\(\int \frac {\arctan (a x)}{x^3 (c+a^2 c x^2)^3} \, dx\) [198]

Optimal result

Integrand size = 20, antiderivative size = 205 \[ \int \frac {\arctan (a x)}{x^3 \left (c+a^2 c x^2\right )^3} \, dx=-\frac {a}{2 c^3 x}+\frac {a^3 x}{16 c^3 \left (1+a^2 x^2\right )^2}+\frac {19 a^3 x}{32 c^3 \left (1+a^2 x^2\right )}+\frac {3 a^2 \arctan (a x)}{32 c^3}-\frac {\arctan (a x)}{2 c^3 x^2}-\frac {a^2 \arctan (a x)}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {a^2 \arctan (a x)}{c^3 \left (1+a^2 x^2\right )}+\frac {3 i a^2 \arctan (a x)^2}{2 c^3}-\frac {3 a^2 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^3}+\frac {3 i a^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c^3} \]

[Out]

-1/2*a/c^3/x+1/16*a^3*x/c^3/(a^2*x^2+1)^2+19/32*a^3*x/c^3/(a^2*x^2+1)+3/32*a^2*arctan(a*x)/c^3-1/2*arctan(a*x)
/c^3/x^2-1/4*a^2*arctan(a*x)/c^3/(a^2*x^2+1)^2-a^2*arctan(a*x)/c^3/(a^2*x^2+1)+3/2*I*a^2*arctan(a*x)^2/c^3-3*a
^2*arctan(a*x)*ln(2-2/(1-I*a*x))/c^3+3/2*I*a^2*polylog(2,-1+2/(1-I*a*x))/c^3

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {5086, 5038, 4946, 331, 209, 5044, 4988, 2497, 5050, 205, 211} \[ \int \frac {\arctan (a x)}{x^3 \left (c+a^2 c x^2\right )^3} \, dx=-\frac {a^2 \arctan (a x)}{c^3 \left (a^2 x^2+1\right )}-\frac {a^2 \arctan (a x)}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {3 i a^2 \arctan (a x)^2}{2 c^3}+\frac {3 a^2 \arctan (a x)}{32 c^3}-\frac {3 a^2 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^3}+\frac {3 i a^2 \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 c^3}+\frac {19 a^3 x}{32 c^3 \left (a^2 x^2+1\right )}+\frac {a^3 x}{16 c^3 \left (a^2 x^2+1\right )^2}-\frac {\arctan (a x)}{2 c^3 x^2}-\frac {a}{2 c^3 x} \]

[In]

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

[Out]

-1/2*a/(c^3*x) + (a^3*x)/(16*c^3*(1 + a^2*x^2)^2) + (19*a^3*x)/(32*c^3*(1 + a^2*x^2)) + (3*a^2*ArcTan[a*x])/(3
2*c^3) - ArcTan[a*x]/(2*c^3*x^2) - (a^2*ArcTan[a*x])/(4*c^3*(1 + a^2*x^2)^2) - (a^2*ArcTan[a*x])/(c^3*(1 + a^2
*x^2)) + (((3*I)/2)*a^2*ArcTan[a*x]^2)/c^3 - (3*a^2*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/c^3 + (((3*I)/2)*a^2*P
olyLog[2, -1 + 2/(1 - I*a*x)])/c^3

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 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)}{x \left (c+a^2 c x^2\right )^3} \, dx\right )+\frac {\int \frac {\arctan (a x)}{x^3 \left (c+a^2 c x^2\right )^2} \, dx}{c} \\ & = a^4 \int \frac {x \arctan (a x)}{\left (c+a^2 c x^2\right )^3} \, dx+\frac {\int \frac {\arctan (a x)}{x^3 \left (c+a^2 c x^2\right )} \, dx}{c^2}-2 \frac {a^2 \int \frac {\arctan (a x)}{x \left (c+a^2 c x^2\right )^2} \, dx}{c} \\ & = -\frac {a^2 \arctan (a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {1}{4} a^3 \int \frac {1}{\left (c+a^2 c x^2\right )^3} \, dx+\frac {\int \frac {\arctan (a x)}{x^3} \, dx}{c^3}-\frac {a^2 \int \frac {\arctan (a x)}{x \left (c+a^2 c x^2\right )} \, dx}{c^2}-2 \left (\frac {a^2 \int \frac {\arctan (a x)}{x \left (c+a^2 c x^2\right )} \, dx}{c^2}-\frac {a^4 \int \frac {x \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{c}\right ) \\ & = \frac {a^3 x}{16 c^3 \left (1+a^2 x^2\right )^2}-\frac {\arctan (a x)}{2 c^3 x^2}-\frac {a^2 \arctan (a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {i a^2 \arctan (a x)^2}{2 c^3}+\frac {a \int \frac {1}{x^2 \left (1+a^2 x^2\right )} \, dx}{2 c^3}-\frac {\left (i a^2\right ) \int \frac {\arctan (a x)}{x (i+a x)} \, dx}{c^3}+\frac {\left (3 a^3\right ) \int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx}{16 c}-2 \left (\frac {a^2 \arctan (a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i a^2 \arctan (a x)^2}{2 c^3}+\frac {\left (i a^2\right ) \int \frac {\arctan (a x)}{x (i+a x)} \, dx}{c^3}-\frac {a^3 \int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx}{2 c}\right ) \\ & = -\frac {a}{2 c^3 x}+\frac {a^3 x}{16 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^3 x}{32 c^3 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)}{2 c^3 x^2}-\frac {a^2 \arctan (a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {i a^2 \arctan (a x)^2}{2 c^3}-\frac {a^2 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {a^3 \int \frac {1}{1+a^2 x^2} \, dx}{2 c^3}+\frac {a^3 \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^3}+\frac {\left (3 a^3\right ) \int \frac {1}{c+a^2 c x^2} \, dx}{32 c^2}-2 \left (-\frac {a^3 x}{4 c^3 \left (1+a^2 x^2\right )}+\frac {a^2 \arctan (a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i a^2 \arctan (a x)^2}{2 c^3}+\frac {a^2 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {a^3 \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^3}-\frac {a^3 \int \frac {1}{c+a^2 c x^2} \, dx}{4 c^2}\right ) \\ & = -\frac {a}{2 c^3 x}+\frac {a^3 x}{16 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^3 x}{32 c^3 \left (1+a^2 x^2\right )}-\frac {13 a^2 \arctan (a x)}{32 c^3}-\frac {\arctan (a x)}{2 c^3 x^2}-\frac {a^2 \arctan (a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {i a^2 \arctan (a x)^2}{2 c^3}-\frac {a^2 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^3}+\frac {i a^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c^3}-2 \left (-\frac {a^3 x}{4 c^3 \left (1+a^2 x^2\right )}-\frac {a^2 \arctan (a x)}{4 c^3}+\frac {a^2 \arctan (a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i a^2 \arctan (a x)^2}{2 c^3}+\frac {a^2 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {i a^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c^3}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.54 \[ \int \frac {\arctan (a x)}{x^3 \left (c+a^2 c x^2\right )^3} \, dx=\frac {a^2 \left (-\frac {64}{a x}+192 i \arctan (a x)^2+\arctan (a x) \left (-64-\frac {64}{a^2 x^2}-80 \cos (2 \arctan (a x))-4 \cos (4 \arctan (a x))-384 \log \left (1-e^{2 i \arctan (a x)}\right )\right )+192 i \operatorname {PolyLog}\left (2,e^{2 i \arctan (a x)}\right )+40 \sin (2 \arctan (a x))+\sin (4 \arctan (a x))\right )}{128 c^3} \]

[In]

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

[Out]

(a^2*(-64/(a*x) + (192*I)*ArcTan[a*x]^2 + ArcTan[a*x]*(-64 - 64/(a^2*x^2) - 80*Cos[2*ArcTan[a*x]] - 4*Cos[4*Ar
cTan[a*x]] - 384*Log[1 - E^((2*I)*ArcTan[a*x])]) + (192*I)*PolyLog[2, E^((2*I)*ArcTan[a*x])] + 40*Sin[2*ArcTan
[a*x]] + Sin[4*ArcTan[a*x]]))/(128*c^3)

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 1.06 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.42

method	result	size
parts	\(-\frac {a^{2} \arctan \left (a x \right )}{4 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {3 \arctan \left (a x \right ) a^{2} \ln \left (a^{2} x^{2}+1\right )}{2 c^{3}}-\frac {a^{2} \arctan \left (a x \right )}{c^{3} \left (a^{2} x^{2}+1\right )}-\frac {\arctan \left (a x \right )}{2 c^{3} x^{2}}-\frac {3 \arctan \left (a x \right ) a^{2} \ln \left (x \right )}{c^{3}}-\frac {a \left (-6 a^{2} \left (-\frac {i \ln \left (x \right ) \left (\ln \left (i a x +1\right )-\ln \left (-i a x +1\right )\right )}{2 a}-\frac {i \left (\operatorname {dilog}\left (i a x +1\right )-\operatorname {dilog}\left (-i a x +1\right )\right )}{2 a}\right )+\frac {a^{2} \left (\frac {-\frac {19}{8} a^{2} x^{3}-\frac {21}{8} x}{\left (a^{2} x^{2}+1\right )^{2}}-\frac {3 \arctan \left (a x \right )}{8 a}\right )}{2}+\frac {1}{x}+\frac {3 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (a^{2} \textit {\_Z}^{2}+1\right )}{\sum }\frac {2 \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (a^{2} x^{2}+1\right )-a^{2} \left (\frac {\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{a^{2} \underline {\hspace {1.25 ex}}\alpha }+2 \underline {\hspace {1.25 ex}}\alpha \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )+2 \underline {\hspace {1.25 ex}}\alpha \operatorname {dilog}\left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )\right )}{\underline {\hspace {1.25 ex}}\alpha }\right )}{4}\right )}{2 c^{3}}\)	\(292\)
derivativedivides	\(a^{2} \left (-\frac {\arctan \left (a x \right )}{2 c^{3} a^{2} x^{2}}-\frac {3 \arctan \left (a x \right ) \ln \left (a x \right )}{c^{3}}-\frac {\arctan \left (a x \right )}{c^{3} \left (a^{2} x^{2}+1\right )}-\frac {\arctan \left (a x \right )}{4 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {3 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{2 c^{3}}-\frac {6 i \ln \left (a x \right ) \ln \left (i a x +1\right )-6 i \ln \left (a x \right ) \ln \left (-i a x +1\right )+6 i \operatorname {dilog}\left (i a x +1\right )-6 i \operatorname {dilog}\left (-i a x +1\right )-3 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )+3 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )+\frac {2}{a x}+\frac {-\frac {19}{8} a^{3} x^{3}-\frac {21}{8} a x}{\left (a^{2} x^{2}+1\right )^{2}}-\frac {3 \arctan \left (a x \right )}{8}}{4 c^{3}}\right )\)	\(318\)
default	\(a^{2} \left (-\frac {\arctan \left (a x \right )}{2 c^{3} a^{2} x^{2}}-\frac {3 \arctan \left (a x \right ) \ln \left (a x \right )}{c^{3}}-\frac {\arctan \left (a x \right )}{c^{3} \left (a^{2} x^{2}+1\right )}-\frac {\arctan \left (a x \right )}{4 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {3 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{2 c^{3}}-\frac {6 i \ln \left (a x \right ) \ln \left (i a x +1\right )-6 i \ln \left (a x \right ) \ln \left (-i a x +1\right )+6 i \operatorname {dilog}\left (i a x +1\right )-6 i \operatorname {dilog}\left (-i a x +1\right )-3 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )+3 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )+\frac {2}{a x}+\frac {-\frac {19}{8} a^{3} x^{3}-\frac {21}{8} a x}{\left (a^{2} x^{2}+1\right )^{2}}-\frac {3 \arctan \left (a x \right )}{8}}{4 c^{3}}\right )\)	\(318\)
risch	\(-\frac {a}{2 c^{3} x}+\frac {19 a^{2} \arctan \left (a x \right )}{64 c^{3}}+\frac {i a^{2}}{64 c^{3} \left (i a x -1\right )}-\frac {i a^{2} \ln \left (i a x \right )}{4 c^{3}}+\frac {i a^{2} \ln \left (i a x +1\right )}{4 c^{3}}+\frac {i \ln \left (i a x +1\right )}{4 c^{3} x^{2}}+\frac {3 i a^{2} \operatorname {dilog}\left (\frac {1}{2}+\frac {i a x}{2}\right )}{4 c^{3}}+\frac {9 i a^{2}}{32 c^{3} \left (i a x +1\right )}+\frac {i a^{2}}{64 c^{3} \left (i a x +1\right )^{2}}-\frac {3 i a^{2} \operatorname {dilog}\left (i a x +1\right )}{2 c^{3}}-\frac {3 i a^{2} \ln \left (i a x +1\right )^{2}}{8 c^{3}}+\frac {3 i a^{2} \ln \left (-i a x +1\right )^{2}}{8 c^{3}}-\frac {i a^{2}}{64 c^{3} \left (-i a x -1\right )}+\frac {i a^{2} \ln \left (-i a x \right )}{4 c^{3}}-\frac {i a^{2} \ln \left (-i a x +1\right )}{4 c^{3}}-\frac {i \ln \left (-i a x +1\right )}{4 c^{3} x^{2}}-\frac {3 i a^{2} \operatorname {dilog}\left (\frac {1}{2}-\frac {i a x}{2}\right )}{4 c^{3}}-\frac {9 i a^{2}}{32 c^{3} \left (-i a x +1\right )}-\frac {i a^{2}}{64 c^{3} \left (-i a x +1\right )^{2}}+\frac {3 i a^{2} \operatorname {dilog}\left (-i a x +1\right )}{2 c^{3}}+\frac {9 a^{3} \ln \left (-i a x +1\right ) x}{64 c^{3} \left (-i a x -1\right )}-\frac {a^{3} \ln \left (-i a x +1\right ) x}{64 c^{3} \left (-i a x -1\right )^{2}}-\frac {9 i a^{2} \ln \left (-i a x +1\right )}{32 c^{3} \left (-i a x +1\right )}-\frac {i a^{2} \ln \left (-i a x +1\right )}{32 c^{3} \left (-i a x +1\right )^{2}}+\frac {9 i a^{2} \ln \left (-i a x +1\right )}{64 c^{3} \left (-i a x -1\right )}-\frac {3 i a^{2} \ln \left (-i a x +1\right )}{128 c^{3} \left (-i a x -1\right )^{2}}-\frac {i a^{4} \ln \left (-i a x +1\right ) x^{2}}{128 c^{3} \left (-i a x -1\right )^{2}}+\frac {i a^{4} \ln \left (i a x +1\right ) x^{2}}{128 c^{3} \left (i a x -1\right )^{2}}+\frac {3 i a^{2} \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (-i a x +1\right )}{4 c^{3}}+\frac {9 a^{3} \ln \left (i a x +1\right ) x}{64 c^{3} \left (i a x -1\right )}-\frac {a^{3} \ln \left (i a x +1\right ) x}{64 c^{3} \left (i a x -1\right )^{2}}+\frac {9 i a^{2} \ln \left (i a x +1\right )}{32 c^{3} \left (i a x +1\right )}+\frac {i a^{2} \ln \left (i a x +1\right )}{32 c^{3} \left (i a x +1\right )^{2}}-\frac {9 i a^{2} \ln \left (i a x +1\right )}{64 c^{3} \left (i a x -1\right )}+\frac {3 i a^{2} \ln \left (i a x +1\right )}{128 c^{3} \left (i a x -1\right )^{2}}-\frac {3 i a^{2} \ln \left (\frac {1}{2}-\frac {i a x}{2}\right ) \ln \left (i a x +1\right )}{4 c^{3}}\)	\(755\)

[In]

int(arctan(a*x)/x^3/(a^2*c*x^2+c)^3,x,method=_RETURNVERBOSE)

[Out]

-1/4*a^2*arctan(a*x)/c^3/(a^2*x^2+1)^2+3/2/c^3*arctan(a*x)*a^2*ln(a^2*x^2+1)-a^2*arctan(a*x)/c^3/(a^2*x^2+1)-1
/2*arctan(a*x)/c^3/x^2-3/c^3*arctan(a*x)*a^2*ln(x)-1/2/c^3*a*(-6*a^2*(-1/2*I*ln(x)*(ln(1+I*a*x)-ln(1-I*a*x))/a
-1/2*I*(dilog(1+I*a*x)-dilog(1-I*a*x))/a)+1/2*a^2*((-19/8*a^2*x^3-21/8*x)/(a^2*x^2+1)^2-3/8/a*arctan(a*x))+1/x
+3/4*sum(1/_alpha*(2*ln(x-_alpha)*ln(a^2*x^2+1)-a^2*(1/a^2/_alpha*ln(x-_alpha)^2+2*_alpha*ln(x-_alpha)*ln(1/2*
(x+_alpha)/_alpha)+2*_alpha*dilog(1/2*(x+_alpha)/_alpha))),_alpha=RootOf(_Z^2*a^2+1)))

Fricas [F]

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

[In]

integrate(arctan(a*x)/x^3/(a^2*c*x^2+c)^3,x, algorithm="fricas")

[Out]

integral(arctan(a*x)/(a^6*c^3*x^9 + 3*a^4*c^3*x^7 + 3*a^2*c^3*x^5 + c^3*x^3), x)

Sympy [F]

\[ \int \frac {\arctan (a x)}{x^3 \left (c+a^2 c x^2\right )^3} \, dx=\frac {\int \frac {\operatorname {atan}{\left (a x \right )}}{a^{6} x^{9} + 3 a^{4} x^{7} + 3 a^{2} x^{5} + x^{3}}\, dx}{c^{3}} \]

[In]

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

[Out]

Integral(atan(a*x)/(a**6*x**9 + 3*a**4*x**7 + 3*a**2*x**5 + x**3), x)/c**3

Maxima [F]

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

[In]

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

[Out]

integrate(arctan(a*x)/((a^2*c*x^2 + c)^3*x^3), x)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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