\(\int \frac {(c+a^2 c x^2)^2 \arctan (a x)^3}{x} \, dx\) [375]

Optimal result

Integrand size = 22, antiderivative size = 370 \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^3}{x} \, dx=-\frac {1}{4} a c^2 x+\frac {1}{4} c^2 \arctan (a x)+\frac {1}{4} a^2 c^2 x^2 \arctan (a x)-2 i c^2 \arctan (a x)^2-\frac {9}{4} a c^2 x \arctan (a x)^2-\frac {1}{4} a^3 c^2 x^3 \arctan (a x)^2+\frac {3}{4} c^2 \arctan (a x)^3+a^2 c^2 x^2 \arctan (a x)^3+\frac {1}{4} a^4 c^2 x^4 \arctan (a x)^3+2 c^2 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-4 c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )-2 i c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )-\frac {3}{2} i c^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {3}{2} c^2 \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {3}{2} c^2 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )+\frac {3}{4} i c^2 \operatorname {PolyLog}\left (4,1-\frac {2}{1+i a x}\right )-\frac {3}{4} i c^2 \operatorname {PolyLog}\left (4,-1+\frac {2}{1+i a x}\right ) \]

[Out]

-1/4*a*c^2*x+1/4*c^2*arctan(a*x)+1/4*a^2*c^2*x^2*arctan(a*x)-2*I*c^2*arctan(a*x)^2-9/4*a*c^2*x*arctan(a*x)^2-1
/4*a^3*c^2*x^3*arctan(a*x)^2+3/4*c^2*arctan(a*x)^3+a^2*c^2*x^2*arctan(a*x)^3+1/4*a^4*c^2*x^4*arctan(a*x)^3-2*c
^2*arctan(a*x)^3*arctanh(-1+2/(1+I*a*x))-4*c^2*arctan(a*x)*ln(2/(1+I*a*x))-2*I*c^2*polylog(2,1-2/(1+I*a*x))+3/
4*I*c^2*polylog(4,1-2/(1+I*a*x))-3/4*I*c^2*polylog(4,-1+2/(1+I*a*x))-3/2*c^2*arctan(a*x)*polylog(3,1-2/(1+I*a*
x))+3/2*c^2*arctan(a*x)*polylog(3,-1+2/(1+I*a*x))-3/2*I*c^2*arctan(a*x)^2*polylog(2,1-2/(1+I*a*x))+3/2*I*c^2*a
rctan(a*x)^2*polylog(2,-1+2/(1+I*a*x))

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.00, number of steps used = 36, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {5068, 4942, 5108, 5004, 5114, 5118, 6745, 4946, 5036, 4930, 5040, 4964, 2449, 2352, 327, 209} \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^3}{x} \, dx=\frac {1}{4} a^4 c^2 x^4 \arctan (a x)^3-\frac {1}{4} a^3 c^2 x^3 \arctan (a x)^2+a^2 c^2 x^2 \arctan (a x)^3+\frac {1}{4} a^2 c^2 x^2 \arctan (a x)+2 c^2 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-\frac {3}{2} i c^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )+\frac {3}{2} i c^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{i a x+1}-1\right )-\frac {3}{2} c^2 \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )+\frac {3}{2} c^2 \arctan (a x) \operatorname {PolyLog}\left (3,\frac {2}{i a x+1}-1\right )-\frac {9}{4} a c^2 x \arctan (a x)^2+\frac {3}{4} c^2 \arctan (a x)^3-2 i c^2 \arctan (a x)^2+\frac {1}{4} c^2 \arctan (a x)-4 c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )-2 i c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )+\frac {3}{4} i c^2 \operatorname {PolyLog}\left (4,1-\frac {2}{i a x+1}\right )-\frac {3}{4} i c^2 \operatorname {PolyLog}\left (4,\frac {2}{i a x+1}-1\right )-\frac {1}{4} a c^2 x \]

[In]

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

[Out]

-1/4*(a*c^2*x) + (c^2*ArcTan[a*x])/4 + (a^2*c^2*x^2*ArcTan[a*x])/4 - (2*I)*c^2*ArcTan[a*x]^2 - (9*a*c^2*x*ArcT
an[a*x]^2)/4 - (a^3*c^2*x^3*ArcTan[a*x]^2)/4 + (3*c^2*ArcTan[a*x]^3)/4 + a^2*c^2*x^2*ArcTan[a*x]^3 + (a^4*c^2*
x^4*ArcTan[a*x]^3)/4 + 2*c^2*ArcTan[a*x]^3*ArcTanh[1 - 2/(1 + I*a*x)] - 4*c^2*ArcTan[a*x]*Log[2/(1 + I*a*x)] -
(2*I)*c^2*PolyLog[2, 1 - 2/(1 + I*a*x)] - ((3*I)/2)*c^2*ArcTan[a*x]^2*PolyLog[2, 1 - 2/(1 + I*a*x)] + ((3*I)/
2)*c^2*ArcTan[a*x]^2*PolyLog[2, -1 + 2/(1 + I*a*x)] - (3*c^2*ArcTan[a*x]*PolyLog[3, 1 - 2/(1 + I*a*x)])/2 + (3
*c^2*ArcTan[a*x]*PolyLog[3, -1 + 2/(1 + I*a*x)])/2 + ((3*I)/4)*c^2*PolyLog[4, 1 - 2/(1 + I*a*x)] - ((3*I)/4)*c
^2*PolyLog[4, -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 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 4942

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

Rule 5108

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

Rule 5114

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

Rule 5118

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

Rule 6745

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c^2 \arctan (a x)^3}{x}+2 a^2 c^2 x \arctan (a x)^3+a^4 c^2 x^3 \arctan (a x)^3\right ) \, dx \\ & = c^2 \int \frac {\arctan (a x)^3}{x} \, dx+\left (2 a^2 c^2\right ) \int x \arctan (a x)^3 \, dx+\left (a^4 c^2\right ) \int x^3 \arctan (a x)^3 \, dx \\ & = a^2 c^2 x^2 \arctan (a x)^3+\frac {1}{4} a^4 c^2 x^4 \arctan (a x)^3+2 c^2 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-\left (6 a c^2\right ) \int \frac {\arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (3 a^3 c^2\right ) \int \frac {x^2 \arctan (a x)^2}{1+a^2 x^2} \, dx-\frac {1}{4} \left (3 a^5 c^2\right ) \int \frac {x^4 \arctan (a x)^2}{1+a^2 x^2} \, dx \\ & = a^2 c^2 x^2 \arctan (a x)^3+\frac {1}{4} a^4 c^2 x^4 \arctan (a x)^3+2 c^2 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-\left (3 a c^2\right ) \int \arctan (a x)^2 \, dx+\left (3 a c^2\right ) \int \frac {\arctan (a x)^2}{1+a^2 x^2} \, dx+\left (3 a c^2\right ) \int \frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (3 a c^2\right ) \int \frac {\arctan (a x)^2 \log \left (2-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac {1}{4} \left (3 a^3 c^2\right ) \int x^2 \arctan (a x)^2 \, dx+\frac {1}{4} \left (3 a^3 c^2\right ) \int \frac {x^2 \arctan (a x)^2}{1+a^2 x^2} \, dx \\ & = -3 a c^2 x \arctan (a x)^2-\frac {1}{4} a^3 c^2 x^3 \arctan (a x)^2+c^2 \arctan (a x)^3+a^2 c^2 x^2 \arctan (a x)^3+\frac {1}{4} a^4 c^2 x^4 \arctan (a x)^3+2 c^2 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-\frac {3}{2} i c^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )+\left (3 i a c^2\right ) \int \frac {\arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (3 i a c^2\right ) \int \frac {\arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\frac {1}{4} \left (3 a c^2\right ) \int \arctan (a x)^2 \, dx-\frac {1}{4} \left (3 a c^2\right ) \int \frac {\arctan (a x)^2}{1+a^2 x^2} \, dx+\left (6 a^2 c^2\right ) \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx+\frac {1}{2} \left (a^4 c^2\right ) \int \frac {x^3 \arctan (a x)}{1+a^2 x^2} \, dx \\ & = -3 i c^2 \arctan (a x)^2-\frac {9}{4} a c^2 x \arctan (a x)^2-\frac {1}{4} a^3 c^2 x^3 \arctan (a x)^2+\frac {3}{4} c^2 \arctan (a x)^3+a^2 c^2 x^2 \arctan (a x)^3+\frac {1}{4} a^4 c^2 x^4 \arctan (a x)^3+2 c^2 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-\frac {3}{2} i c^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {3}{2} c^2 \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {3}{2} c^2 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )+\frac {1}{2} \left (3 a c^2\right ) \int \frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac {1}{2} \left (3 a c^2\right ) \int \frac {\operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (6 a c^2\right ) \int \frac {\arctan (a x)}{i-a x} \, dx+\frac {1}{2} \left (a^2 c^2\right ) \int x \arctan (a x) \, dx-\frac {1}{2} \left (a^2 c^2\right ) \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{2} \left (3 a^2 c^2\right ) \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx \\ & = \frac {1}{4} a^2 c^2 x^2 \arctan (a x)-2 i c^2 \arctan (a x)^2-\frac {9}{4} a c^2 x \arctan (a x)^2-\frac {1}{4} a^3 c^2 x^3 \arctan (a x)^2+\frac {3}{4} c^2 \arctan (a x)^3+a^2 c^2 x^2 \arctan (a x)^3+\frac {1}{4} a^4 c^2 x^4 \arctan (a x)^3+2 c^2 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-6 c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )-\frac {3}{2} i c^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {3}{2} c^2 \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {3}{2} c^2 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )+\frac {3}{4} i c^2 \operatorname {PolyLog}\left (4,1-\frac {2}{1+i a x}\right )-\frac {3}{4} i c^2 \operatorname {PolyLog}\left (4,-1+\frac {2}{1+i a x}\right )+\frac {1}{2} \left (a c^2\right ) \int \frac {\arctan (a x)}{i-a x} \, dx+\frac {1}{2} \left (3 a c^2\right ) \int \frac {\arctan (a x)}{i-a x} \, dx+\left (6 a c^2\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac {1}{4} \left (a^3 c^2\right ) \int \frac {x^2}{1+a^2 x^2} \, dx \\ & = -\frac {1}{4} a c^2 x+\frac {1}{4} a^2 c^2 x^2 \arctan (a x)-2 i c^2 \arctan (a x)^2-\frac {9}{4} a c^2 x \arctan (a x)^2-\frac {1}{4} a^3 c^2 x^3 \arctan (a x)^2+\frac {3}{4} c^2 \arctan (a x)^3+a^2 c^2 x^2 \arctan (a x)^3+\frac {1}{4} a^4 c^2 x^4 \arctan (a x)^3+2 c^2 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-4 c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )-\frac {3}{2} i c^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {3}{2} c^2 \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {3}{2} c^2 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )+\frac {3}{4} i c^2 \operatorname {PolyLog}\left (4,1-\frac {2}{1+i a x}\right )-\frac {3}{4} i c^2 \operatorname {PolyLog}\left (4,-1+\frac {2}{1+i a x}\right )-\left (6 i 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}{4} \left (a c^2\right ) \int \frac {1}{1+a^2 x^2} \, dx-\frac {1}{2} \left (a c^2\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac {1}{2} \left (3 a c^2\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx \\ & = -\frac {1}{4} a c^2 x+\frac {1}{4} c^2 \arctan (a x)+\frac {1}{4} a^2 c^2 x^2 \arctan (a x)-2 i c^2 \arctan (a x)^2-\frac {9}{4} a c^2 x \arctan (a x)^2-\frac {1}{4} a^3 c^2 x^3 \arctan (a x)^2+\frac {3}{4} c^2 \arctan (a x)^3+a^2 c^2 x^2 \arctan (a x)^3+\frac {1}{4} a^4 c^2 x^4 \arctan (a x)^3+2 c^2 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-4 c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )-3 i c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )-\frac {3}{2} i c^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {3}{2} c^2 \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {3}{2} c^2 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )+\frac {3}{4} i c^2 \operatorname {PolyLog}\left (4,1-\frac {2}{1+i a x}\right )-\frac {3}{4} i c^2 \operatorname {PolyLog}\left (4,-1+\frac {2}{1+i a x}\right )+\frac {1}{2} \left (i 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}{2} \left (3 i 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}{4} a c^2 x+\frac {1}{4} c^2 \arctan (a x)+\frac {1}{4} a^2 c^2 x^2 \arctan (a x)-2 i c^2 \arctan (a x)^2-\frac {9}{4} a c^2 x \arctan (a x)^2-\frac {1}{4} a^3 c^2 x^3 \arctan (a x)^2+\frac {3}{4} c^2 \arctan (a x)^3+a^2 c^2 x^2 \arctan (a x)^3+\frac {1}{4} a^4 c^2 x^4 \arctan (a x)^3+2 c^2 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-4 c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )-2 i c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )-\frac {3}{2} i c^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {3}{2} c^2 \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {3}{2} c^2 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )+\frac {3}{4} i c^2 \operatorname {PolyLog}\left (4,1-\frac {2}{1+i a x}\right )-\frac {3}{4} i c^2 \operatorname {PolyLog}\left (4,-1+\frac {2}{1+i a x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.82 \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^3}{x} \, dx=\frac {1}{64} c^2 \left (-i \pi ^4-16 a x+16 \arctan (a x)+16 a^2 x^2 \arctan (a x)+128 i \arctan (a x)^2-144 a x \arctan (a x)^2-16 a^3 x^3 \arctan (a x)^2+48 \arctan (a x)^3+64 a^2 x^2 \arctan (a x)^3+16 a^4 x^4 \arctan (a x)^3+32 i \arctan (a x)^4+64 \arctan (a x)^3 \log \left (1-e^{-2 i \arctan (a x)}\right )-256 \arctan (a x) \log \left (1+e^{2 i \arctan (a x)}\right )-64 \arctan (a x)^3 \log \left (1+e^{2 i \arctan (a x)}\right )+96 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )+32 i \left (4+3 \arctan (a x)^2\right ) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )+96 \arctan (a x) \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )-96 \arctan (a x) \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )-48 i \operatorname {PolyLog}\left (4,e^{-2 i \arctan (a x)}\right )-48 i \operatorname {PolyLog}\left (4,-e^{2 i \arctan (a x)}\right )\right ) \]

[In]

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

[Out]

(c^2*((-I)*Pi^4 - 16*a*x + 16*ArcTan[a*x] + 16*a^2*x^2*ArcTan[a*x] + (128*I)*ArcTan[a*x]^2 - 144*a*x*ArcTan[a*
x]^2 - 16*a^3*x^3*ArcTan[a*x]^2 + 48*ArcTan[a*x]^3 + 64*a^2*x^2*ArcTan[a*x]^3 + 16*a^4*x^4*ArcTan[a*x]^3 + (32
*I)*ArcTan[a*x]^4 + 64*ArcTan[a*x]^3*Log[1 - E^((-2*I)*ArcTan[a*x])] - 256*ArcTan[a*x]*Log[1 + E^((2*I)*ArcTan
[a*x])] - 64*ArcTan[a*x]^3*Log[1 + E^((2*I)*ArcTan[a*x])] + (96*I)*ArcTan[a*x]^2*PolyLog[2, E^((-2*I)*ArcTan[a
*x])] + (32*I)*(4 + 3*ArcTan[a*x]^2)*PolyLog[2, -E^((2*I)*ArcTan[a*x])] + 96*ArcTan[a*x]*PolyLog[3, E^((-2*I)*
ArcTan[a*x])] - 96*ArcTan[a*x]*PolyLog[3, -E^((2*I)*ArcTan[a*x])] - (48*I)*PolyLog[4, E^((-2*I)*ArcTan[a*x])]
- (48*I)*PolyLog[4, -E^((2*I)*ArcTan[a*x])]))/64

Maple [A] (verified)

Time = 24.32 (sec) , antiderivative size = 566, normalized size of antiderivative = 1.53

method	result	size
derivativedivides	\(\frac {c^{2} \left (-3 i \arctan \left (a x \right )^{3}+3 \arctan \left (a x \right )^{3} a x -i \arctan \left (a x \right )^{3} a^{2} x^{2}+\arctan \left (a x \right )^{3} a^{3} x^{3}-8 \arctan \left (a x \right )^{2}+i \arctan \left (a x \right )^{2} a x -x^{2} \arctan \left (a x \right )^{2} a^{2}-i \arctan \left (a x \right )+x \arctan \left (a x \right ) a -1\right ) \left (a x +i\right )}{4}+c^{2} \arctan \left (a x \right )^{3} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )+2 i c^{2} \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )+6 c^{2} \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 i c^{2} \operatorname {polylog}\left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-c^{2} \arctan \left (a x \right )^{3} \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )+4 i c^{2} \arctan \left (a x \right )^{2}-\frac {3 c^{2} \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}-3 i c^{2} \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+c^{2} \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 i c^{2} \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 c^{2} \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {3 i c^{2} \operatorname {polylog}\left (4, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{4}+\frac {3 i c^{2} \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}-4 c^{2} \arctan \left (a x \right ) \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )+6 i c^{2} \operatorname {polylog}\left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\)	\(566\)
default	\(\frac {c^{2} \left (-3 i \arctan \left (a x \right )^{3}+3 \arctan \left (a x \right )^{3} a x -i \arctan \left (a x \right )^{3} a^{2} x^{2}+\arctan \left (a x \right )^{3} a^{3} x^{3}-8 \arctan \left (a x \right )^{2}+i \arctan \left (a x \right )^{2} a x -x^{2} \arctan \left (a x \right )^{2} a^{2}-i \arctan \left (a x \right )+x \arctan \left (a x \right ) a -1\right ) \left (a x +i\right )}{4}+c^{2} \arctan \left (a x \right )^{3} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )+2 i c^{2} \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )+6 c^{2} \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 i c^{2} \operatorname {polylog}\left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-c^{2} \arctan \left (a x \right )^{3} \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )+4 i c^{2} \arctan \left (a x \right )^{2}-\frac {3 c^{2} \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}-3 i c^{2} \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+c^{2} \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 i c^{2} \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 c^{2} \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {3 i c^{2} \operatorname {polylog}\left (4, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{4}+\frac {3 i c^{2} \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}-4 c^{2} \arctan \left (a x \right ) \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )+6 i c^{2} \operatorname {polylog}\left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\)	\(566\)

[In]

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

[Out]

1/4*c^2*(-3*I*arctan(a*x)^3+3*arctan(a*x)^3*a*x-I*arctan(a*x)^3*a^2*x^2+arctan(a*x)^3*a^3*x^3-8*arctan(a*x)^2+
I*arctan(a*x)^2*a*x-x^2*arctan(a*x)^2*a^2-I*arctan(a*x)+x*arctan(a*x)*a-1)*(I+a*x)+c^2*arctan(a*x)^3*ln((1+I*a
*x)/(a^2*x^2+1)^(1/2)+1)+2*I*c^2*polylog(2,-(1+I*a*x)^2/(a^2*x^2+1))+6*c^2*arctan(a*x)*polylog(3,-(1+I*a*x)/(a
^2*x^2+1)^(1/2))+6*I*c^2*polylog(4,(1+I*a*x)/(a^2*x^2+1)^(1/2))-c^2*arctan(a*x)^3*ln((1+I*a*x)^2/(a^2*x^2+1)+1
)+4*I*c^2*arctan(a*x)^2-3/2*c^2*arctan(a*x)*polylog(3,-(1+I*a*x)^2/(a^2*x^2+1))-3*I*c^2*arctan(a*x)^2*polylog(
2,(1+I*a*x)/(a^2*x^2+1)^(1/2))+c^2*arctan(a*x)^3*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))-3*I*c^2*arctan(a*x)^2*polyl
og(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*c^2*arctan(a*x)*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))-3/4*I*c^2*polylog(
4,-(1+I*a*x)^2/(a^2*x^2+1))+3/2*I*c^2*arctan(a*x)^2*polylog(2,-(1+I*a*x)^2/(a^2*x^2+1))-4*c^2*arctan(a*x)*ln((
1+I*a*x)^2/(a^2*x^2+1)+1)+6*I*c^2*polylog(4,-(1+I*a*x)/(a^2*x^2+1)^(1/2))

Fricas [F]

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

[In]

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

[Out]

integral((a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2)*arctan(a*x)^3/x, x)

Sympy [F]

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

[In]

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

[Out]

c**2*(Integral(atan(a*x)**3/x, x) + Integral(2*a**2*x*atan(a*x)**3, x) + Integral(a**4*x**3*atan(a*x)**3, x))

Maxima [F]

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

[In]

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

[Out]

1/32*(a^4*c^2*x^4 + 4*a^2*c^2*x^2)*arctan(a*x)^3 - 3/128*(a^4*c^2*x^4 + 4*a^2*c^2*x^2)*arctan(a*x)*log(a^2*x^2
+ 1)^2 + integrate(1/128*(112*(a^6*c^2*x^6 + 3*a^4*c^2*x^4 + 3*a^2*c^2*x^2 + c^2)*arctan(a*x)^3 - 12*(a^5*c^2
*x^5 + 4*a^3*c^2*x^3)*arctan(a*x)^2 + 12*(a^6*c^2*x^6 + 4*a^4*c^2*x^4)*arctan(a*x)*log(a^2*x^2 + 1) + 3*(a^5*c
^2*x^5 + 4*a^3*c^2*x^3 + 4*(a^6*c^2*x^6 + 3*a^4*c^2*x^4 + 3*a^2*c^2*x^2 + c^2)*arctan(a*x))*log(a^2*x^2 + 1)^2
)/(a^2*x^3 + x), x)

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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