\(\int \frac {(c+a^2 c x^2)^2 \arctan (a x)^2}{x^2} \, dx\) [271]

Optimal result

Integrand size = 22, antiderivative size = 205 \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^2}{x^2} \, dx=\frac {1}{3} a^2 c^2 x-\frac {1}{3} a c^2 \arctan (a x)-\frac {1}{3} a^3 c^2 x^2 \arctan (a x)+\frac {2}{3} i a c^2 \arctan (a x)^2-\frac {c^2 \arctan (a x)^2}{x}+2 a^2 c^2 x \arctan (a x)^2+\frac {1}{3} a^4 c^2 x^3 \arctan (a x)^2+\frac {10}{3} a c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )+2 a c^2 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-i a c^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+\frac {5}{3} i a c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right ) \]

[Out]

1/3*a^2*c^2*x-1/3*a*c^2*arctan(a*x)-1/3*a^3*c^2*x^2*arctan(a*x)+2/3*I*a*c^2*arctan(a*x)^2-c^2*arctan(a*x)^2/x+
2*a^2*c^2*x*arctan(a*x)^2+1/3*a^4*c^2*x^3*arctan(a*x)^2+10/3*a*c^2*arctan(a*x)*ln(2/(1+I*a*x))+2*a*c^2*arctan(
a*x)*ln(2-2/(1-I*a*x))-I*a*c^2*polylog(2,-1+2/(1-I*a*x))+5/3*I*a*c^2*polylog(2,1-2/(1+I*a*x))

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {5068, 4930, 5040, 4964, 2449, 2352, 4946, 5044, 4988, 2497, 5036, 327, 209} \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^2}{x^2} \, dx=\frac {1}{3} a^4 c^2 x^3 \arctan (a x)^2-\frac {1}{3} a^3 c^2 x^2 \arctan (a x)+2 a^2 c^2 x \arctan (a x)^2+\frac {1}{3} a^2 c^2 x+\frac {2}{3} i a c^2 \arctan (a x)^2-\frac {1}{3} a c^2 \arctan (a x)-\frac {c^2 \arctan (a x)^2}{x}+\frac {10}{3} a c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )+2 a c^2 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-i a c^2 \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )+\frac {5}{3} i a c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right ) \]

[In]

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

[Out]

(a^2*c^2*x)/3 - (a*c^2*ArcTan[a*x])/3 - (a^3*c^2*x^2*ArcTan[a*x])/3 + ((2*I)/3)*a*c^2*ArcTan[a*x]^2 - (c^2*Arc
Tan[a*x]^2)/x + 2*a^2*c^2*x*ArcTan[a*x]^2 + (a^4*c^2*x^3*ArcTan[a*x]^2)/3 + (10*a*c^2*ArcTan[a*x]*Log[2/(1 + I
*a*x)])/3 + 2*a*c^2*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)] - I*a*c^2*PolyLog[2, -1 + 2/(1 - I*a*x)] + ((5*I)/3)*a*
c^2*PolyLog[2, 1 - 2/(1 + I*a*x)]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 327

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

Rule 2352

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

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-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 2497

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[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rule 4930

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTan[c*x^n])^p, x] - Dist[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 4946

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] - Dist[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 4964

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] + Dist[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 4988

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] - Dist[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 5036

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

Rule 5040

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*((a + b*ArcT
an[c*x])^(p + 1)/(b*e*(p + 1))), x] - Dist[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 5044

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[(-I)*((a + b*ArcT
an[c*x])^(p + 1)/(b*d*(p + 1))), x] + Dist[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 5068

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

Rubi steps \begin{align*} \text {integral}& = \int \left (2 a^2 c^2 \arctan (a x)^2+\frac {c^2 \arctan (a x)^2}{x^2}+a^4 c^2 x^2 \arctan (a x)^2\right ) \, dx \\ & = c^2 \int \frac {\arctan (a x)^2}{x^2} \, dx+\left (2 a^2 c^2\right ) \int \arctan (a x)^2 \, dx+\left (a^4 c^2\right ) \int x^2 \arctan (a x)^2 \, dx \\ & = -\frac {c^2 \arctan (a x)^2}{x}+2 a^2 c^2 x \arctan (a x)^2+\frac {1}{3} a^4 c^2 x^3 \arctan (a x)^2+\left (2 a c^2\right ) \int \frac {\arctan (a x)}{x \left (1+a^2 x^2\right )} \, dx-\left (4 a^3 c^2\right ) \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{3} \left (2 a^5 c^2\right ) \int \frac {x^3 \arctan (a x)}{1+a^2 x^2} \, dx \\ & = i a c^2 \arctan (a x)^2-\frac {c^2 \arctan (a x)^2}{x}+2 a^2 c^2 x \arctan (a x)^2+\frac {1}{3} a^4 c^2 x^3 \arctan (a x)^2+\left (2 i a c^2\right ) \int \frac {\arctan (a x)}{x (i+a x)} \, dx+\left (4 a^2 c^2\right ) \int \frac {\arctan (a x)}{i-a x} \, dx-\frac {1}{3} \left (2 a^3 c^2\right ) \int x \arctan (a x) \, dx+\frac {1}{3} \left (2 a^3 c^2\right ) \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx \\ & = -\frac {1}{3} a^3 c^2 x^2 \arctan (a x)+\frac {2}{3} i a c^2 \arctan (a x)^2-\frac {c^2 \arctan (a x)^2}{x}+2 a^2 c^2 x \arctan (a x)^2+\frac {1}{3} a^4 c^2 x^3 \arctan (a x)^2+4 a c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )+2 a c^2 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{3} \left (2 a^2 c^2\right ) \int \frac {\arctan (a x)}{i-a x} \, dx-\left (2 a^2 c^2\right ) \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx-\left (4 a^2 c^2\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\frac {1}{3} \left (a^4 c^2\right ) \int \frac {x^2}{1+a^2 x^2} \, dx \\ & = \frac {1}{3} a^2 c^2 x-\frac {1}{3} a^3 c^2 x^2 \arctan (a x)+\frac {2}{3} i a c^2 \arctan (a x)^2-\frac {c^2 \arctan (a x)^2}{x}+2 a^2 c^2 x \arctan (a x)^2+\frac {1}{3} a^4 c^2 x^3 \arctan (a x)^2+\frac {10}{3} a c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )+2 a c^2 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-i a c^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+\left (4 i a c^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )-\frac {1}{3} \left (a^2 c^2\right ) \int \frac {1}{1+a^2 x^2} \, dx+\frac {1}{3} \left (2 a^2 c^2\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx \\ & = \frac {1}{3} a^2 c^2 x-\frac {1}{3} a c^2 \arctan (a x)-\frac {1}{3} a^3 c^2 x^2 \arctan (a x)+\frac {2}{3} i a c^2 \arctan (a x)^2-\frac {c^2 \arctan (a x)^2}{x}+2 a^2 c^2 x \arctan (a x)^2+\frac {1}{3} a^4 c^2 x^3 \arctan (a x)^2+\frac {10}{3} a c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )+2 a c^2 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-i a c^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+2 i a c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )-\frac {1}{3} \left (2 i a c^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right ) \\ & = \frac {1}{3} a^2 c^2 x-\frac {1}{3} a c^2 \arctan (a x)-\frac {1}{3} a^3 c^2 x^2 \arctan (a x)+\frac {2}{3} i a c^2 \arctan (a x)^2-\frac {c^2 \arctan (a x)^2}{x}+2 a^2 c^2 x \arctan (a x)^2+\frac {1}{3} a^4 c^2 x^3 \arctan (a x)^2+\frac {10}{3} a c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )+2 a c^2 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-i a c^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+\frac {5}{3} i a c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.81 \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^2}{x^2} \, dx=\frac {c^2 \left (a^2 x^2-a x \arctan (a x)-a^3 x^3 \arctan (a x)-3 \arctan (a x)^2-8 i a x \arctan (a x)^2+6 a^2 x^2 \arctan (a x)^2+a^4 x^4 \arctan (a x)^2+6 a x \arctan (a x) \log \left (1-e^{2 i \arctan (a x)}\right )+10 a x \arctan (a x) \log \left (1+e^{2 i \arctan (a x)}\right )-5 i a x \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )-3 i a x \operatorname {PolyLog}\left (2,e^{2 i \arctan (a x)}\right )\right )}{3 x} \]

[In]

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

[Out]

(c^2*(a^2*x^2 - a*x*ArcTan[a*x] - a^3*x^3*ArcTan[a*x] - 3*ArcTan[a*x]^2 - (8*I)*a*x*ArcTan[a*x]^2 + 6*a^2*x^2*
ArcTan[a*x]^2 + a^4*x^4*ArcTan[a*x]^2 + 6*a*x*ArcTan[a*x]*Log[1 - E^((2*I)*ArcTan[a*x])] + 10*a*x*ArcTan[a*x]*
Log[1 + E^((2*I)*ArcTan[a*x])] - (5*I)*a*x*PolyLog[2, -E^((2*I)*ArcTan[a*x])] - (3*I)*a*x*PolyLog[2, E^((2*I)*
ArcTan[a*x])]))/(3*x)

Maple [A] (verified)

Time = 1.18 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.39

method	result	size
derivativedivides	\(a \left (\frac {a^{3} c^{2} x^{3} \arctan \left (a x \right )^{2}}{3}+2 a \,c^{2} x \arctan \left (a x \right )^{2}-\frac {c^{2} \arctan \left (a x \right )^{2}}{a x}-\frac {2 c^{2} \left (\frac {a^{2} \arctan \left (a x \right ) x^{2}}{2}-3 \arctan \left (a x \right ) \ln \left (a x \right )+4 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {a x}{2}+\frac {\arctan \left (a x \right )}{2}-\frac {3 i \ln \left (a x \right ) \ln \left (i a x +1\right )}{2}+\frac {3 i \ln \left (a x \right ) \ln \left (-i a x +1\right )}{2}-\frac {3 i \operatorname {dilog}\left (i a x +1\right )}{2}+\frac {3 i \operatorname {dilog}\left (-i a x +1\right )}{2}+2 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )-2 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )\right )}{3}\right )\)	\(285\)
default	\(a \left (\frac {a^{3} c^{2} x^{3} \arctan \left (a x \right )^{2}}{3}+2 a \,c^{2} x \arctan \left (a x \right )^{2}-\frac {c^{2} \arctan \left (a x \right )^{2}}{a x}-\frac {2 c^{2} \left (\frac {a^{2} \arctan \left (a x \right ) x^{2}}{2}-3 \arctan \left (a x \right ) \ln \left (a x \right )+4 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {a x}{2}+\frac {\arctan \left (a x \right )}{2}-\frac {3 i \ln \left (a x \right ) \ln \left (i a x +1\right )}{2}+\frac {3 i \ln \left (a x \right ) \ln \left (-i a x +1\right )}{2}-\frac {3 i \operatorname {dilog}\left (i a x +1\right )}{2}+\frac {3 i \operatorname {dilog}\left (-i a x +1\right )}{2}+2 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )-2 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )\right )}{3}\right )\)	\(285\)
parts	\(\frac {a^{4} c^{2} x^{3} \arctan \left (a x \right )^{2}}{3}+2 a^{2} c^{2} x \arctan \left (a x \right )^{2}-\frac {c^{2} \arctan \left (a x \right )^{2}}{x}-\frac {2 c^{2} \left (\frac {\arctan \left (a x \right ) x^{2} a^{3}}{2}-3 a \arctan \left (a x \right ) \ln \left (a x \right )+4 a \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {a \left (a x -\arctan \left (a x \right )+3 i \ln \left (a x \right ) \ln \left (i a x +1\right )-3 i \ln \left (a x \right ) \ln \left (-i a x +1\right )+3 i \operatorname {dilog}\left (i a x +1\right )-3 i \operatorname {dilog}\left (-i a x +1\right )-4 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )+4 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )\right )}{2}\right )}{3}\)	\(287\)

[In]

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

[Out]

a*(1/3*a^3*c^2*x^3*arctan(a*x)^2+2*a*c^2*x*arctan(a*x)^2-c^2*arctan(a*x)^2/a/x-2/3*c^2*(1/2*a^2*arctan(a*x)*x^
2-3*arctan(a*x)*ln(a*x)+4*arctan(a*x)*ln(a^2*x^2+1)-1/2*a*x+1/2*arctan(a*x)-3/2*I*ln(a*x)*ln(1+I*a*x)+3/2*I*ln
(a*x)*ln(1-I*a*x)-3/2*I*dilog(1+I*a*x)+3/2*I*dilog(1-I*a*x)+2*I*(ln(a*x-I)*ln(a^2*x^2+1)-dilog(-1/2*I*(I+a*x))
-ln(a*x-I)*ln(-1/2*I*(I+a*x))-1/2*ln(a*x-I)^2)-2*I*(ln(I+a*x)*ln(a^2*x^2+1)-dilog(1/2*I*(a*x-I))-ln(I+a*x)*ln(
1/2*I*(a*x-I))-1/2*ln(I+a*x)^2)))

Fricas [F]

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

[In]

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

[Out]

integral((a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2)*arctan(a*x)^2/x^2, x)

Sympy [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^2}{x^2} \, dx=c^{2} \left (\int 2 a^{2} \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{x^{2}}\, dx + \int a^{4} x^{2} \operatorname {atan}^{2}{\left (a x \right )}\, dx\right ) \]

[In]

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

[Out]

c**2*(Integral(2*a**2*atan(a*x)**2, x) + Integral(atan(a*x)**2/x**2, x) + Integral(a**4*x**2*atan(a*x)**2, x))

Maxima [F]

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

[In]

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

[Out]

1/48*(4*(a^4*c^2*x^4 + 6*a^2*c^2*x^2 - 3*c^2)*arctan(a*x)^2 - (a^4*c^2*x^4 + 6*a^2*c^2*x^2 - 3*c^2)*log(a^2*x^
2 + 1)^2 + 12*(144*a^6*c^2*integrate(1/48*x^6*arctan(a*x)^2/(a^2*x^4 + x^2), x) + 12*a^6*c^2*integrate(1/48*x^
6*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) + 16*a^6*c^2*integrate(1/48*x^6*log(a^2*x^2 + 1)/(a^2*x^4 + x^2), x)
- 32*a^5*c^2*integrate(1/48*x^5*arctan(a*x)/(a^2*x^4 + x^2), x) + 432*a^4*c^2*integrate(1/48*x^4*arctan(a*x)^2
/(a^2*x^4 + x^2), x) + 36*a^4*c^2*integrate(1/48*x^4*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) + 96*a^4*c^2*integ
rate(1/48*x^4*log(a^2*x^2 + 1)/(a^2*x^4 + x^2), x) + 3*a*c^2*arctan(a*x)^3 - 192*a^3*c^2*integrate(1/48*x^3*ar
ctan(a*x)/(a^2*x^4 + x^2), x) + 36*a^2*c^2*integrate(1/48*x^2*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) - 48*a^2*
c^2*integrate(1/48*x^2*log(a^2*x^2 + 1)/(a^2*x^4 + x^2), x) + 96*a*c^2*integrate(1/48*x*arctan(a*x)/(a^2*x^4 +
x^2), x) + 144*c^2*integrate(1/48*arctan(a*x)^2/(a^2*x^4 + x^2), x) + 12*c^2*integrate(1/48*log(a^2*x^2 + 1)^
2/(a^2*x^4 + x^2), x))*x)/x

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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