))])/(-d)^(5/2) + ((I/4)*b^2*c*e*PolyLog[2, 1 - 
(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(...
 

3.13.73.3.1 Defintions of rubi rules used

rule 2009

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

rule 5515

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])
 

3.13.73.4 Maple [F]

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

input

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

output

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

3.13.73.5 Fricas [F]

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

input

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

output

integral((b^2*arctan(c*x)^2 + 2*a*b*arctan(c*x) + a^2)/(e^2*x^6 + 2*d*e*x^ 
4 + d^2*x^2), x)
 

3.13.73.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.13.73.7 Maxima [F(-2)]

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

input

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

output

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

3.13.73.8 Giac [F]

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

input

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

output

sage0*x
 

3.13.73.9 Mupad [F(-1)]

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

input

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

output

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