3.13.74 \(\int \frac {(a+b \arctan (c x))^2}{x^3 (d+e x^2)^2} \, dx\) [1274]

3.13.74.1 Optimal result

Integrand size = 23, antiderivative size = 1181 \[ \int \frac {(a+b \arctan (c x))^2}{x^3 \left (d+e x^2\right )^2} \, dx=-\frac {b c (a+b \arctan (c x))}{d^2 x}-\frac {c^2 (a+b \arctan (c x))^2}{2 d^2}+\frac {c^2 e (a+b \arctan (c x))^2}{2 d^2 \left (c^2 d-e\right )}-\frac {(a+b \arctan (c x))^2}{2 d^2 x^2}-\frac {e (a+b \arctan (c x))^2}{4 d^3 \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {e (a+b \arctan (c x))^2}{4 d^3 \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {4 e (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^3}+\frac {b^2 c^2 \log (x)}{d^2}-\frac {2 e (a+b \arctan (c x))^2 \log \left (\frac {2}{1-i c x}\right )}{d^3}-\frac {b c e^{3/2} (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)^{5/2} \left (c^2 d-e\right )}+\frac {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 )}{d^3}+\frac {b c e^{3/2} (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)^{5/2} \left (c^2 d-e\right )}+\frac {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 )}{d^3}-\frac {b^2 c^2 \log \left (1+c^2 x^2\right )}{2 d^2}+\frac {2 i b e (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{d^3}+\frac {2 i b e (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{d^3}-\frac {2 i b e (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{d^3}+\frac {i b^2 c e^{3/2} \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} \left (c^2 d-e\right )}-\frac {i b 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 )}{d^3}-\frac {i b^2 c e^{3/2} \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} \left (c^2 d-e\right )}-\frac {i b 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 )}{d^3}-\frac {b^2 e \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{d^3}+\frac {b^2 e \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{d^3}-\frac {b^2 e \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{d^3}+\frac {b^2 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 )}{2 d^3}+\frac {b^2 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 )}{2 d^3} \]

output

e*(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)))/d^3+b^2*e*polylog(3,1-2/(1+I*c*x))/d^3-b^2*e*polylog(3,-1+2/ 
(1+I*c*x))/d^3+e*(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)))/d^3-1/4*e*(a+b*arctan(c*x))^2/d^3/(1-x*e^(1/2 
)/(-d)^(1/2))-1/4*e*(a+b*arctan(c*x))^2/d^3/(1+x*e^(1/2)/(-d)^(1/2))+4*e*( 
a+b*arctan(c*x))^2*arctanh(-1+2/(1+I*c*x))/d^3-1/2*b^2*c^2*ln(c^2*x^2+1)/d 
^2-b^2*e*polylog(3,1-2/(1-I*c*x))/d^3-1/2*b*c*e^(3/2)*(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)^(5/2) 
/(c^2*d-e)+1/2*b*c*e^(3/2)*(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)^(5/2)/(c^2*d-e)-1/4*I*b^2*c*e^(3 
/2)*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)^(5/2)/(c^2*d-e)-1/2*c^2*(a+b*arctan(c*x))^2/d^2-1/2*(a+b*arctan( 
c*x))^2/d^2/x^2-2*I*b*e*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))/d^3-I* 
b*e*(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)))/d^3-I*b*e*(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)))/d^3+1/4*I*b^2*c*e^(3/2 
)*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)^(5/2)/(c^2*d-e)+2*I*b*e*(a+b*arctan(c*x))*polylog(2,1-2/(1-I*c*x)) 
/d^3+2*I*b*e*(a+b*arctan(c*x))*polylog(2,1-2/(1+I*c*x))/d^3-b*c*(a+b*arcta 
n(c*x))/d^2/x+b^2*c^2*ln(x)/d^2-2*e*(a+b*arctan(c*x))^2*ln(2/(1-I*c*x))...
 

3.13.74.2 Mathematica [F]

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

input

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

output

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

3.13.74.3 Rubi [A] (verified)

Time = 2.26 (sec) , antiderivative size = 1181, 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.

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

\(\Big \downarrow \) 5515

\(\displaystyle \int \left (\frac {2 e^2 x (a+b \arctan (c x))^2}{d^3 \left (d+e x^2\right )}-\frac {2 e (a+b \arctan (c x))^2}{d^3 x}+\frac {e^2 x (a+b \arctan (c x))^2}{d^2 \left (d+e x^2\right )^2}+\frac {(a+b \arctan (c x))^2}{d^2 x^3}\right )dx\)

\(\Big \downarrow \) 2009

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

input

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

output

-((b*c*(a + b*ArcTan[c*x]))/(d^2*x)) - (c^2*(a + b*ArcTan[c*x])^2)/(2*d^2) 
 + (c^2*e*(a + b*ArcTan[c*x])^2)/(2*d^2*(c^2*d - e)) - (a + b*ArcTan[c*x]) 
^2/(2*d^2*x^2) - (e*(a + b*ArcTan[c*x])^2)/(4*d^3*(1 - (Sqrt[e]*x)/Sqrt[-d 
])) - (e*(a + b*ArcTan[c*x])^2)/(4*d^3*(1 + (Sqrt[e]*x)/Sqrt[-d])) - (4*e* 
(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)])/d^3 + (b^2*c^2*Log[x])/d 
^2 - (2*e*(a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/d^3 - (b*c*e^(3/2)*(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)^(5/2)*(c^2*d - e)) + (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))])/d^3 
 + (b*c*e^(3/2)*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*S 
qrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*(-d)^(5/2)*(c^2*d - e)) + (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))])/d^3 - (b^2*c^2*Log[1 + c^2*x^2])/(2*d^2) + ((2*I)*b*e*(a + 
 b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/d^3 + ((2*I)*b*e*(a + b*Arc 
Tan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/d^3 - ((2*I)*b*e*(a + b*ArcTan[c* 
x])*PolyLog[2, -1 + 2/(1 + I*c*x)])/d^3 + ((I/4)*b^2*c*e^(3/2)*PolyLog[2, 
1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/ 
((-d)^(5/2)*(c^2*d - e)) - (I*b*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^3 - ((I 
/4)*b^2*c*e^(3/2)*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[...
 

3.13.74.3.1 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])
 

3.13.74.4 Maple [F]

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

input

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

output

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

3.13.74.5 Fricas [F]

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

input

integrate((a+b*arctan(c*x))^2/x^3/(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^7 + 2*d*e*x^ 
5 + d^2*x^3), x)
 

3.13.74.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.13.74.7 Maxima [F]

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

input

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

output

-1/2*a^2*((2*e*x^2 + d)/(d^2*e*x^4 + d^3*x^2) - 2*e*log(e*x^2 + d)/d^3 + 4 
*e*log(x)/d^3) + integrate((b^2*arctan(c*x)^2 + 2*a*b*arctan(c*x))/(e^2*x^ 
7 + 2*d*e*x^5 + d^2*x^3), x)
 

3.13.74.8 Giac [F]

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

input

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

output

sage0*x
 

3.13.74.9 Mupad [F(-1)]

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

input

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

output

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