3.13.72 \(\int \frac {(a+b \arctan (c x))^2}{x (d+e x^2)^2} \, dx\) [1272]

3.13.72.1 Optimal result

Integrand size = 23, antiderivative size = 1087 \[ \int \frac {(a+b \arctan (c x))^2}{x \left (d+e x^2\right )^2} \, dx=-\frac {c^2 (a+b \arctan (c x))^2}{2 d \left (c^2 d-e\right )}+\frac {(a+b \arctan (c x))^2}{4 d^2 \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}+\frac {(a+b \arctan (c x))^2}{4 d^2 \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}+\frac {2 (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^2}+\frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{1-i c x}\right )}{d^2}-\frac {b c \sqrt {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)^{3/2} \left (c^2 d-e\right )}-\frac {(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 )}{2 d^2}+\frac {b c \sqrt {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)^{3/2} \left (c^2 d-e\right )}-\frac {(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 )}{2 d^2}-\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{d^2}-\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{d^2}+\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{d^2}+\frac {i b^2 c \sqrt {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)^{3/2} \left (c^2 d-e\right )}+\frac {i b (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 )}{2 d^2}-\frac {i b^2 c \sqrt {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)^{3/2} \left (c^2 d-e\right )}+\frac {i b (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 )}{2 d^2}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 d^2}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 d^2}+\frac {b^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{2 d^2}-\frac {b^2 \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 )}{4 d^2}-\frac {b^2 \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 )}{4 d^2} \]

output

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

3.13.72.2 Mathematica [F]

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

input

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

output

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

3.13.72.3 Rubi [A] (verified)

Time = 2.11 (sec) , antiderivative size = 1087, 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.

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

\(\Big \downarrow \) 5515

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

\(\Big \downarrow \) 2009

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

input

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

output

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

3.13.72.3.1 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])
 

3.13.72.4 Maple [F]

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

input

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

output

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

3.13.72.5 Fricas [F]

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

input

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

3.13.72.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.13.72.7 Maxima [F]

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

input

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

output

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

3.13.72.8 Giac [F]

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

input

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

output

sage0*x
 

3.13.72.9 Mupad [F(-1)]

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

input

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

output

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