\(\int \frac {(c+a^2 c x^2) \arctan (a x)^2}{x^2} \, dx\) [263]

Optimal result

Integrand size = 20, antiderivative size = 113 \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^2}{x^2} \, dx=-\frac {c \arctan (a x)^2}{x}+a^2 c x \arctan (a x)^2+2 a c \arctan (a x) \log \left (\frac {2}{1+i a x}\right )+2 a c \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-i a c \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+i a c \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.83 \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^2}{x^2} \, dx=c \left (\frac {(-i+a x)^2 \arctan (a x)^2}{x}+2 a \arctan (a x) \left (\log \left (1-e^{2 i \arctan (a x)}\right )+\log \left (1+e^{2 i \arctan (a x)}\right )\right )-i a \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )-i a \operatorname {PolyLog}\left (2,e^{2 i \arctan (a x)}\right )\right ) \] Input:

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

Output:

c*(((-I + a*x)^2*ArcTan[a*x]^2)/x + 2*a*ArcTan[a*x]*(Log[1 - E^((2*I)*ArcT 
an[a*x])] + Log[1 + E^((2*I)*ArcTan[a*x])]) - I*a*PolyLog[2, -E^((2*I)*Arc 
Tan[a*x])] - I*a*PolyLog[2, E^((2*I)*ArcTan[a*x])])
 

Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.45, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5485, 5345, 5361, 5455, 5379, 2849, 2752, 5459, 5403, 2897}

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.

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

\(\Big \downarrow \) 5485

\(\displaystyle a^2 c \int \arctan (a x)^2dx+c \int \frac {\arctan (a x)^2}{x^2}dx\)

\(\Big \downarrow \) 5345

\(\displaystyle a^2 c \left (x \arctan (a x)^2-2 a \int \frac {x \arctan (a x)}{a^2 x^2+1}dx\right )+c \int \frac {\arctan (a x)^2}{x^2}dx\)

\(\Big \downarrow \) 5361

\(\displaystyle a^2 c \left (x \arctan (a x)^2-2 a \int \frac {x \arctan (a x)}{a^2 x^2+1}dx\right )+c \left (2 a \int \frac {\arctan (a x)}{x \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)^2}{x}\right )\)

\(\Big \downarrow \) 5455

\(\displaystyle c \left (2 a \int \frac {\arctan (a x)}{x \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)^2}{x}\right )+a^2 c \left (x \arctan (a x)^2-2 a \left (-\frac {\int \frac {\arctan (a x)}{i-a x}dx}{a}-\frac {i \arctan (a x)^2}{2 a^2}\right )\right )\)

\(\Big \downarrow \) 5379

\(\displaystyle c \left (2 a \int \frac {\arctan (a x)}{x \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)^2}{x}\right )+a^2 c \left (x \arctan (a x)^2-2 a \left (-\frac {\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a}-\int \frac {\log \left (\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx}{a}-\frac {i \arctan (a x)^2}{2 a^2}\right )\right )\)

\(\Big \downarrow \) 2849

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

\(\Big \downarrow \) 2752

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

\(\Big \downarrow \) 5459

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

\(\Big \downarrow \) 5403

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

\(\Big \downarrow \) 2897

\(\displaystyle a^2 c \left (x \arctan (a x)^2-2 a \left (-\frac {i \arctan (a x)^2}{2 a^2}-\frac {\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}}{a}\right )\right )+c \left (-\frac {\arctan (a x)^2}{x}+2 a \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )\right )\)

Input:

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

Output:

c*(-(ArcTan[a*x]^2/x) + 2*a*((-1/2*I)*ArcTan[a*x]^2 + I*((-I)*ArcTan[a*x]* 
Log[2 - 2/(1 - I*a*x)] - PolyLog[2, -1 + 2/(1 - I*a*x)]/2))) + a^2*c*(x*Ar 
cTan[a*x]^2 - 2*a*(((-1/2*I)*ArcTan[a*x]^2)/a^2 - ((ArcTan[a*x]*Log[2/(1 + 
 I*a*x)])/a + ((I/2)*PolyLog[2, 1 - 2/(1 + I*a*x)])/a)/a))
 

Defintions of rubi rules used

rule 2752

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo 
g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
 

rule 2849

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp 
[-e/g   Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ 
{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
 

rule 2897

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/ 
D[u, x])]}, Simp[C*PolyLog[2, 1 - u], x] /; FreeQ[C, x]] /; IntegerQ[m] && 
PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponents[u, 
 x][[2]], Expon[Pq, x]]
 

rule 5345

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a 
+ b*ArcTan[c*x^n])^p, x] - Simp[b*c*n*p   Int[x^n*((a + b*ArcTan[c*x^n])^(p 
 - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && 
 (EqQ[n, 1] || EqQ[p, 1])
 

rule 5361

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> 
 Simp[x^(m + 1)*((a + b*ArcTan[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 
1))   Int[x^(m + n)*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], 
x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] & 
& IntegerQ[m])) && NeQ[m, -1]
 

rule 5379

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] 
 :> Simp[(-(a + b*ArcTan[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c*( 
p/e)   Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[2/(1 + e*(x/d))]/(1 + c^2*x^2)) 
, x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0 
]
 

rule 5403

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_ 
Symbol] :> Simp[(a + b*ArcTan[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Si 
mp[b*c*(p/d)   Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[2 - 2/(1 + e*(x/d))]/(1 
 + c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2* 
d^2 + e^2, 0]
 

rule 5455

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), 
x_Symbol] :> Simp[(-I)*((a + b*ArcTan[c*x])^(p + 1)/(b*e*(p + 1))), x] - Si 
mp[1/(c*d)   Int[(a + b*ArcTan[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b, c, 
 d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
 

rule 5459

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), 
x_Symbol] :> Simp[(-I)*((a + b*ArcTan[c*x])^(p + 1)/(b*d*(p + 1))), x] + Si 
mp[I/d   Int[(a + b*ArcTan[c*x])^p/(x*(I + c*x)), x], x] /; FreeQ[{a, b, c, 
 d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]
 

rule 5485

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_. 
)*(x_)^2)^(q_.), x_Symbol] :> Simp[d   Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + 
 b*ArcTan[c*x])^p, x], x] + Simp[c^2*(d/f^2)   Int[(f*x)^(m + 2)*(d + e*x^2 
)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] 
&& EqQ[e, c^2*d] && GtQ[q, 0] && IGtQ[p, 0] && (RationalQ[m] || (EqQ[p, 1] 
&& IntegerQ[q]))
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 237 vs. \(2 (107 ) = 214\).

Time = 0.48 (sec) , antiderivative size = 238, normalized size of antiderivative = 2.11

method	result	size
derivativedivides	\(a \left (a c x \arctan \left (a x \right )^{2}-\frac {c \arctan \left (a x \right )^{2}}{a x}-2 c \left (-\arctan \left (a x \right ) \ln \left (a x \right )+\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )+\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )}{2}-\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )}{2}-\frac {i \ln \left (a x \right ) \ln \left (i a x +1\right )}{2}+\frac {i \ln \left (a x \right ) \ln \left (-i a x +1\right )}{2}-\frac {i \operatorname {dilog}\left (i a x +1\right )}{2}+\frac {i \operatorname {dilog}\left (-i a x +1\right )}{2}\right )\right )\)	\(238\)
default	\(a \left (a c x \arctan \left (a x \right )^{2}-\frac {c \arctan \left (a x \right )^{2}}{a x}-2 c \left (-\arctan \left (a x \right ) \ln \left (a x \right )+\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )+\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )}{2}-\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )}{2}-\frac {i \ln \left (a x \right ) \ln \left (i a x +1\right )}{2}+\frac {i \ln \left (a x \right ) \ln \left (-i a x +1\right )}{2}-\frac {i \operatorname {dilog}\left (i a x +1\right )}{2}+\frac {i \operatorname {dilog}\left (-i a x +1\right )}{2}\right )\right )\)	\(238\)
parts	\(a^{2} c x \arctan \left (a x \right )^{2}-\frac {c \arctan \left (a x \right )^{2}}{x}-2 c \left (-a \arctan \left (a x \right ) \ln \left (a x \right )+a \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-a \left (-\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )}{2}+\frac {i \ln \left (a x \right ) \ln \left (i a x +1\right )}{2}-\frac {i \ln \left (a x \right ) \ln \left (-i a x +1\right )}{2}+\frac {i \operatorname {dilog}\left (i a x +1\right )}{2}-\frac {i \operatorname {dilog}\left (-i a x +1\right )}{2}\right )\right )\)	\(241\)

Input:

int((a^2*c*x^2+c)*arctan(a*x)^2/x^2,x,method=_RETURNVERBOSE)
 

Output:

a*(a*c*x*arctan(a*x)^2-c*arctan(a*x)^2/a/x-2*c*(-arctan(a*x)*ln(a*x)+arcta 
n(a*x)*ln(a^2*x^2+1)+1/2*I*(ln(a*x-I)*ln(a^2*x^2+1)-1/2*ln(a*x-I)^2-dilog( 
-1/2*I*(a*x+I))-ln(a*x-I)*ln(-1/2*I*(a*x+I)))-1/2*I*(ln(a*x+I)*ln(a^2*x^2+ 
1)-1/2*ln(a*x+I)^2-dilog(1/2*I*(a*x-I))-ln(a*x+I)*ln(1/2*I*(a*x-I)))-1/2*I 
*ln(a*x)*ln(1+I*a*x)+1/2*I*ln(a*x)*ln(1-I*a*x)-1/2*I*dilog(1+I*a*x)+1/2*I* 
dilog(1-I*a*x)))
 

Fricas [F]

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

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

Output:

integral((a^2*c*x^2 + c)*arctan(a*x)^2/x^2, x)
 

Sympy [F]

\[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^2}{x^2} \, dx=c \left (\int a^{2} \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{x^{2}}\, dx\right ) \] Input:

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

Output:

c*(Integral(a**2*atan(a*x)**2, x) + Integral(atan(a*x)**2/x**2, x))
 

Maxima [F]

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

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

Output:

1/16*(4*(a^2*c*x^2 - c)*arctan(a*x)^2 - (a^2*c*x^2 - c)*log(a^2*x^2 + 1)^2 
 + 8*(24*a^4*c*integrate(1/16*x^4*arctan(a*x)^2/(a^2*x^4 + x^2), x) + 2*a^ 
4*c*integrate(1/16*x^4*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) + 8*a^4*c*in 
tegrate(1/16*x^4*log(a^2*x^2 + 1)/(a^2*x^4 + x^2), x) + a*c*arctan(a*x)^3 
- 16*a^3*c*integrate(1/16*x^3*arctan(a*x)/(a^2*x^4 + x^2), x) + 4*a^2*c*in 
tegrate(1/16*x^2*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) - 8*a^2*c*integrat 
e(1/16*x^2*log(a^2*x^2 + 1)/(a^2*x^4 + x^2), x) + 16*a*c*integrate(1/16*x* 
arctan(a*x)/(a^2*x^4 + x^2), x) + 24*c*integrate(1/16*arctan(a*x)^2/(a^2*x 
^4 + x^2), x) + 2*c*integrate(1/16*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x)) 
*x)/x
 

Giac [F]

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

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

Output:

integrate((a^2*c*x^2 + c)*arctan(a*x)^2/x^2, x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^2}{x^2} \, dx=\frac {c \left (-\mathit {atan} \left (a x \right )^{2}+\left (\int \mathit {atan} \left (a x \right )^{2}d x \right ) a^{2} x +2 \left (\int \frac {\mathit {atan} \left (a x \right )}{a^{2} x^{3}+x}d x \right ) a x \right )}{x} \] Input:

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

Output:

(c*( - atan(a*x)**2 + int(atan(a*x)**2,x)*a**2*x + 2*int(atan(a*x)/(a**2*x 
**3 + x),x)*a*x))/x