3.272 \(\int \frac {(c+a^2 c x^2)^2 \tan ^{-1}(a x)^2}{x^3} \, dx\)

Optimal. Leaf size=207 \[ \frac {1}{2} a^4 c^2 x^2 \tan ^{-1}(a x)^2-a^3 c^2 x \tan ^{-1}(a x)-a^2 c^2 \text {Li}_3\left (1-\frac {2}{i a x+1}\right )+a^2 c^2 \text {Li}_3\left (\frac {2}{i a x+1}-1\right )-2 i a^2 c^2 \text {Li}_2\left (1-\frac {2}{i a x+1}\right ) \tan ^{-1}(a x)+2 i a^2 c^2 \text {Li}_2\left (\frac {2}{i a x+1}-1\right ) \tan ^{-1}(a x)+a^2 c^2 \log (x)+4 a^2 c^2 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-\frac {c^2 \tan ^{-1}(a x)^2}{2 x^2}-\frac {a c^2 \tan ^{-1}(a x)}{x} \]

[Out]

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

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Rubi [A]  time = 0.45, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 15, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used = {4948, 4852, 4918, 266, 36, 29, 31, 4884, 4850, 4988, 4994, 6610, 4916, 4846, 260} \[ -a^2 c^2 \text {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+a^2 c^2 \text {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )-2 i a^2 c^2 \tan ^{-1}(a x) \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+2 i a^2 c^2 \tan ^{-1}(a x) \text {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )+\frac {1}{2} a^4 c^2 x^2 \tan ^{-1}(a x)^2+a^2 c^2 \log (x)-a^3 c^2 x \tan ^{-1}(a x)+4 a^2 c^2 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-\frac {c^2 \tan ^{-1}(a x)^2}{2 x^2}-\frac {a c^2 \tan ^{-1}(a x)}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 4846

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

Rule 4850

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 4852

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcTa
n[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcTan[c*x])^(p - 1))/(1 + c^
2*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 4884

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 4916

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 4918

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 4948

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 4988

Int[(ArcTanh[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/2, Int[(
Log[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 4994

Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(I*(a + b*Arc
Tan[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 6610

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*} \int \frac {\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2}{x^3} \, dx &=\int \left (\frac {c^2 \tan ^{-1}(a x)^2}{x^3}+\frac {2 a^2 c^2 \tan ^{-1}(a x)^2}{x}+a^4 c^2 x \tan ^{-1}(a x)^2\right ) \, dx\\ &=c^2 \int \frac {\tan ^{-1}(a x)^2}{x^3} \, dx+\left (2 a^2 c^2\right ) \int \frac {\tan ^{-1}(a x)^2}{x} \, dx+\left (a^4 c^2\right ) \int x \tan ^{-1}(a x)^2 \, dx\\ &=-\frac {c^2 \tan ^{-1}(a x)^2}{2 x^2}+\frac {1}{2} a^4 c^2 x^2 \tan ^{-1}(a x)^2+4 a^2 c^2 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )+\left (a c^2\right ) \int \frac {\tan ^{-1}(a x)}{x^2 \left (1+a^2 x^2\right )} \, dx-\left (8 a^3 c^2\right ) \int \frac {\tan ^{-1}(a x) \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (a^5 c^2\right ) \int \frac {x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-\frac {c^2 \tan ^{-1}(a x)^2}{2 x^2}+\frac {1}{2} a^4 c^2 x^2 \tan ^{-1}(a x)^2+4 a^2 c^2 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )+\left (a c^2\right ) \int \frac {\tan ^{-1}(a x)}{x^2} \, dx-\left (a^3 c^2\right ) \int \tan ^{-1}(a x) \, dx+\left (4 a^3 c^2\right ) \int \frac {\tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (4 a^3 c^2\right ) \int \frac {\tan ^{-1}(a x) \log \left (2-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac {a c^2 \tan ^{-1}(a x)}{x}-a^3 c^2 x \tan ^{-1}(a x)-\frac {c^2 \tan ^{-1}(a x)^2}{2 x^2}+\frac {1}{2} a^4 c^2 x^2 \tan ^{-1}(a x)^2+4 a^2 c^2 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-2 i a^2 c^2 \tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+2 i a^2 c^2 \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )+\left (a^2 c^2\right ) \int \frac {1}{x \left (1+a^2 x^2\right )} \, dx+\left (2 i a^3 c^2\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (2 i a^3 c^2\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\left (a^4 c^2\right ) \int \frac {x}{1+a^2 x^2} \, dx\\ &=-\frac {a c^2 \tan ^{-1}(a x)}{x}-a^3 c^2 x \tan ^{-1}(a x)-\frac {c^2 \tan ^{-1}(a x)^2}{2 x^2}+\frac {1}{2} a^4 c^2 x^2 \tan ^{-1}(a x)^2+4 a^2 c^2 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )+\frac {1}{2} a^2 c^2 \log \left (1+a^2 x^2\right )-2 i a^2 c^2 \tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+2 i a^2 c^2 \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )-a^2 c^2 \text {Li}_3\left (1-\frac {2}{1+i a x}\right )+a^2 c^2 \text {Li}_3\left (-1+\frac {2}{1+i a x}\right )+\frac {1}{2} \left (a^2 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac {a c^2 \tan ^{-1}(a x)}{x}-a^3 c^2 x \tan ^{-1}(a x)-\frac {c^2 \tan ^{-1}(a x)^2}{2 x^2}+\frac {1}{2} a^4 c^2 x^2 \tan ^{-1}(a x)^2+4 a^2 c^2 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )+\frac {1}{2} a^2 c^2 \log \left (1+a^2 x^2\right )-2 i a^2 c^2 \tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+2 i a^2 c^2 \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )-a^2 c^2 \text {Li}_3\left (1-\frac {2}{1+i a x}\right )+a^2 c^2 \text {Li}_3\left (-1+\frac {2}{1+i a x}\right )+\frac {1}{2} \left (a^2 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\frac {1}{2} \left (a^4 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{1+a^2 x} \, dx,x,x^2\right )\\ &=-\frac {a c^2 \tan ^{-1}(a x)}{x}-a^3 c^2 x \tan ^{-1}(a x)-\frac {c^2 \tan ^{-1}(a x)^2}{2 x^2}+\frac {1}{2} a^4 c^2 x^2 \tan ^{-1}(a x)^2+4 a^2 c^2 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )+a^2 c^2 \log (x)-2 i a^2 c^2 \tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+2 i a^2 c^2 \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )-a^2 c^2 \text {Li}_3\left (1-\frac {2}{1+i a x}\right )+a^2 c^2 \text {Li}_3\left (-1+\frac {2}{1+i a x}\right )\\ \end {align*}

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Mathematica [A]  time = 0.33, size = 226, normalized size = 1.09 \[ a^2 c^2 \left (\log \left (\frac {a x}{\sqrt {a^2 x^2+1}}\right )+\frac {1}{2} \log \left (a^2 x^2+1\right )+\frac {1}{2} a^2 x^2 \tan ^{-1}(a x)^2-\frac {\tan ^{-1}(a x)^2}{2 a^2 x^2}+2 i \tan ^{-1}(a x) \text {Li}_2\left (e^{-2 i \tan ^{-1}(a x)}\right )+2 i \tan ^{-1}(a x) \text {Li}_2\left (-e^{2 i \tan ^{-1}(a x)}\right )+\text {Li}_3\left (e^{-2 i \tan ^{-1}(a x)}\right )-\text {Li}_3\left (-e^{2 i \tan ^{-1}(a x)}\right )+\frac {4}{3} i \tan ^{-1}(a x)^3-a x \tan ^{-1}(a x)-\frac {\tan ^{-1}(a x)}{a x}+2 \tan ^{-1}(a x)^2 \log \left (1-e^{-2 i \tan ^{-1}(a x)}\right )-2 \tan ^{-1}(a x)^2 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-\frac {i \pi ^3}{12}\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

a^2*c^2*((-1/12*I)*Pi^3 - ArcTan[a*x]/(a*x) - a*x*ArcTan[a*x] - ArcTan[a*x]^2/(2*a^2*x^2) + (a^2*x^2*ArcTan[a*
x]^2)/2 + ((4*I)/3)*ArcTan[a*x]^3 + 2*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] - 2*ArcTan[a*x]^2*Log[1 +
E^((2*I)*ArcTan[a*x])] + Log[(a*x)/Sqrt[1 + a^2*x^2]] + Log[1 + a^2*x^2]/2 + (2*I)*ArcTan[a*x]*PolyLog[2, E^((
-2*I)*ArcTan[a*x])] + (2*I)*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] + PolyLog[3, E^((-2*I)*ArcTan[a*x])
] - PolyLog[3, -E^((2*I)*ArcTan[a*x])])

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fricas [F]  time = 1.03, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \arctan \left (a x\right )^{2}}{x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [C]  time = 5.53, size = 1255, normalized size = 6.06 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^4*c^2*x^2*arctan(a*x)^2-a*c^2*arctan(a*x)/x-a^3*c^2*x*arctan(a*x)-1/2*c^2*arctan(a*x)^2/x^2+I*a^2*c^2*ar
ctan(a*x)^2*Pi*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))+I*a^2*c^2*arctan(a*x)^2*Pi*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+I*a^2*c^2*arctan(a*x)^2*Pi*csgn(((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1
))^3-I*a^2*c^2*arctan(a*x)^2*Pi*csgn(((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^2+a^2*c^2*ln(1+(
1+I*a*x)/(a^2*x^2+1)^(1/2))-a^2*c^2*polylog(3,-(1+I*a*x)^2/(a^2*x^2+1))-a^2*c^2*ln((1+I*a*x)^2/(a^2*x^2+1)+1)+
a^2*c^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)-1)+4*a^2*c^2*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+4*a^2*c^2*polylog(
3,(1+I*a*x)/(a^2*x^2+1)^(1/2))+I*a^2*c^2*arctan(a*x)^2*Pi+2*I*a^2*c^2*arctan(a*x)*polylog(2,-(1+I*a*x)^2/(a^2*
x^2+1))-4*I*a^2*c^2*arctan(a*x)*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))-4*I*a^2*c^2*arctan(a*x)*polylog(2,-(1+I
*a*x)/(a^2*x^2+1)^(1/2))-2*a^2*c^2*arctan(a*x)^2*ln((1+I*a*x)^2/(a^2*x^2+1)-1)+2*a^2*c^2*arctan(a*x)^2*ln(1+(1
+I*a*x)/(a^2*x^2+1)^(1/2))+2*a^2*c^2*arctan(a*x)^2*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))+2*a^2*c^2*arctan(a*x)^2*l
n(a*x)+I*a^2*c^2*arctan(a*x)^2*Pi*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))-I*a^2*c^2*arctan(a*x)^2*Pi*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-I*a^2*c^2*arctan(a*x)^2*Pi*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-I*a^2*c^2*arctan(a*x)^2*Pi*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

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^2}{x^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ c^{2} \left (\int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{x^{3}}\, dx + \int \frac {2 a^{2} \operatorname {atan}^{2}{\left (a x \right )}}{x}\, dx + \int a^{4} x \operatorname {atan}^{2}{\left (a x \right )}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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