3.4.0 ∫ (c+a2cx2)2 arctan(ax)3 __________________________________

Integrand size = 22, antiderivative size = 284 \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^3}{x^2} \, dx=a^2 c^2 x \arctan (a x)-\frac {1}{2} a c^2 \arctan (a x)^2-\frac {1}{2} a^3 c^2 x^2 \arctan (a x)^2+\frac {2}{3} i a c^2 \arctan (a x)^3-\frac {c^2 \arctan (a x)^3}{x}+2 a^2 c^2 x \arctan (a x)^3+\frac {1}{3} a^4 c^2 x^3 \arctan (a x)^3+5 a c^2 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )-\frac {1}{2} a c^2 \log \left (1+a^2 x^2\right )+3 a c^2 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-3 i a c^2 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+5 i a c^2 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} a c^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )+\frac {5}{2} a c^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right ) \]

a^2*c^2*x*arctan(a*x)-1/2*a*c^2*arctan(a*x)^2-1/2*a^3*c^2*x^2*arctan(a*x)^2+2/3*I*a*c^2*arctan(a*x)^3-c^2*arct

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

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

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

Rubi [A] (veriﬁed)

Time = 0.57 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {5068, 4930, 5040, 4964, 5004, 5114, 6745, 4946, 5044, 4988, 5112, 5036, 266} \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^3}{x^2} \, dx=\frac {1}{3} a^4 c^2 x^3 \arctan (a x)^3-\frac {1}{2} a^3 c^2 x^2 \arctan (a x)^2+2 a^2 c^2 x \arctan (a x)^3+a^2 c^2 x \arctan (a x)-\frac {1}{2} a c^2 \log \left (a^2 x^2+1\right )-3 i a c^2 \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )+5 i a c^2 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )+\frac {2}{3} i a c^2 \arctan (a x)^3-\frac {1}{2} a c^2 \arctan (a x)^2-\frac {c^2 \arctan (a x)^3}{x}+5 a c^2 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )+3 a c^2 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )+\frac {3}{2} a c^2 \operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )+\frac {5}{2} a c^2 \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right ) \]

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

a^2*c^2*x*ArcTan[a*x] - (a*c^2*ArcTan[a*x]^2)/2 - (a^3*c^2*x^2*ArcTan[a*x]^2)/2 + ((2*I)/3)*a*c^2*ArcTan[a*x]^

3 - (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)/3 + 5*a*c^2*ArcTan[a*x]^2*

Log[2/(1 + I*a*x)] - (a*c^2*Log[1 + a^2*x^2])/2 + 3*a*c^2*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)] - (3*I)*a*c^2*A

rcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)] + (5*I)*a*c^2*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)] + (3*a*c^2*P

olyLog[3, -1 + 2/(1 - I*a*x)])/2 + (5*a*c^2*PolyLog[3, 1 - 2/(1 + I*a*x)])/2

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ

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

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]

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]

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]

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]

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]

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]

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]

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])

Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[I*(a + b*ArcTa

n[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

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

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

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

Mathematica [A] (veriﬁed)

Time = 0.32 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.87 \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^3}{x^2} \, dx=\frac {c^2 \left (-3 i a \pi ^3 x+24 a^2 x^2 \arctan (a x)-12 a x \arctan (a x)^2-12 a^3 x^3 \arctan (a x)^2-24 \arctan (a x)^3-16 i a x \arctan (a x)^3+48 a^2 x^2 \arctan (a x)^3+8 a^4 x^4 \arctan (a x)^3+72 a x \arctan (a x)^2 \log \left (1-e^{-2 i \arctan (a x)}\right )+120 a x \arctan (a x)^2 \log \left (1+e^{2 i \arctan (a x)}\right )-12 a x \log \left (1+a^2 x^2\right )+72 i a x \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )-120 i a x \arctan (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )+36 a x \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )+60 a x \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )\right )}{24 x} \]

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

(c^2*((-3*I)*a*Pi^3*x + 24*a^2*x^2*ArcTan[a*x] - 12*a*x*ArcTan[a*x]^2 - 12*a^3*x^3*ArcTan[a*x]^2 - 24*ArcTan[a

*x]^3 - (16*I)*a*x*ArcTan[a*x]^3 + 48*a^2*x^2*ArcTan[a*x]^3 + 8*a^4*x^4*ArcTan[a*x]^3 + 72*a*x*ArcTan[a*x]^2*L

og[1 - E^((-2*I)*ArcTan[a*x])] + 120*a*x*ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] - 12*a*x*Log[1 + a^2*x^2

] + (72*I)*a*x*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] - (120*I)*a*x*ArcTan[a*x]*PolyLog[2, -E^((2*I)*A

rcTan[a*x])] + 36*a*x*PolyLog[3, E^((-2*I)*ArcTan[a*x])] + 60*a*x*PolyLog[3, -E^((2*I)*ArcTan[a*x])]))/(24*x)

Maple [C] (warning: unable to verify)

Time = 76.25 (sec) , antiderivative size = 1793, normalized size of antiderivative = 6.31


method	result	size

derivativedivides	\(\text {Expression too large to display}\)	\(1793\)
default	\(\text {Expression too large to display}\)	\(1793\)
parts	\(\text {Expression too large to display}\)	\(1794\)

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

a*(1/3*c^2*arctan(a*x)^3*a^3*x^3+2*c^2*arctan(a*x)^3*a*x-c^2*arctan(a*x)^3/a/x-c^2*(1/2*x^2*arctan(a*x)^2*a^2-

3*arctan(a*x)^2*ln(a*x)+4*arctan(a*x)^2*ln(a^2*x^2+1)-8*arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))+1/6*I*ar

ctan(a*x)*(-9*arctan(a*x)*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1))*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(I*((1+I

*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))*Pi+9*arctan(a*x)*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1))*csgn(

I*((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^2*Pi-12*arctan(a*x)*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2

+1)+1)^2)^3+24*arctan(a*x)*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))-12*a

rctan(a*x)*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))^2-12*arctan(a*x)*Pi*cs

gn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)+12*arctan(

a*x)*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))*csgn(I/(

(1+I*a*x)^2/(a^2*x^2+1)+1)^2)+12*arctan(a*x)*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^

3-12*arctan(a*x)*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2*csgn(I*(1+I*a*x)^2/(a^2*x^

2+1))+12*arctan(a*x)*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^3-24*arctan(a*x)*Pi*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2)

)*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^2+12*arctan(a*x)*Pi*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))^2*csgn(I*(1+I*a*x)^2

/(a^2*x^2+1))+9*arctan(a*x)*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^

2/(a^2*x^2+1)+1))^2*Pi-9*arctan(a*x)*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^3*Pi+9*ar

ctan(a*x)*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(((1+I*a*x)^2/(a^2*x^2+1)-1)/((1

+I*a*x)^2/(a^2*x^2+1)+1))^2*Pi-9*arctan(a*x)*csgn(((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^3*P

i-9*arctan(a*x)*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(((1+I*a*x)^2/(a^2*x^2+1)-

1)/((1+I*a*x)^2/(a^2*x^2+1)+1))*Pi+9*arctan(a*x)*csgn(((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))

^2*Pi+16*arctan(a*x)^2-9*arctan(a*x)*Pi+6*I*a*x-3*I*arctan(a*x)+6+48*I*arctan(a*x)*ln(2))-ln((1+I*a*x)^2/(a^2*

x^2+1)+1)+3*arctan(a*x)^2*ln((1+I*a*x)^2/(a^2*x^2+1)-1)-3*arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)+6*I*

arctan(a*x)*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-6*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-3*arctan(a*x)^2*

ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*I*arctan(a*x)*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))-6*polylog(3,(1+I*a*x)

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

Fricas [F]

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

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

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

Sympy [F]

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

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

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

Maxima [F]

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

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

1/96*(4*(a^4*c^2*x^4 + 6*a^2*c^2*x^2 - 3*c^2)*arctan(a*x)^3 - 3*(a^4*c^2*x^4 + 6*a^2*c^2*x^2 - 3*c^2)*arctan(a

*x)*log(a^2*x^2 + 1)^2 + 3*(896*a^6*c^2*integrate(1/32*x^6*arctan(a*x)^3/(a^2*x^4 + x^2), x) + 96*a^6*c^2*inte

grate(1/32*x^6*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) + 128*a^6*c^2*integrate(1/32*x^6*arctan(a*x)

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

a^5*c^2*integrate(1/32*x^5*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) + 21*a*c^2*arctan(a*x)^4 + 2688*a^4*c^2*inte

grate(1/32*x^4*arctan(a*x)^3/(a^2*x^4 + x^2), x) + 288*a^4*c^2*integrate(1/32*x^4*arctan(a*x)*log(a^2*x^2 + 1)

^2/(a^2*x^4 + x^2), x) + 768*a^4*c^2*integrate(1/32*x^4*arctan(a*x)*log(a^2*x^2 + 1)/(a^2*x^4 + x^2), x) - 768

*a^3*c^2*integrate(1/32*x^3*arctan(a*x)^2/(a^2*x^4 + x^2), x) + a*c^2*log(a^2*x^2 + 1)^3 + 288*a^2*c^2*integra

te(1/32*x^2*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) - 384*a^2*c^2*integrate(1/32*x^2*arctan(a*x)*lo

g(a^2*x^2 + 1)/(a^2*x^4 + x^2), x) + 384*a*c^2*integrate(1/32*x*arctan(a*x)^2/(a^2*x^4 + x^2), x) - 96*a*c^2*i

ntegrate(1/32*x*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) + 896*c^2*integrate(1/32*arctan(a*x)^3/(a^2*x^4 + x^2),

x) + 96*c^2*integrate(1/32*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x))*x)/x

Giac [F(-1)]

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

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

Mupad [F(-1)]

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

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

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

\(\int \frac {(c+a^2 c x^2)^2 \arctan (a x)^3}{x^2} \, dx\) [376]

Optimal result