3.298 \(\int \frac {\tan ^{-1}(a x)^2}{x^4 (c+a^2 c x^2)^2} \, dx\)

Optimal. Leaf size=242 \[ \frac {7 i a^3 \text {Li}_2\left (\frac {2}{1-i a x}-1\right )}{3 c^2}+\frac {5 a^3 \tan ^{-1}(a x)^3}{6 c^2}+\frac {7 i a^3 \tan ^{-1}(a x)^2}{3 c^2}-\frac {7 a^3 \tan ^{-1}(a x)}{12 c^2}-\frac {14 a^3 \log \left (2-\frac {2}{1-i a x}\right ) \tan ^{-1}(a x)}{3 c^2}-\frac {a^2}{3 c^2 x}+\frac {2 a^2 \tan ^{-1}(a x)^2}{c^2 x}-\frac {a^4 x}{4 c^2 \left (a^2 x^2+1\right )}+\frac {a^4 x \tan ^{-1}(a x)^2}{2 c^2 \left (a^2 x^2+1\right )}+\frac {a^3 \tan ^{-1}(a x)}{2 c^2 \left (a^2 x^2+1\right )}-\frac {\tan ^{-1}(a x)^2}{3 c^2 x^3}-\frac {a \tan ^{-1}(a x)}{3 c^2 x^2} \]

[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

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Rubi [A]  time = 0.87, antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 13, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {4966, 4918, 4852, 325, 203, 4924, 4868, 2447, 4884, 4892, 4930, 199, 205} \[ \frac {7 i a^3 \text {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{3 c^2}-\frac {a^4 x}{4 c^2 \left (a^2 x^2+1\right )}+\frac {a^4 x \tan ^{-1}(a x)^2}{2 c^2 \left (a^2 x^2+1\right )}+\frac {a^3 \tan ^{-1}(a x)}{2 c^2 \left (a^2 x^2+1\right )}-\frac {a^2}{3 c^2 x}+\frac {5 a^3 \tan ^{-1}(a x)^3}{6 c^2}+\frac {7 i a^3 \tan ^{-1}(a x)^2}{3 c^2}-\frac {7 a^3 \tan ^{-1}(a x)}{12 c^2}+\frac {2 a^2 \tan ^{-1}(a x)^2}{c^2 x}-\frac {14 a^3 \log \left (2-\frac {2}{1-i a x}\right ) \tan ^{-1}(a x)}{3 c^2}-\frac {a \tan ^{-1}(a x)}{3 c^2 x^2}-\frac {\tan ^{-1}(a x)^2}{3 c^2 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-a^2/(3*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) - (14*
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 199

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] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 205

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

Rule 325

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 2447

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 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 4868

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 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 4892

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 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 4924

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> -Simp[(I*(a + b*ArcTan
[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 4930

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 4966

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*} \int \frac {\tan ^{-1}(a x)^2}{x^4 \left (c+a^2 c x^2\right )^2} \, dx &=-\left (a^2 \int \frac {\tan ^{-1}(a x)^2}{x^2 \left (c+a^2 c x^2\right )^2} \, dx\right )+\frac {\int \frac {\tan ^{-1}(a x)^2}{x^4 \left (c+a^2 c x^2\right )} \, dx}{c}\\ &=a^4 \int \frac {\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx+\frac {\int \frac {\tan ^{-1}(a x)^2}{x^4} \, dx}{c^2}-2 \frac {a^2 \int \frac {\tan ^{-1}(a x)^2}{x^2 \left (c+a^2 c x^2\right )} \, dx}{c}\\ &=-\frac {\tan ^{-1}(a x)^2}{3 c^2 x^3}+\frac {a^4 x \tan ^{-1}(a x)^2}{2 c^2 \left (1+a^2 x^2\right )}+\frac {a^3 \tan ^{-1}(a x)^3}{6 c^2}-a^5 \int \frac {x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx+\frac {(2 a) \int \frac {\tan ^{-1}(a x)}{x^3 \left (1+a^2 x^2\right )} \, dx}{3 c^2}-2 \left (\frac {a^2 \int \frac {\tan ^{-1}(a x)^2}{x^2} \, dx}{c^2}-\frac {a^4 \int \frac {\tan ^{-1}(a x)^2}{c+a^2 c x^2} \, dx}{c}\right )\\ &=\frac {a^3 \tan ^{-1}(a x)}{2 c^2 \left (1+a^2 x^2\right )}-\frac {\tan ^{-1}(a x)^2}{3 c^2 x^3}+\frac {a^4 x \tan ^{-1}(a x)^2}{2 c^2 \left (1+a^2 x^2\right )}+\frac {a^3 \tan ^{-1}(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 {\tan ^{-1}(a x)}{x^3} \, dx}{3 c^2}-\frac {\left (2 a^3\right ) \int \frac {\tan ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx}{3 c^2}-2 \left (-\frac {a^2 \tan ^{-1}(a x)^2}{c^2 x}-\frac {a^3 \tan ^{-1}(a x)^3}{3 c^2}+\frac {\left (2 a^3\right ) \int \frac {\tan ^{-1}(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 \tan ^{-1}(a x)}{3 c^2 x^2}+\frac {a^3 \tan ^{-1}(a x)}{2 c^2 \left (1+a^2 x^2\right )}+\frac {i a^3 \tan ^{-1}(a x)^2}{3 c^2}-\frac {\tan ^{-1}(a x)^2}{3 c^2 x^3}+\frac {a^4 x \tan ^{-1}(a x)^2}{2 c^2 \left (1+a^2 x^2\right )}+\frac {a^3 \tan ^{-1}(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 {\tan ^{-1}(a x)}{x (i+a x)} \, dx}{3 c^2}-2 \left (-\frac {i a^3 \tan ^{-1}(a x)^2}{c^2}-\frac {a^2 \tan ^{-1}(a x)^2}{c^2 x}-\frac {a^3 \tan ^{-1}(a x)^3}{3 c^2}+\frac {\left (2 i a^3\right ) \int \frac {\tan ^{-1}(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 \tan ^{-1}(a x)}{4 c^2}-\frac {a \tan ^{-1}(a x)}{3 c^2 x^2}+\frac {a^3 \tan ^{-1}(a x)}{2 c^2 \left (1+a^2 x^2\right )}+\frac {i a^3 \tan ^{-1}(a x)^2}{3 c^2}-\frac {\tan ^{-1}(a x)^2}{3 c^2 x^3}+\frac {a^4 x \tan ^{-1}(a x)^2}{2 c^2 \left (1+a^2 x^2\right )}+\frac {a^3 \tan ^{-1}(a x)^3}{6 c^2}-\frac {2 a^3 \tan ^{-1}(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 \tan ^{-1}(a x)^2}{c^2}-\frac {a^2 \tan ^{-1}(a x)^2}{c^2 x}-\frac {a^3 \tan ^{-1}(a x)^3}{3 c^2}+\frac {2 a^3 \tan ^{-1}(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 \tan ^{-1}(a x)}{12 c^2}-\frac {a \tan ^{-1}(a x)}{3 c^2 x^2}+\frac {a^3 \tan ^{-1}(a x)}{2 c^2 \left (1+a^2 x^2\right )}+\frac {i a^3 \tan ^{-1}(a x)^2}{3 c^2}-\frac {\tan ^{-1}(a x)^2}{3 c^2 x^3}+\frac {a^4 x \tan ^{-1}(a x)^2}{2 c^2 \left (1+a^2 x^2\right )}+\frac {a^3 \tan ^{-1}(a x)^3}{6 c^2}-\frac {2 a^3 \tan ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )}{3 c^2}+\frac {i a^3 \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{3 c^2}-2 \left (-\frac {i a^3 \tan ^{-1}(a x)^2}{c^2}-\frac {a^2 \tan ^{-1}(a x)^2}{c^2 x}-\frac {a^3 \tan ^{-1}(a x)^3}{3 c^2}+\frac {2 a^3 \tan ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-\frac {i a^3 \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{c^2}\right )\\ \end {align*}

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Mathematica [A]  time = 0.45, size = 166, normalized size = 0.69 \[ \frac {56 i a^3 x^3 \text {Li}_2\left (e^{2 i \tan ^{-1}(a x)}\right )+20 a^3 x^3 \tan ^{-1}(a x)^3-a^2 x^2 \left (3 a x \sin \left (2 \tan ^{-1}(a x)\right )+8\right )+2 a x \tan ^{-1}(a x) \left (-4 a^2 x^2-56 a^2 x^2 \log \left (1-e^{2 i \tan ^{-1}(a x)}\right )+3 a^2 x^2 \cos \left (2 \tan ^{-1}(a x)\right )-4\right )+\tan ^{-1}(a x)^2 \left (56 i a^3 x^3+6 a^3 x^3 \sin \left (2 \tan ^{-1}(a x)\right )+48 a^2 x^2-8\right )}{24 c^2 x^3} \]

Warning: Unable to verify antiderivative.

[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)

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fricas [F]  time = 1.05, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\arctan \left (a x\right )^{2}}{a^{4} c^{2} x^{8} + 2 \, a^{2} c^{2} x^{6} + c^{2} x^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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maple [B]  time = 0.17, size = 444, normalized size = 1.83 \[ -\frac {\arctan \left (a x \right )^{2}}{3 c^{2} x^{3}}+\frac {2 a^{2} \arctan \left (a x \right )^{2}}{c^{2} x}+\frac {a^{4} x \arctan \left (a x \right )^{2}}{2 c^{2} \left (a^{2} x^{2}+1\right )}+\frac {5 a^{3} \arctan \left (a x \right )^{3}}{6 c^{2}}-\frac {a \arctan \left (a x \right )}{3 c^{2} x^{2}}-\frac {14 a^{3} \arctan \left (a x \right ) \ln \left (a x \right )}{3 c^{2}}+\frac {7 a^{3} \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{3 c^{2}}+\frac {a^{3} \arctan \left (a x \right )}{2 c^{2} \left (a^{2} x^{2}+1\right )}-\frac {7 i a^{3} \dilog \left (i a x +1\right )}{3 c^{2}}+\frac {7 i a^{3} \dilog \left (-i a x +1\right )}{3 c^{2}}-\frac {7 i a^{3} \dilog \left (-\frac {i \left (a x +i\right )}{2}\right )}{6 c^{2}}+\frac {7 i a^{3} \dilog \left (\frac {i \left (a x -i\right )}{2}\right )}{6 c^{2}}-\frac {7 i a^{3} \ln \left (a x -i\right )^{2}}{12 c^{2}}+\frac {7 i a^{3} \ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )}{6 c^{2}}+\frac {7 i a^{3} \ln \left (a x +i\right )^{2}}{12 c^{2}}+\frac {7 i a^{3} \ln \left (a x \right ) \ln \left (-i a x +1\right )}{3 c^{2}}-\frac {7 i a^{3} \ln \left (a x \right ) \ln \left (i a x +1\right )}{3 c^{2}}+\frac {7 i a^{3} \ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )}{6 c^{2}}-\frac {7 i a^{3} \ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )}{6 c^{2}}-\frac {7 i a^{3} \ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )}{6 c^{2}}-\frac {a^{2}}{3 c^{2} x}-\frac {a^{4} x}{4 c^{2} \left (a^{2} x^{2}+1\right )}-\frac {7 a^{3} \arctan \left (a x \right )}{12 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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*arctan(a*x)^2/c^2/(a^2*x^2+1)+5/6*a^3*arctan(a*
x)^3/c^2-1/3*a*arctan(a*x)/c^2/x^2-14/3*a^3/c^2*arctan(a*x)*ln(a*x)+7/3*a^3/c^2*arctan(a*x)*ln(a^2*x^2+1)+1/2*
a^3*arctan(a*x)/c^2/(a^2*x^2+1)-7/12*I*a^3/c^2*ln(a*x-I)^2+7/3*I*a^3/c^2*dilog(1-I*a*x)-7/3*I*a^3/c^2*dilog(1+
I*a*x)+7/6*I*a^3/c^2*ln(a*x-I)*ln(a^2*x^2+1)+7/12*I*a^3/c^2*ln(I+a*x)^2-7/6*I*a^3/c^2*dilog(-1/2*I*(I+a*x))+7/
6*I*a^3/c^2*dilog(1/2*I*(a*x-I))+7/3*I*a^3/c^2*ln(a*x)*ln(1-I*a*x)-7/3*I*a^3/c^2*ln(a*x)*ln(1+I*a*x)+7/6*I*a^3
/c^2*ln(I+a*x)*ln(1/2*I*(a*x-I))-7/6*I*a^3/c^2*ln(a*x-I)*ln(-1/2*I*(I+a*x))-7/6*I*a^3/c^2*ln(I+a*x)*ln(a^2*x^2
+1)-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

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

[Out]

Timed out

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {atan}\left (a\,x\right )}^2}{x^4\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{a^{4} x^{8} + 2 a^{2} x^{6} + x^{4}}\, dx}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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