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

Optimal. Leaf size=1360 \[ \frac {1}{3} (-1)^{3/4} c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2 b^2-\frac {1}{3} \sqrt [4]{-1} c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2 b^2+\frac {\log ^2\left (i c x^2+1\right ) b^2}{12 x^3}-\frac {4}{3} \sqrt [4]{-1} c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) b^2-\frac {4}{3} \sqrt [4]{-1} c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) b^2+\frac {2}{3} \sqrt [4]{-1} c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right ) b^2-\frac {2}{3} \sqrt [4]{-1} c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{\sqrt [4]{-1} \sqrt {c} x+1}\right ) b^2+\frac {1}{3} \sqrt [4]{-1} c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {\sqrt {2} \left (\sqrt {c} x+\sqrt [4]{-1}\right )}{\sqrt [4]{-1} \sqrt {c} x+1}\right ) b^2+\frac {2}{3} \sqrt [4]{-1} c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right ) b^2-\frac {2}{3} \sqrt [4]{-1} c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{(-1)^{3/4} \sqrt {c} x+1}\right ) b^2+\frac {1}{3} \sqrt [4]{-1} c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (-\frac {\sqrt {2} \left (\sqrt {c} x+(-1)^{3/4}\right )}{(-1)^{3/4} \sqrt {c} x+1}\right ) b^2+\frac {1}{3} \sqrt [4]{-1} c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {(1+i) \left (\sqrt [4]{-1} \sqrt {c} x+1\right )}{(-1)^{3/4} \sqrt {c} x+1}\right ) b^2+\frac {1}{3} \sqrt [4]{-1} c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {(1-i) \left ((-1)^{3/4} \sqrt {c} x+1\right )}{\sqrt [4]{-1} \sqrt {c} x+1}\right ) b^2-\frac {1}{3} \sqrt [4]{-1} c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right ) b^2-\frac {i c \log \left (1-i c x^2\right ) b^2}{3 x}-\frac {1}{3} \sqrt [4]{-1} c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (i c x^2+1\right ) b^2+\frac {1}{3} \sqrt [4]{-1} c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (i c x^2+1\right ) b^2-\frac {\log \left (1-i c x^2\right ) \log \left (i c x^2+1\right ) b^2}{6 x^3}+\frac {2 i c \log \left (i c x^2+1\right ) b^2}{3 x}+\frac {1}{3} (-1)^{3/4} c^{3/2} \text {Li}_2\left (1-\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right ) b^2+\frac {1}{3} (-1)^{3/4} c^{3/2} \text {Li}_2\left (1-\frac {2}{\sqrt [4]{-1} \sqrt {c} x+1}\right ) b^2-\frac {1}{6} (-1)^{3/4} c^{3/2} \text {Li}_2\left (1-\frac {\sqrt {2} \left (\sqrt {c} x+\sqrt [4]{-1}\right )}{\sqrt [4]{-1} \sqrt {c} x+1}\right ) b^2+\frac {1}{3} \sqrt [4]{-1} c^{3/2} \text {Li}_2\left (1-\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right ) b^2+\frac {1}{3} \sqrt [4]{-1} c^{3/2} \text {Li}_2\left (1-\frac {2}{(-1)^{3/4} \sqrt {c} x+1}\right ) b^2-\frac {1}{6} \sqrt [4]{-1} c^{3/2} \text {Li}_2\left (\frac {\sqrt {2} \left (\sqrt {c} x+(-1)^{3/4}\right )}{(-1)^{3/4} \sqrt {c} x+1}+1\right ) b^2-\frac {1}{6} \sqrt [4]{-1} c^{3/2} \text {Li}_2\left (1-\frac {(1+i) \left (\sqrt [4]{-1} \sqrt {c} x+1\right )}{(-1)^{3/4} \sqrt {c} x+1}\right ) b^2-\frac {1}{6} (-1)^{3/4} c^{3/2} \text {Li}_2\left (1-\frac {(1-i) \left ((-1)^{3/4} \sqrt {c} x+1\right )}{\sqrt [4]{-1} \sqrt {c} x+1}\right ) b^2+\frac {2}{3} (-1)^{3/4} a c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) b-\frac {1}{3} (-1)^{3/4} c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right ) b-\frac {c \left (2 a+i b \log \left (1-i c x^2\right )\right ) b}{3 x}+\frac {i a \log \left (i c x^2+1\right ) b}{3 x^3}-\frac {2 a c b}{3 x}-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{12 x^3} \]

[Out]

-1/12*(2*a+I*b*ln(1-I*c*x^2))^2/x^3-2/3*a*b*c/x+2/3*(-1)^(3/4)*a*b*c^(3/2)*arctanh((-1)^(3/4)*x*c^(1/2))-1/3*(
-1)^(1/4)*b^2*c^(3/2)*arctanh((-1)^(3/4)*x*c^(1/2))*ln(1-I*c*x^2)-1/3*(-1)^(3/4)*b*c^(3/2)*arctan((-1)^(3/4)*x
*c^(1/2))*(2*a+I*b*ln(1-I*c*x^2))-1/3*(-1)^(1/4)*b^2*c^(3/2)*arctan((-1)^(3/4)*x*c^(1/2))*ln(1+I*c*x^2)+1/3*(-
1)^(1/4)*b^2*c^(3/2)*arctanh((-1)^(3/4)*x*c^(1/2))*ln(1+I*c*x^2)+2/3*(-1)^(1/4)*b^2*c^(3/2)*arctan((-1)^(3/4)*
x*c^(1/2))*ln(2/(1-(-1)^(1/4)*x*c^(1/2)))-2/3*(-1)^(1/4)*b^2*c^(3/2)*arctan((-1)^(3/4)*x*c^(1/2))*ln(2/(1+(-1)
^(1/4)*x*c^(1/2)))+1/3*(-1)^(1/4)*b^2*c^(3/2)*arctan((-1)^(3/4)*x*c^(1/2))*ln(2^(1/2)*((-1)^(1/4)+x*c^(1/2))/(
1+(-1)^(1/4)*x*c^(1/2)))+2/3*(-1)^(1/4)*b^2*c^(3/2)*arctanh((-1)^(3/4)*x*c^(1/2))*ln(2/(1-(-1)^(3/4)*x*c^(1/2)
))-2/3*(-1)^(1/4)*b^2*c^(3/2)*arctanh((-1)^(3/4)*x*c^(1/2))*ln(2/(1+(-1)^(3/4)*x*c^(1/2)))+1/3*(-1)^(1/4)*b^2*
c^(3/2)*arctanh((-1)^(3/4)*x*c^(1/2))*ln(-2^(1/2)*((-1)^(3/4)+x*c^(1/2))/(1+(-1)^(3/4)*x*c^(1/2)))+1/3*(-1)^(1
/4)*b^2*c^(3/2)*arctanh((-1)^(3/4)*x*c^(1/2))*ln((1+I)*(1+(-1)^(1/4)*x*c^(1/2))/(1+(-1)^(3/4)*x*c^(1/2)))+1/3*
(-1)^(1/4)*b^2*c^(3/2)*arctan((-1)^(3/4)*x*c^(1/2))*ln((1-I)*(1+(-1)^(3/4)*x*c^(1/2))/(1+(-1)^(1/4)*x*c^(1/2))
)-1/3*I*b^2*c*ln(1-I*c*x^2)/x+1/3*I*a*b*ln(1+I*c*x^2)/x^3+2/3*I*b^2*c*ln(1+I*c*x^2)/x-4/3*(-1)^(1/4)*b^2*c^(3/
2)*arctan((-1)^(3/4)*x*c^(1/2))+1/3*(-1)^(3/4)*b^2*c^(3/2)*arctan((-1)^(3/4)*x*c^(1/2))^2-4/3*(-1)^(1/4)*b^2*c
^(3/2)*arctanh((-1)^(3/4)*x*c^(1/2))-1/3*(-1)^(1/4)*b^2*c^(3/2)*arctanh((-1)^(3/4)*x*c^(1/2))^2-1/3*b*c*(2*a+I
*b*ln(1-I*c*x^2))/x-1/6*b^2*ln(1-I*c*x^2)*ln(1+I*c*x^2)/x^3+1/3*(-1)^(3/4)*b^2*c^(3/2)*polylog(2,1-2/(1-(-1)^(
1/4)*x*c^(1/2)))+1/3*(-1)^(3/4)*b^2*c^(3/2)*polylog(2,1-2/(1+(-1)^(1/4)*x*c^(1/2)))-1/6*(-1)^(3/4)*b^2*c^(3/2)
*polylog(2,1-2^(1/2)*((-1)^(1/4)+x*c^(1/2))/(1+(-1)^(1/4)*x*c^(1/2)))+1/3*(-1)^(1/4)*b^2*c^(3/2)*polylog(2,1-2
/(1-(-1)^(3/4)*x*c^(1/2)))+1/3*(-1)^(1/4)*b^2*c^(3/2)*polylog(2,1-2/(1+(-1)^(3/4)*x*c^(1/2)))-1/6*(-1)^(1/4)*b
^2*c^(3/2)*polylog(2,1+2^(1/2)*((-1)^(3/4)+x*c^(1/2))/(1+(-1)^(3/4)*x*c^(1/2)))-1/6*(-1)^(1/4)*b^2*c^(3/2)*pol
ylog(2,1-(1+I)*(1+(-1)^(1/4)*x*c^(1/2))/(1+(-1)^(3/4)*x*c^(1/2)))-1/6*(-1)^(3/4)*b^2*c^(3/2)*polylog(2,1+(-1+I
)*(1+(-1)^(3/4)*x*c^(1/2))/(1+(-1)^(1/4)*x*c^(1/2)))+1/12*b^2*ln(1+I*c*x^2)^2/x^3

________________________________________________________________________________________

Rubi [A]  time = 2.23, antiderivative size = 1360, normalized size of antiderivative = 1.00, number of steps used = 64, number of rules used = 25, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.562, Rules used = {5035, 2457, 2476, 2455, 203, 205, 2470, 12, 4920, 4854, 2402, 2315, 325, 6742, 206, 30, 2557, 4928, 4856, 2447, 208, 5992, 5920, 5984, 5918} \[ \text {result too large to display} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTan[c*x^2])^2/x^4,x]

[Out]

(-2*a*b*c)/(3*x) - (4*(-1)^(1/4)*b^2*c^(3/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x])/3 + ((-1)^(3/4)*b^2*c^(3/2)*ArcTan[
(-1)^(3/4)*Sqrt[c]*x]^2)/3 + (2*(-1)^(3/4)*a*b*c^(3/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x])/3 - (4*(-1)^(1/4)*b^2*c^
(3/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x])/3 - ((-1)^(1/4)*b^2*c^(3/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]^2)/3 + (2*(-1)^
(1/4)*b^2*c^(3/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 - (-1)^(1/4)*Sqrt[c]*x)])/3 - (2*(-1)^(1/4)*b^2*c^(3/2
)*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 + (-1)^(1/4)*Sqrt[c]*x)])/3 + ((-1)^(1/4)*b^2*c^(3/2)*ArcTan[(-1)^(3/4
)*Sqrt[c]*x]*Log[(Sqrt[2]*((-1)^(1/4) + Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/3 + (2*(-1)^(1/4)*b^2*c^(3/2)
*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 - (-1)^(3/4)*Sqrt[c]*x)])/3 - (2*(-1)^(1/4)*b^2*c^(3/2)*ArcTanh[(-1)^(
3/4)*Sqrt[c]*x]*Log[2/(1 + (-1)^(3/4)*Sqrt[c]*x)])/3 + ((-1)^(1/4)*b^2*c^(3/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*L
og[-((Sqrt[2]*((-1)^(3/4) + Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x))])/3 + ((-1)^(1/4)*b^2*c^(3/2)*ArcTanh[(-1)
^(3/4)*Sqrt[c]*x]*Log[((1 + I)*(1 + (-1)^(1/4)*Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x)])/3 + ((-1)^(1/4)*b^2*c^
(3/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[((1 - I)*(1 + (-1)^(3/4)*Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/3 - (
(I/3)*b^2*c*Log[1 - I*c*x^2])/x - ((-1)^(1/4)*b^2*c^(3/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[1 - I*c*x^2])/3 -
(b*c*(2*a + I*b*Log[1 - I*c*x^2]))/(3*x) - ((-1)^(3/4)*b*c^(3/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*(2*a + I*b*Log[1
- I*c*x^2]))/3 - (2*a + I*b*Log[1 - I*c*x^2])^2/(12*x^3) + ((I/3)*a*b*Log[1 + I*c*x^2])/x^3 + (((2*I)/3)*b^2*
c*Log[1 + I*c*x^2])/x - ((-1)^(1/4)*b^2*c^(3/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[1 + I*c*x^2])/3 + ((-1)^(1/4)
*b^2*c^(3/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[1 + I*c*x^2])/3 - (b^2*Log[1 - I*c*x^2]*Log[1 + I*c*x^2])/(6*x^
3) + (b^2*Log[1 + I*c*x^2]^2)/(12*x^3) + ((-1)^(3/4)*b^2*c^(3/2)*PolyLog[2, 1 - 2/(1 - (-1)^(1/4)*Sqrt[c]*x)])
/3 + ((-1)^(3/4)*b^2*c^(3/2)*PolyLog[2, 1 - 2/(1 + (-1)^(1/4)*Sqrt[c]*x)])/3 - ((-1)^(3/4)*b^2*c^(3/2)*PolyLog
[2, 1 - (Sqrt[2]*((-1)^(1/4) + Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/6 + ((-1)^(1/4)*b^2*c^(3/2)*PolyLog[2,
1 - 2/(1 - (-1)^(3/4)*Sqrt[c]*x)])/3 + ((-1)^(1/4)*b^2*c^(3/2)*PolyLog[2, 1 - 2/(1 + (-1)^(3/4)*Sqrt[c]*x)])/
3 - ((-1)^(1/4)*b^2*c^(3/2)*PolyLog[2, 1 + (Sqrt[2]*((-1)^(3/4) + Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x)])/6 -
((-1)^(1/4)*b^2*c^(3/2)*PolyLog[2, 1 - ((1 + I)*(1 + (-1)^(1/4)*Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x)])/6 -
((-1)^(3/4)*b^2*c^(3/2)*PolyLog[2, 1 - ((1 - I)*(1 + (-1)^(3/4)*Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/6

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 206

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

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[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 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2402

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 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 2455

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[((f*x)^(m
+ 1)*(a + b*Log[c*(d + e*x^n)^p]))/(f*(m + 1)), x] - Dist[(b*e*n*p)/(f*(m + 1)), Int[(x^(n - 1)*(f*x)^(m + 1))
/(d + e*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]

Rule 2457

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_)*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[((f*x
)^(m + 1)*(a + b*Log[c*(d + e*x^n)^p])^q)/(f*(m + 1)), x] - Dist[(b*e*n*p*q)/(f^n*(m + 1)), Int[((f*x)^(m + n)
*(a + b*Log[c*(d + e*x^n)^p])^(q - 1))/(d + e*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && IGtQ[q, 1]
&& IntegerQ[n] && NeQ[m, -1]

Rule 2470

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_) + (g_.)*(x_)^2), x_Symbol] :> With[{u = In
tHide[1/(f + g*x^2), x]}, Simp[u*(a + b*Log[c*(d + e*x^n)^p]), x] - Dist[b*e*n*p, Int[(u*x^(n - 1))/(d + e*x^n
), x], x]] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IntegerQ[n]

Rule 2476

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x^s)^r, x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, n, p, q, r, s}, x] && IGtQ[q, 0] && IntegerQ[m] && IntegerQ[r] && IntegerQ[s]

Rule 2557

Int[Log[v_]*Log[w_]*(u_), x_Symbol] :> With[{z = IntHide[u, x]}, Dist[Log[v]*Log[w], z, x] + (-Int[SimplifyInt
egrand[(z*Log[w]*D[v, x])/v, x], x] - Int[SimplifyIntegrand[(z*Log[v]*D[w, x])/w, x], x]) /; InverseFunctionFr
eeQ[z, x]] /; InverseFunctionFreeQ[v, x] && InverseFunctionFreeQ[w, x]

Rule 4854

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTan[c*x])^p*Lo
g[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 4856

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

Rule 4920

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(I*(a + b*ArcTan
[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 4928

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*(x_)^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[a
+ b*ArcTan[c*x], x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] &&  !(EqQ[m, 1] && NeQ[a,
0])

Rule 5035

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

Rule 5918

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTanh[c*x])^p*
Log[2/(1 + (e*x)/d)])/e, x] + Dist[(b*c*p)/e, Int[((a + b*ArcTanh[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 5920

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTanh[c*x])*Log[2/(1
+ c*x)])/e, x] + (Dist[(b*c)/e, Int[Log[2/(1 + c*x)]/(1 - c^2*x^2), x], x] - Dist[(b*c)/e, Int[Log[(2*c*(d +
e*x))/((c*d + e)*(1 + c*x))]/(1 - c^2*x^2), x], x] + Simp[((a + b*ArcTanh[c*x])*Log[(2*c*(d + e*x))/((c*d + e)
*(1 + c*x))])/e, x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2, 0]

Rule 5984

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

Rule 5992

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*(x_)^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[a
+ b*ArcTanh[c*x], x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] &&  !(EqQ[m, 1] && NeQ[
a, 0])

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {\left (a+b \tan ^{-1}\left (c x^2\right )\right )^2}{x^4} \, dx &=\int \left (\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{4 x^4}+\frac {b \left (-2 i a+b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{2 x^4}-\frac {b^2 \log ^2\left (1+i c x^2\right )}{4 x^4}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{x^4} \, dx+\frac {1}{2} b \int \frac {\left (-2 i a+b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{x^4} \, dx-\frac {1}{4} b^2 \int \frac {\log ^2\left (1+i c x^2\right )}{x^4} \, dx\\ &=-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{12 x^3}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{12 x^3}+\frac {1}{2} b \int \left (-\frac {2 i a \log \left (1+i c x^2\right )}{x^4}+\frac {b \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{x^4}\right ) \, dx+\frac {1}{3} (b c) \int \frac {2 a+i b \log \left (1-i c x^2\right )}{x^2 \left (1-i c x^2\right )} \, dx-\frac {1}{3} \left (i b^2 c\right ) \int \frac {\log \left (1+i c x^2\right )}{x^2 \left (1+i c x^2\right )} \, dx\\ &=-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{12 x^3}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{12 x^3}-(i a b) \int \frac {\log \left (1+i c x^2\right )}{x^4} \, dx+\frac {1}{2} b^2 \int \frac {\log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{x^4} \, dx+\frac {1}{3} (b c) \int \left (\frac {2 a+i b \log \left (1-i c x^2\right )}{x^2}-\frac {c \left (2 a+i b \log \left (1-i c x^2\right )\right )}{i+c x^2}\right ) \, dx-\frac {1}{3} \left (i b^2 c\right ) \int \left (\frac {\log \left (1+i c x^2\right )}{x^2}-\frac {c \log \left (1+i c x^2\right )}{-i+c x^2}\right ) \, dx\\ &=-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{12 x^3}+\frac {i a b \log \left (1+i c x^2\right )}{3 x^3}-\frac {b^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{6 x^3}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{12 x^3}-\frac {1}{2} b^2 \int \frac {2 c \log \left (1-i c x^2\right )}{x^2 \left (3 i-3 c x^2\right )} \, dx-\frac {1}{2} b^2 \int \frac {2 c \log \left (1+i c x^2\right )}{x^2 \left (-3 i-3 c x^2\right )} \, dx+\frac {1}{3} (b c) \int \frac {2 a+i b \log \left (1-i c x^2\right )}{x^2} \, dx+\frac {1}{3} (2 a b c) \int \frac {1}{x^2 \left (1+i c x^2\right )} \, dx-\frac {1}{3} \left (i b^2 c\right ) \int \frac {\log \left (1+i c x^2\right )}{x^2} \, dx-\frac {1}{3} \left (b c^2\right ) \int \frac {2 a+i b \log \left (1-i c x^2\right )}{i+c x^2} \, dx+\frac {1}{3} \left (i b^2 c^2\right ) \int \frac {\log \left (1+i c x^2\right )}{-i+c x^2} \, dx\\ &=-\frac {2 a b c}{3 x}-\frac {b c \left (2 a+i b \log \left (1-i c x^2\right )\right )}{3 x}-\frac {1}{3} (-1)^{3/4} b c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{12 x^3}+\frac {i a b \log \left (1+i c x^2\right )}{3 x^3}+\frac {i b^2 c \log \left (1+i c x^2\right )}{3 x}+\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )-\frac {b^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{6 x^3}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{12 x^3}-\left (b^2 c\right ) \int \frac {\log \left (1-i c x^2\right )}{x^2 \left (3 i-3 c x^2\right )} \, dx-\left (b^2 c\right ) \int \frac {\log \left (1+i c x^2\right )}{x^2 \left (-3 i-3 c x^2\right )} \, dx-\frac {1}{3} \left (2 i a b c^2\right ) \int \frac {1}{1+i c x^2} \, dx+\frac {1}{3} \left (2 b^2 c^2\right ) \int \frac {1}{1-i c x^2} \, dx+\frac {1}{3} \left (2 b^2 c^2\right ) \int \frac {1}{1+i c x^2} \, dx+\frac {1}{3} \left (2 b^2 c^3\right ) \int \frac {(-1)^{3/4} x \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c} \left (1-i c x^2\right )} \, dx-\frac {1}{3} \left (2 b^2 c^3\right ) \int \frac {(-1)^{3/4} x \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c} \left (1+i c x^2\right )} \, dx\\ &=-\frac {2 a b c}{3 x}-\frac {2}{3} \sqrt [4]{-1} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )+\frac {2}{3} (-1)^{3/4} a b c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )-\frac {2}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )-\frac {b c \left (2 a+i b \log \left (1-i c x^2\right )\right )}{3 x}-\frac {1}{3} (-1)^{3/4} b c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{12 x^3}+\frac {i a b \log \left (1+i c x^2\right )}{3 x^3}+\frac {i b^2 c \log \left (1+i c x^2\right )}{3 x}+\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )-\frac {b^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{6 x^3}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{12 x^3}-\left (b^2 c\right ) \int \left (-\frac {i \log \left (1-i c x^2\right )}{3 x^2}+\frac {i c \log \left (1-i c x^2\right )}{3 \left (-i+c x^2\right )}\right ) \, dx-\left (b^2 c\right ) \int \left (\frac {i \log \left (1+i c x^2\right )}{3 x^2}-\frac {i c \log \left (1+i c x^2\right )}{3 \left (i+c x^2\right )}\right ) \, dx+\frac {1}{3} \left (2 (-1)^{3/4} b^2 c^{5/2}\right ) \int \frac {x \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{1-i c x^2} \, dx-\frac {1}{3} \left (2 (-1)^{3/4} b^2 c^{5/2}\right ) \int \frac {x \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{1+i c x^2} \, dx\\ &=-\frac {2 a b c}{3 x}-\frac {2}{3} \sqrt [4]{-1} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )+\frac {1}{3} (-1)^{3/4} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{3} (-1)^{3/4} a b c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )-\frac {2}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )-\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2-\frac {b c \left (2 a+i b \log \left (1-i c x^2\right )\right )}{3 x}-\frac {1}{3} (-1)^{3/4} b c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{12 x^3}+\frac {i a b \log \left (1+i c x^2\right )}{3 x^3}+\frac {i b^2 c \log \left (1+i c x^2\right )}{3 x}+\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )-\frac {b^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{6 x^3}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{12 x^3}+\frac {1}{3} \left (i b^2 c\right ) \int \frac {\log \left (1-i c x^2\right )}{x^2} \, dx-\frac {1}{3} \left (i b^2 c\right ) \int \frac {\log \left (1+i c x^2\right )}{x^2} \, dx-\frac {1}{3} \left (i b^2 c^2\right ) \int \frac {\log \left (1-i c x^2\right )}{-i+c x^2} \, dx+\frac {1}{3} \left (i b^2 c^2\right ) \int \frac {\log \left (1+i c x^2\right )}{i+c x^2} \, dx-\frac {1}{3} \left (2 b^2 c^2\right ) \int \frac {\tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{i-(-1)^{3/4} \sqrt {c} x} \, dx-\frac {1}{3} \left (2 b^2 c^2\right ) \int \frac {\tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{1-(-1)^{3/4} \sqrt {c} x} \, dx\\ &=-\frac {2 a b c}{3 x}-\frac {2}{3} \sqrt [4]{-1} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )+\frac {1}{3} (-1)^{3/4} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{3} (-1)^{3/4} a b c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )-\frac {2}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )-\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{3} \sqrt [4]{-1} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )+\frac {2}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )-\frac {i b^2 c \log \left (1-i c x^2\right )}{3 x}-\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )-\frac {b c \left (2 a+i b \log \left (1-i c x^2\right )\right )}{3 x}-\frac {1}{3} (-1)^{3/4} b c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{12 x^3}+\frac {i a b \log \left (1+i c x^2\right )}{3 x^3}+\frac {2 i b^2 c \log \left (1+i c x^2\right )}{3 x}-\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )+\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )-\frac {b^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{6 x^3}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{12 x^3}+\frac {1}{3} \left (2 b^2 c^2\right ) \int \frac {1}{1-i c x^2} \, dx+\frac {1}{3} \left (2 b^2 c^2\right ) \int \frac {1}{1+i c x^2} \, dx+\frac {1}{3} \left (2 b^2 c^2\right ) \int \frac {\log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )}{1-i c x^2} \, dx+\frac {1}{3} \left (2 b^2 c^2\right ) \int \frac {\log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )}{1+i c x^2} \, dx+\frac {1}{3} \left (2 b^2 c^3\right ) \int \frac {(-1)^{3/4} x \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c} \left (1+i c x^2\right )} \, dx-\frac {1}{3} \left (2 b^2 c^3\right ) \int \frac {(-1)^{3/4} x \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c} \left (1-i c x^2\right )} \, dx\\ &=-\frac {2 a b c}{3 x}-\frac {4}{3} \sqrt [4]{-1} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )+\frac {1}{3} (-1)^{3/4} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{3} (-1)^{3/4} a b c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )-\frac {4}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )-\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{3} \sqrt [4]{-1} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )+\frac {2}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )-\frac {i b^2 c \log \left (1-i c x^2\right )}{3 x}-\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )-\frac {b c \left (2 a+i b \log \left (1-i c x^2\right )\right )}{3 x}-\frac {1}{3} (-1)^{3/4} b c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{12 x^3}+\frac {i a b \log \left (1+i c x^2\right )}{3 x^3}+\frac {2 i b^2 c \log \left (1+i c x^2\right )}{3 x}-\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )+\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )-\frac {b^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{6 x^3}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{12 x^3}+\frac {1}{3} \left (2 \sqrt [4]{-1} b^2 c^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-(-1)^{3/4} \sqrt {c} x}\right )+\frac {1}{3} \left (2 (-1)^{3/4} b^2 c^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\sqrt [4]{-1} \sqrt {c} x}\right )+\frac {1}{3} \left (2 (-1)^{3/4} b^2 c^{5/2}\right ) \int \frac {x \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{1+i c x^2} \, dx-\frac {1}{3} \left (2 (-1)^{3/4} b^2 c^{5/2}\right ) \int \frac {x \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{1-i c x^2} \, dx\\ &=-\frac {2 a b c}{3 x}-\frac {4}{3} \sqrt [4]{-1} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )+\frac {1}{3} (-1)^{3/4} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{3} (-1)^{3/4} a b c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )-\frac {4}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )-\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{3} \sqrt [4]{-1} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )+\frac {2}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )-\frac {i b^2 c \log \left (1-i c x^2\right )}{3 x}-\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )-\frac {b c \left (2 a+i b \log \left (1-i c x^2\right )\right )}{3 x}-\frac {1}{3} (-1)^{3/4} b c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{12 x^3}+\frac {i a b \log \left (1+i c x^2\right )}{3 x^3}+\frac {2 i b^2 c \log \left (1+i c x^2\right )}{3 x}-\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )+\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )-\frac {b^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{6 x^3}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{12 x^3}+\frac {1}{3} (-1)^{3/4} b^2 c^{3/2} \text {Li}_2\left (1-\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )+\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \text {Li}_2\left (1-\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )+\frac {1}{3} \left (2 (-1)^{3/4} b^2 c^{5/2}\right ) \int \left (\frac {i \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{2 \sqrt {c} \left (\sqrt [4]{-1}-\sqrt {c} x\right )}-\frac {i \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{2 \sqrt {c} \left (\sqrt [4]{-1}+\sqrt {c} x\right )}\right ) \, dx-\frac {1}{3} \left (2 (-1)^{3/4} b^2 c^{5/2}\right ) \int \left (-\frac {i \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{2 \sqrt {c} \left (-(-1)^{3/4}-\sqrt {c} x\right )}+\frac {i \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{2 \sqrt {c} \left (-(-1)^{3/4}+\sqrt {c} x\right )}\right ) \, dx\\ &=-\frac {2 a b c}{3 x}-\frac {4}{3} \sqrt [4]{-1} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )+\frac {1}{3} (-1)^{3/4} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{3} (-1)^{3/4} a b c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )-\frac {4}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )-\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{3} \sqrt [4]{-1} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )+\frac {2}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )-\frac {i b^2 c \log \left (1-i c x^2\right )}{3 x}-\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )-\frac {b c \left (2 a+i b \log \left (1-i c x^2\right )\right )}{3 x}-\frac {1}{3} (-1)^{3/4} b c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{12 x^3}+\frac {i a b \log \left (1+i c x^2\right )}{3 x^3}+\frac {2 i b^2 c \log \left (1+i c x^2\right )}{3 x}-\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )+\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )-\frac {b^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{6 x^3}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{12 x^3}+\frac {1}{3} (-1)^{3/4} b^2 c^{3/2} \text {Li}_2\left (1-\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )+\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \text {Li}_2\left (1-\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )-\frac {1}{3} \left (\sqrt [4]{-1} b^2 c^2\right ) \int \frac {\tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt [4]{-1}-\sqrt {c} x} \, dx+\frac {1}{3} \left (\sqrt [4]{-1} b^2 c^2\right ) \int \frac {\tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt [4]{-1}+\sqrt {c} x} \, dx-\frac {1}{3} \left (\sqrt [4]{-1} b^2 c^2\right ) \int \frac {\tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{-(-1)^{3/4}-\sqrt {c} x} \, dx+\frac {1}{3} \left (\sqrt [4]{-1} b^2 c^2\right ) \int \frac {\tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{-(-1)^{3/4}+\sqrt {c} x} \, dx\\ &=-\frac {2 a b c}{3 x}-\frac {4}{3} \sqrt [4]{-1} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )+\frac {1}{3} (-1)^{3/4} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{3} (-1)^{3/4} a b c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )-\frac {4}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )-\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{3} \sqrt [4]{-1} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {2}{3} \sqrt [4]{-1} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1+\sqrt [4]{-1} \sqrt {c} x}\right )+\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {\sqrt {2} \left (\sqrt [4]{-1}+\sqrt {c} x\right )}{1+\sqrt [4]{-1} \sqrt {c} x}\right )+\frac {2}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )-\frac {2}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1+(-1)^{3/4} \sqrt {c} x}\right )+\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (-\frac {\sqrt {2} \left ((-1)^{3/4}+\sqrt {c} x\right )}{1+(-1)^{3/4} \sqrt {c} x}\right )+\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {\sqrt [4]{-1} \sqrt {2} \left (1+\sqrt [4]{-1} \sqrt {c} x\right )}{1+(-1)^{3/4} \sqrt {c} x}\right )+\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {(1-i) \left (1+(-1)^{3/4} \sqrt {c} x\right )}{1+\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {i b^2 c \log \left (1-i c x^2\right )}{3 x}-\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )-\frac {b c \left (2 a+i b \log \left (1-i c x^2\right )\right )}{3 x}-\frac {1}{3} (-1)^{3/4} b c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{12 x^3}+\frac {i a b \log \left (1+i c x^2\right )}{3 x^3}+\frac {2 i b^2 c \log \left (1+i c x^2\right )}{3 x}-\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )+\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )-\frac {b^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{6 x^3}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{12 x^3}+\frac {1}{3} (-1)^{3/4} b^2 c^{3/2} \text {Li}_2\left (1-\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )+\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \text {Li}_2\left (1-\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )-2 \left (\frac {1}{3} \left (b^2 c^2\right ) \int \frac {\log \left (\frac {2}{1+\sqrt [4]{-1} \sqrt {c} x}\right )}{1-i c x^2} \, dx\right )+\frac {1}{3} \left (b^2 c^2\right ) \int \frac {\log \left (-\frac {(1-i) (-1)^{3/4} \left (\sqrt [4]{-1}-\sqrt {c} x\right )}{1+\sqrt [4]{-1} \sqrt {c} x}\right )}{1-i c x^2} \, dx+\frac {1}{3} \left (b^2 c^2\right ) \int \frac {\log \left (-\frac {(1+i) (-1)^{3/4} \left (\sqrt [4]{-1}+\sqrt {c} x\right )}{1+\sqrt [4]{-1} \sqrt {c} x}\right )}{1-i c x^2} \, dx-2 \left (\frac {1}{3} \left (b^2 c^2\right ) \int \frac {\log \left (\frac {2}{1+(-1)^{3/4} \sqrt {c} x}\right )}{1+i c x^2} \, dx\right )+\frac {1}{3} \left (b^2 c^2\right ) \int \frac {\log \left (-\frac {(1+i) (-1)^{3/4} \left (-(-1)^{3/4}-\sqrt {c} x\right )}{1+(-1)^{3/4} \sqrt {c} x}\right )}{1+i c x^2} \, dx+\frac {1}{3} \left (b^2 c^2\right ) \int \frac {\log \left (\frac {(1-i) (-1)^{3/4} \left (-(-1)^{3/4}+\sqrt {c} x\right )}{1+(-1)^{3/4} \sqrt {c} x}\right )}{1+i c x^2} \, dx\\ &=-\frac {2 a b c}{3 x}-\frac {4}{3} \sqrt [4]{-1} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )+\frac {1}{3} (-1)^{3/4} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{3} (-1)^{3/4} a b c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )-\frac {4}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )-\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{3} \sqrt [4]{-1} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {2}{3} \sqrt [4]{-1} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1+\sqrt [4]{-1} \sqrt {c} x}\right )+\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {\sqrt {2} \left (\sqrt [4]{-1}+\sqrt {c} x\right )}{1+\sqrt [4]{-1} \sqrt {c} x}\right )+\frac {2}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )-\frac {2}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1+(-1)^{3/4} \sqrt {c} x}\right )+\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (-\frac {\sqrt {2} \left ((-1)^{3/4}+\sqrt {c} x\right )}{1+(-1)^{3/4} \sqrt {c} x}\right )+\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {\sqrt [4]{-1} \sqrt {2} \left (1+\sqrt [4]{-1} \sqrt {c} x\right )}{1+(-1)^{3/4} \sqrt {c} x}\right )+\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {(1-i) \left (1+(-1)^{3/4} \sqrt {c} x\right )}{1+\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {i b^2 c \log \left (1-i c x^2\right )}{3 x}-\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )-\frac {b c \left (2 a+i b \log \left (1-i c x^2\right )\right )}{3 x}-\frac {1}{3} (-1)^{3/4} b c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{12 x^3}+\frac {i a b \log \left (1+i c x^2\right )}{3 x^3}+\frac {2 i b^2 c \log \left (1+i c x^2\right )}{3 x}-\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )+\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )-\frac {b^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{6 x^3}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{12 x^3}+\frac {1}{3} (-1)^{3/4} b^2 c^{3/2} \text {Li}_2\left (1-\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {1}{6} (-1)^{3/4} b^2 c^{3/2} \text {Li}_2\left (1-\frac {\sqrt {2} \left (\sqrt [4]{-1}+\sqrt {c} x\right )}{1+\sqrt [4]{-1} \sqrt {c} x}\right )+\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \text {Li}_2\left (1-\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )-\frac {1}{6} \sqrt [4]{-1} b^2 c^{3/2} \text {Li}_2\left (1+\frac {\sqrt {2} \left ((-1)^{3/4}+\sqrt {c} x\right )}{1+(-1)^{3/4} \sqrt {c} x}\right )-\frac {1}{6} \sqrt [4]{-1} b^2 c^{3/2} \text {Li}_2\left (1-\frac {\sqrt [4]{-1} \sqrt {2} \left (1+\sqrt [4]{-1} \sqrt {c} x\right )}{1+(-1)^{3/4} \sqrt {c} x}\right )-\frac {1}{6} (-1)^{3/4} b^2 c^{3/2} \text {Li}_2\left (1-\frac {(1-i) \left (1+(-1)^{3/4} \sqrt {c} x\right )}{1+\sqrt [4]{-1} \sqrt {c} x}\right )+2 \left (\frac {1}{3} \left (\sqrt [4]{-1} b^2 c^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+(-1)^{3/4} \sqrt {c} x}\right )\right )+2 \left (\frac {1}{3} \left ((-1)^{3/4} b^2 c^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\sqrt [4]{-1} \sqrt {c} x}\right )\right )\\ &=-\frac {2 a b c}{3 x}-\frac {4}{3} \sqrt [4]{-1} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )+\frac {1}{3} (-1)^{3/4} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{3} (-1)^{3/4} a b c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )-\frac {4}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )-\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{3} \sqrt [4]{-1} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {2}{3} \sqrt [4]{-1} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1+\sqrt [4]{-1} \sqrt {c} x}\right )+\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {\sqrt {2} \left (\sqrt [4]{-1}+\sqrt {c} x\right )}{1+\sqrt [4]{-1} \sqrt {c} x}\right )+\frac {2}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )-\frac {2}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1+(-1)^{3/4} \sqrt {c} x}\right )+\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (-\frac {\sqrt {2} \left ((-1)^{3/4}+\sqrt {c} x\right )}{1+(-1)^{3/4} \sqrt {c} x}\right )+\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {\sqrt [4]{-1} \sqrt {2} \left (1+\sqrt [4]{-1} \sqrt {c} x\right )}{1+(-1)^{3/4} \sqrt {c} x}\right )+\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {(1-i) \left (1+(-1)^{3/4} \sqrt {c} x\right )}{1+\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {i b^2 c \log \left (1-i c x^2\right )}{3 x}-\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )-\frac {b c \left (2 a+i b \log \left (1-i c x^2\right )\right )}{3 x}-\frac {1}{3} (-1)^{3/4} b c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{12 x^3}+\frac {i a b \log \left (1+i c x^2\right )}{3 x^3}+\frac {2 i b^2 c \log \left (1+i c x^2\right )}{3 x}-\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )+\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )-\frac {b^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{6 x^3}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{12 x^3}+\frac {1}{3} (-1)^{3/4} b^2 c^{3/2} \text {Li}_2\left (1-\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )+\frac {1}{3} (-1)^{3/4} b^2 c^{3/2} \text {Li}_2\left (1-\frac {2}{1+\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {1}{6} (-1)^{3/4} b^2 c^{3/2} \text {Li}_2\left (1-\frac {\sqrt {2} \left (\sqrt [4]{-1}+\sqrt {c} x\right )}{1+\sqrt [4]{-1} \sqrt {c} x}\right )+\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \text {Li}_2\left (1-\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )+\frac {1}{3} \sqrt [4]{-1} b^2 c^{3/2} \text {Li}_2\left (1-\frac {2}{1+(-1)^{3/4} \sqrt {c} x}\right )-\frac {1}{6} \sqrt [4]{-1} b^2 c^{3/2} \text {Li}_2\left (1+\frac {\sqrt {2} \left ((-1)^{3/4}+\sqrt {c} x\right )}{1+(-1)^{3/4} \sqrt {c} x}\right )-\frac {1}{6} \sqrt [4]{-1} b^2 c^{3/2} \text {Li}_2\left (1-\frac {\sqrt [4]{-1} \sqrt {2} \left (1+\sqrt [4]{-1} \sqrt {c} x\right )}{1+(-1)^{3/4} \sqrt {c} x}\right )-\frac {1}{6} (-1)^{3/4} b^2 c^{3/2} \text {Li}_2\left (1-\frac {(1-i) \left (1+(-1)^{3/4} \sqrt {c} x\right )}{1+\sqrt [4]{-1} \sqrt {c} x}\right )\\ \end {align*}

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Mathematica [F]  time = 2.94, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \tan ^{-1}\left (c x^2\right )\right )^2}{x^4} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [F]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \arctan \left (c x^{2}\right )^{2} + 2 \, a b \arctan \left (c x^{2}\right ) + a^{2}}{x^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \arctan \left (c x^{2}\right ) + a\right )}^{2}}{x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.33, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \arctan \left (c \,x^{2}\right )\right )^{2}}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{6} \, {\left ({\left (c^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, c x + \sqrt {2} \sqrt {c}\right )}}{2 \, \sqrt {c}}\right )}{c^{\frac {3}{2}}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, c x - \sqrt {2} \sqrt {c}\right )}}{2 \, \sqrt {c}}\right )}{c^{\frac {3}{2}}} - \frac {\sqrt {2} \log \left (c x^{2} + \sqrt {2} \sqrt {c} x + 1\right )}{c^{\frac {3}{2}}} + \frac {\sqrt {2} \log \left (c x^{2} - \sqrt {2} \sqrt {c} x + 1\right )}{c^{\frac {3}{2}}}\right )} + \frac {8}{x}\right )} c + \frac {4 \, \arctan \left (c x^{2}\right )}{x^{3}}\right )} a b + \frac {\frac {1}{4} \, {\left (4 \, x^{3} \int -\frac {24 \, c^{2} x^{4} \log \left (c^{2} x^{4} + 1\right ) - 112 \, c x^{2} \arctan \left (c x^{2}\right ) - 108 \, {\left (c^{2} x^{4} + 1\right )} \arctan \left (c x^{2}\right )^{2} - 9 \, {\left (c^{2} x^{4} + 1\right )} \log \left (c^{2} x^{4} + 1\right )^{2}}{4 \, {\left (c^{2} x^{8} + x^{4}\right )}}\,{d x} - 28 \, \arctan \left (c x^{2}\right )^{2} + 3 \, \log \left (c^{2} x^{4} + 1\right )^{2}\right )} b^{2}}{48 \, x^{3}} - \frac {a^{2}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*((c^2*(2*sqrt(2)*arctan(1/2*sqrt(2)*(2*c*x + sqrt(2)*sqrt(c))/sqrt(c))/c^(3/2) + 2*sqrt(2)*arctan(1/2*sqr
t(2)*(2*c*x - sqrt(2)*sqrt(c))/sqrt(c))/c^(3/2) - sqrt(2)*log(c*x^2 + sqrt(2)*sqrt(c)*x + 1)/c^(3/2) + sqrt(2)
*log(c*x^2 - sqrt(2)*sqrt(c)*x + 1)/c^(3/2)) + 8/x)*c + 4*arctan(c*x^2)/x^3)*a*b + 1/48*(48*x^3*integrate(-1/4
8*(8*c^2*x^4*log(c^2*x^4 + 1) - 16*c*x^2*arctan(c*x^2) - 36*(c^2*x^4 + 1)*arctan(c*x^2)^2 - 3*(c^2*x^4 + 1)*lo
g(c^2*x^4 + 1)^2)/(c^2*x^8 + x^4), x) - 4*arctan(c*x^2)^2 + log(c^2*x^4 + 1)^2)*b^2/x^3 - 1/3*a^2/x^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atan(c*x^2))^2/x^4,x)

[Out]

int((a + b*atan(c*x^2))^2/x^4, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*atan(c*x**2))**2/x**4,x)

[Out]

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

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