\(\int \frac {\arctan (a x)^2}{x^4 (c+a^2 c x^2)^2} \, dx\) [298]

Optimal result

Integrand size = 22, antiderivative size = 242 \[ \int \frac {\arctan (a x)^2}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=-\frac {a^2}{3 c^2 x}-\frac {a^4 x}{4 c^2 \left (1+a^2 x^2\right )}-\frac {7 a^3 \arctan (a x)}{12 c^2}-\frac {a \arctan (a x)}{3 c^2 x^2}+\frac {a^3 \arctan (a x)}{2 c^2 \left (1+a^2 x^2\right )}+\frac {7 i a^3 \arctan (a x)^2}{3 c^2}-\frac {\arctan (a x)^2}{3 c^2 x^3}+\frac {2 a^2 \arctan (a x)^2}{c^2 x}+\frac {a^4 x \arctan (a x)^2}{2 c^2 \left (1+a^2 x^2\right )}+\frac {5 a^3 \arctan (a x)^3}{6 c^2}-\frac {14 a^3 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{3 c^2}+\frac {7 i a^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{3 c^2} \]

[Out]

-1/3*a^2/c^2/x-1/4*a^4*x/c^2/(a^2*x^2+1)-7/12*a^3*arctan(a*x)/c^2-1/3*a*arctan(a*x)/c^2/x^2+1/2*a^3*arctan(a*x
)/c^2/(a^2*x^2+1)+7/3*I*a^3*arctan(a*x)^2/c^2-1/3*arctan(a*x)^2/c^2/x^3+2*a^2*arctan(a*x)^2/c^2/x+1/2*a^4*x*ar
ctan(a*x)^2/c^2/(a^2*x^2+1)+5/6*a^3*arctan(a*x)^3/c^2-14/3*a^3*arctan(a*x)*ln(2-2/(1-I*a*x))/c^2+7/3*I*a^3*pol
ylog(2,-1+2/(1-I*a*x))/c^2

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {5086, 5038, 4946, 331, 209, 5044, 4988, 2497, 5004, 5012, 5050, 205, 211} \[ \int \frac {\arctan (a x)^2}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=\frac {5 a^3 \arctan (a x)^3}{6 c^2}+\frac {7 i a^3 \arctan (a x)^2}{3 c^2}-\frac {7 a^3 \arctan (a x)}{12 c^2}-\frac {14 a^3 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{3 c^2}+\frac {7 i a^3 \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{3 c^2}+\frac {2 a^2 \arctan (a x)^2}{c^2 x}-\frac {a^2}{3 c^2 x}+\frac {a^4 x \arctan (a x)^2}{2 c^2 \left (a^2 x^2+1\right )}-\frac {a^4 x}{4 c^2 \left (a^2 x^2+1\right )}+\frac {a^3 \arctan (a x)}{2 c^2 \left (a^2 x^2+1\right )}-\frac {\arctan (a x)^2}{3 c^2 x^3}-\frac {a \arctan (a x)}{3 c^2 x^2} \]

[In]

Int[ArcTan[a*x]^2/(x^4*(c + a^2*c*x^2)^2),x]

[Out]

-1/3*a^2/(c^2*x) - (a^4*x)/(4*c^2*(1 + a^2*x^2)) - (7*a^3*ArcTan[a*x])/(12*c^2) - (a*ArcTan[a*x])/(3*c^2*x^2)
+ (a^3*ArcTan[a*x])/(2*c^2*(1 + a^2*x^2)) + (((7*I)/3)*a^3*ArcTan[a*x]^2)/c^2 - ArcTan[a*x]^2/(3*c^2*x^3) + (2
*a^2*ArcTan[a*x]^2)/(c^2*x) + (a^4*x*ArcTan[a*x]^2)/(2*c^2*(1 + a^2*x^2)) + (5*a^3*ArcTan[a*x]^3)/(6*c^2) - (1
4*a^3*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/(3*c^2) + (((7*I)/3)*a^3*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^2

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

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 211

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

Rule 331

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

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 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 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 5004

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

Rule 5012

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

Rule 5038

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

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 5050

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

Rule 5086

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

Rubi steps \begin{align*} \text {integral}& = -\left (a^2 \int \frac {\arctan (a x)^2}{x^2 \left (c+a^2 c x^2\right )^2} \, dx\right )+\frac {\int \frac {\arctan (a x)^2}{x^4 \left (c+a^2 c x^2\right )} \, dx}{c} \\ & = a^4 \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx+\frac {\int \frac {\arctan (a x)^2}{x^4} \, dx}{c^2}-2 \frac {a^2 \int \frac {\arctan (a x)^2}{x^2 \left (c+a^2 c x^2\right )} \, dx}{c} \\ & = -\frac {\arctan (a x)^2}{3 c^2 x^3}+\frac {a^4 x \arctan (a x)^2}{2 c^2 \left (1+a^2 x^2\right )}+\frac {a^3 \arctan (a x)^3}{6 c^2}-a^5 \int \frac {x \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx+\frac {(2 a) \int \frac {\arctan (a x)}{x^3 \left (1+a^2 x^2\right )} \, dx}{3 c^2}-2 \left (\frac {a^2 \int \frac {\arctan (a x)^2}{x^2} \, dx}{c^2}-\frac {a^4 \int \frac {\arctan (a x)^2}{c+a^2 c x^2} \, dx}{c}\right ) \\ & = \frac {a^3 \arctan (a x)}{2 c^2 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)^2}{3 c^2 x^3}+\frac {a^4 x \arctan (a x)^2}{2 c^2 \left (1+a^2 x^2\right )}+\frac {a^3 \arctan (a x)^3}{6 c^2}-\frac {1}{2} a^4 \int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx+\frac {(2 a) \int \frac {\arctan (a x)}{x^3} \, dx}{3 c^2}-\frac {\left (2 a^3\right ) \int \frac {\arctan (a x)}{x \left (1+a^2 x^2\right )} \, dx}{3 c^2}-2 \left (-\frac {a^2 \arctan (a x)^2}{c^2 x}-\frac {a^3 \arctan (a x)^3}{3 c^2}+\frac {\left (2 a^3\right ) \int \frac {\arctan (a x)}{x \left (1+a^2 x^2\right )} \, dx}{c^2}\right ) \\ & = -\frac {a^4 x}{4 c^2 \left (1+a^2 x^2\right )}-\frac {a \arctan (a x)}{3 c^2 x^2}+\frac {a^3 \arctan (a x)}{2 c^2 \left (1+a^2 x^2\right )}+\frac {i a^3 \arctan (a x)^2}{3 c^2}-\frac {\arctan (a x)^2}{3 c^2 x^3}+\frac {a^4 x \arctan (a x)^2}{2 c^2 \left (1+a^2 x^2\right )}+\frac {a^3 \arctan (a x)^3}{6 c^2}+\frac {a^2 \int \frac {1}{x^2 \left (1+a^2 x^2\right )} \, dx}{3 c^2}-\frac {\left (2 i a^3\right ) \int \frac {\arctan (a x)}{x (i+a x)} \, dx}{3 c^2}-2 \left (-\frac {i a^3 \arctan (a x)^2}{c^2}-\frac {a^2 \arctan (a x)^2}{c^2 x}-\frac {a^3 \arctan (a x)^3}{3 c^2}+\frac {\left (2 i a^3\right ) \int \frac {\arctan (a x)}{x (i+a x)} \, dx}{c^2}\right )-\frac {a^4 \int \frac {1}{c+a^2 c x^2} \, dx}{4 c} \\ & = -\frac {a^2}{3 c^2 x}-\frac {a^4 x}{4 c^2 \left (1+a^2 x^2\right )}-\frac {a^3 \arctan (a x)}{4 c^2}-\frac {a \arctan (a x)}{3 c^2 x^2}+\frac {a^3 \arctan (a x)}{2 c^2 \left (1+a^2 x^2\right )}+\frac {i a^3 \arctan (a x)^2}{3 c^2}-\frac {\arctan (a x)^2}{3 c^2 x^3}+\frac {a^4 x \arctan (a x)^2}{2 c^2 \left (1+a^2 x^2\right )}+\frac {a^3 \arctan (a x)^3}{6 c^2}-\frac {2 a^3 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{3 c^2}-\frac {a^4 \int \frac {1}{1+a^2 x^2} \, dx}{3 c^2}+\frac {\left (2 a^4\right ) \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{3 c^2}-2 \left (-\frac {i a^3 \arctan (a x)^2}{c^2}-\frac {a^2 \arctan (a x)^2}{c^2 x}-\frac {a^3 \arctan (a x)^3}{3 c^2}+\frac {2 a^3 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-\frac {\left (2 a^4\right ) \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^2}\right ) \\ & = -\frac {a^2}{3 c^2 x}-\frac {a^4 x}{4 c^2 \left (1+a^2 x^2\right )}-\frac {7 a^3 \arctan (a x)}{12 c^2}-\frac {a \arctan (a x)}{3 c^2 x^2}+\frac {a^3 \arctan (a x)}{2 c^2 \left (1+a^2 x^2\right )}+\frac {i a^3 \arctan (a x)^2}{3 c^2}-\frac {\arctan (a x)^2}{3 c^2 x^3}+\frac {a^4 x \arctan (a x)^2}{2 c^2 \left (1+a^2 x^2\right )}+\frac {a^3 \arctan (a x)^3}{6 c^2}-\frac {2 a^3 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{3 c^2}+\frac {i a^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{3 c^2}-2 \left (-\frac {i a^3 \arctan (a x)^2}{c^2}-\frac {a^2 \arctan (a x)^2}{c^2 x}-\frac {a^3 \arctan (a x)^3}{3 c^2}+\frac {2 a^3 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-\frac {i a^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{c^2}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.69 \[ \int \frac {\arctan (a x)^2}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=\frac {20 a^3 x^3 \arctan (a x)^3+2 a x \arctan (a x) \left (-4-4 a^2 x^2+3 a^2 x^2 \cos (2 \arctan (a x))-56 a^2 x^2 \log \left (1-e^{2 i \arctan (a x)}\right )\right )+56 i a^3 x^3 \operatorname {PolyLog}\left (2,e^{2 i \arctan (a x)}\right )-a^2 x^2 (8+3 a x \sin (2 \arctan (a x)))+\arctan (a x)^2 \left (-8+48 a^2 x^2+56 i a^3 x^3+6 a^3 x^3 \sin (2 \arctan (a x))\right )}{24 c^2 x^3} \]

[In]

Integrate[ArcTan[a*x]^2/(x^4*(c + a^2*c*x^2)^2),x]

[Out]

(20*a^3*x^3*ArcTan[a*x]^3 + 2*a*x*ArcTan[a*x]*(-4 - 4*a^2*x^2 + 3*a^2*x^2*Cos[2*ArcTan[a*x]] - 56*a^2*x^2*Log[
1 - E^((2*I)*ArcTan[a*x])]) + (56*I)*a^3*x^3*PolyLog[2, E^((2*I)*ArcTan[a*x])] - a^2*x^2*(8 + 3*a*x*Sin[2*ArcT
an[a*x]]) + ArcTan[a*x]^2*(-8 + 48*a^2*x^2 + (56*I)*a^3*x^3 + 6*a^3*x^3*Sin[2*ArcTan[a*x]]))/(24*c^2*x^3)

Maple [A] (verified)

Time = 1.16 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.45

method	result	size
derivativedivides	\(a^{3} \left (-\frac {\arctan \left (a x \right )^{2}}{3 c^{2} a^{3} x^{3}}+\frac {2 \arctan \left (a x \right )^{2}}{c^{2} a x}+\frac {a x \arctan \left (a x \right )^{2}}{2 c^{2} \left (a^{2} x^{2}+1\right )}+\frac {5 \arctan \left (a x \right )^{3}}{2 c^{2}}-\frac {\frac {\arctan \left (a x \right )}{a^{2} x^{2}}+14 \arctan \left (a x \right ) \ln \left (a x \right )-\frac {3 \arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )}-7 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )+7 i \ln \left (a x \right ) \ln \left (i a x +1\right )-7 i \ln \left (a x \right ) \ln \left (-i a x +1\right )+7 i \operatorname {dilog}\left (i a x +1\right )-7 i \operatorname {dilog}\left (-i a x +1\right )-\frac {7 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{2}+\frac {7 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{2}+\frac {1}{a x}+\frac {3 a x}{4 \left (a^{2} x^{2}+1\right )}+\frac {7 \arctan \left (a x \right )}{4}+5 \arctan \left (a x \right )^{3}}{3 c^{2}}\right )\)	\(351\)
default	\(a^{3} \left (-\frac {\arctan \left (a x \right )^{2}}{3 c^{2} a^{3} x^{3}}+\frac {2 \arctan \left (a x \right )^{2}}{c^{2} a x}+\frac {a x \arctan \left (a x \right )^{2}}{2 c^{2} \left (a^{2} x^{2}+1\right )}+\frac {5 \arctan \left (a x \right )^{3}}{2 c^{2}}-\frac {\frac {\arctan \left (a x \right )}{a^{2} x^{2}}+14 \arctan \left (a x \right ) \ln \left (a x \right )-\frac {3 \arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )}-7 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )+7 i \ln \left (a x \right ) \ln \left (i a x +1\right )-7 i \ln \left (a x \right ) \ln \left (-i a x +1\right )+7 i \operatorname {dilog}\left (i a x +1\right )-7 i \operatorname {dilog}\left (-i a x +1\right )-\frac {7 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{2}+\frac {7 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{2}+\frac {1}{a x}+\frac {3 a x}{4 \left (a^{2} x^{2}+1\right )}+\frac {7 \arctan \left (a x \right )}{4}+5 \arctan \left (a x \right )^{3}}{3 c^{2}}\right )\)	\(351\)
parts	\(\frac {a^{4} x \arctan \left (a x \right )^{2}}{2 c^{2} \left (a^{2} x^{2}+1\right )}+\frac {5 a^{3} \arctan \left (a x \right )^{3}}{2 c^{2}}-\frac {\arctan \left (a x \right )^{2}}{3 c^{2} x^{3}}+\frac {2 a^{2} \arctan \left (a x \right )^{2}}{c^{2} x}-\frac {2 \left (\frac {5 a^{3} \arctan \left (a x \right )^{3}}{2}+\frac {a^{3} \left (\frac {\arctan \left (a x \right )}{a^{2} x^{2}}+14 \arctan \left (a x \right ) \ln \left (a x \right )-\frac {3 \arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )}-7 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )+7 i \ln \left (a x \right ) \ln \left (i a x +1\right )-7 i \ln \left (a x \right ) \ln \left (-i a x +1\right )+7 i \operatorname {dilog}\left (i a x +1\right )-7 i \operatorname {dilog}\left (-i a x +1\right )-\frac {7 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{2}+\frac {7 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{2}+\frac {1}{a x}+\frac {3 a x}{4 \left (a^{2} x^{2}+1\right )}+\frac {7 \arctan \left (a x \right )}{4}\right )}{2}\right )}{3 c^{2}}\)	\(358\)

[In]

int(arctan(a*x)^2/x^4/(a^2*c*x^2+c)^2,x,method=_RETURNVERBOSE)

[Out]

a^3*(-1/3/c^2*arctan(a*x)^2/a^3/x^3+2/c^2*arctan(a*x)^2/a/x+1/2*a*x*arctan(a*x)^2/c^2/(a^2*x^2+1)+5/2*arctan(a
*x)^3/c^2-1/3/c^2*(arctan(a*x)/a^2/x^2+14*arctan(a*x)*ln(a*x)-3/2*arctan(a*x)/(a^2*x^2+1)-7*arctan(a*x)*ln(a^2
*x^2+1)+7*I*ln(a*x)*ln(1+I*a*x)-7*I*ln(a*x)*ln(1-I*a*x)+7*I*dilog(1+I*a*x)-7*I*dilog(1-I*a*x)-7/2*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)+7/2*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)+1/a/x+3/4*a*x/(a^2*x^2+1)+7/4*arctan(a*x
)+5*arctan(a*x)^3))

Fricas [F]

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

[In]

integrate(arctan(a*x)^2/x^4/(a^2*c*x^2+c)^2,x, algorithm="fricas")

[Out]

integral(arctan(a*x)^2/(a^4*c^2*x^8 + 2*a^2*c^2*x^6 + c^2*x^4), x)

Sympy [F]

\[ \int \frac {\arctan (a x)^2}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=\frac {\int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{a^{4} x^{8} + 2 a^{2} x^{6} + x^{4}}\, dx}{c^{2}} \]

[In]

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

[Out]

Integral(atan(a*x)**2/(a**4*x**8 + 2*a**2*x**6 + x**4), x)/c**2

Maxima [F(-1)]

Timed out. \[ \int \frac {\arctan (a x)^2}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=\text {Timed out} \]

[In]

integrate(arctan(a*x)^2/x^4/(a^2*c*x^2+c)^2,x, algorithm="maxima")

[Out]

Timed out

Giac [F]

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

[In]

integrate(arctan(a*x)^2/x^4/(a^2*c*x^2+c)^2,x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {\arctan (a x)^2}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^2}{x^4\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]

[In]

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

[Out]

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