3.1.83 \(\int \frac {(a+b \text {ArcTan}(c x^2))^2}{x^2} \, dx\) [83]

Optimal. Leaf size=1164 \[ \sqrt [4]{-1} b^2 \sqrt {c} \text {ArcTan}\left ((-1)^{3/4} \sqrt {c} x\right )^2-2 \sqrt [4]{-1} a b \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )-(-1)^{3/4} b^2 \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2-2 (-1)^{3/4} b^2 \sqrt {c} \text {ArcTan}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )+2 (-1)^{3/4} b^2 \sqrt {c} \text {ArcTan}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1+\sqrt [4]{-1} \sqrt {c} x}\right )-(-1)^{3/4} b^2 \sqrt {c} \text {ArcTan}\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 )+2 (-1)^{3/4} b^2 \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )-2 (-1)^{3/4} b^2 \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1+(-1)^{3/4} \sqrt {c} x}\right )+(-1)^{3/4} b^2 \sqrt {c} \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 )+(-1)^{3/4} b^2 \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {(1+i) \left (1+\sqrt [4]{-1} \sqrt {c} x\right )}{1+(-1)^{3/4} \sqrt {c} x}\right )-(-1)^{3/4} b^2 \sqrt {c} \text {ArcTan}\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 )-(-1)^{3/4} b^2 \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )-\sqrt [4]{-1} b \sqrt {c} \text {ArcTan}\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}{4 x}+\frac {i a b \log \left (1+i c x^2\right )}{x}+(-1)^{3/4} b^2 \sqrt {c} \text {ArcTan}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )+(-1)^{3/4} b^2 \sqrt {c} \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 )}{2 x}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{4 x}+\sqrt [4]{-1} b^2 \sqrt {c} \text {PolyLog}\left (2,1-\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )+\sqrt [4]{-1} b^2 \sqrt {c} \text {PolyLog}\left (2,1-\frac {2}{1+\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {1}{2} \sqrt [4]{-1} b^2 \sqrt {c} \text {PolyLog}\left (2,1-\frac {\sqrt {2} \left (\sqrt [4]{-1}+\sqrt {c} x\right )}{1+\sqrt [4]{-1} \sqrt {c} x}\right )+(-1)^{3/4} b^2 \sqrt {c} \text {PolyLog}\left (2,1-\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )+(-1)^{3/4} b^2 \sqrt {c} \text {PolyLog}\left (2,1-\frac {2}{1+(-1)^{3/4} \sqrt {c} x}\right )-\frac {1}{2} (-1)^{3/4} b^2 \sqrt {c} \text {PolyLog}\left (2,1+\frac {\sqrt {2} \left ((-1)^{3/4}+\sqrt {c} x\right )}{1+(-1)^{3/4} \sqrt {c} x}\right )-\frac {1}{2} (-1)^{3/4} b^2 \sqrt {c} \text {PolyLog}\left (2,1-\frac {(1+i) \left (1+\sqrt [4]{-1} \sqrt {c} x\right )}{1+(-1)^{3/4} \sqrt {c} x}\right )-\frac {1}{2} \sqrt [4]{-1} b^2 \sqrt {c} \text {PolyLog}\left (2,1-\frac {(1-i) \left (1+(-1)^{3/4} \sqrt {c} x\right )}{1+\sqrt [4]{-1} \sqrt {c} x}\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 1.15, antiderivative size = 1164, normalized size of antiderivative = 1.00, number of steps used = 47, number of rules used = 23, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.438, Rules used = {4950, 2507, 209, 2520, 12, 5040, 4964, 2449, 2352, 2505, 6874, 212, 30, 2637, 211, 5048, 4966, 2497, 214, 6139, 6057, 6131, 6055} \begin {gather*} \sqrt [4]{-1} \sqrt {c} \text {ArcTan}\left ((-1)^{3/4} \sqrt {c} x\right )^2 b^2-(-1)^{3/4} \sqrt {c} \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}{4 x}-2 (-1)^{3/4} \sqrt {c} \text {ArcTan}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right ) b^2+2 (-1)^{3/4} \sqrt {c} \text {ArcTan}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{\sqrt [4]{-1} \sqrt {c} x+1}\right ) b^2-(-1)^{3/4} \sqrt {c} \text {ArcTan}\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+2 (-1)^{3/4} \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right ) b^2-2 (-1)^{3/4} \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{(-1)^{3/4} \sqrt {c} x+1}\right ) b^2+(-1)^{3/4} \sqrt {c} \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+(-1)^{3/4} \sqrt {c} \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-(-1)^{3/4} \sqrt {c} \text {ArcTan}\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-(-1)^{3/4} \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right ) b^2+(-1)^{3/4} \sqrt {c} \text {ArcTan}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (i c x^2+1\right ) b^2+(-1)^{3/4} \sqrt {c} \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}{2 x}+\sqrt [4]{-1} \sqrt {c} \text {Li}_2\left (1-\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right ) b^2+\sqrt [4]{-1} \sqrt {c} \text {Li}_2\left (1-\frac {2}{\sqrt [4]{-1} \sqrt {c} x+1}\right ) b^2-\frac {1}{2} \sqrt [4]{-1} \sqrt {c} \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+(-1)^{3/4} \sqrt {c} \text {Li}_2\left (1-\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right ) b^2+(-1)^{3/4} \sqrt {c} \text {Li}_2\left (1-\frac {2}{(-1)^{3/4} \sqrt {c} x+1}\right ) b^2-\frac {1}{2} (-1)^{3/4} \sqrt {c} \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}{2} (-1)^{3/4} \sqrt {c} \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}{2} \sqrt [4]{-1} \sqrt {c} \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^2-2 \sqrt [4]{-1} a \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) b-\sqrt [4]{-1} \sqrt {c} \text {ArcTan}\left ((-1)^{3/4} \sqrt {c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right ) b+\frac {i a \log \left (i c x^2+1\right ) b}{x}-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{4 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTan[c*x^2])^2/x^2,x]

[Out]

(-1)^(1/4)*b^2*Sqrt[c]*ArcTan[(-1)^(3/4)*Sqrt[c]*x]^2 - 2*(-1)^(1/4)*a*b*Sqrt[c]*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]
- (-1)^(3/4)*b^2*Sqrt[c]*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]^2 - 2*(-1)^(3/4)*b^2*Sqrt[c]*ArcTan[(-1)^(3/4)*Sqrt[c]
*x]*Log[2/(1 - (-1)^(1/4)*Sqrt[c]*x)] + 2*(-1)^(3/4)*b^2*Sqrt[c]*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 + (-1)^
(1/4)*Sqrt[c]*x)] - (-1)^(3/4)*b^2*Sqrt[c]*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[(Sqrt[2]*((-1)^(1/4) + Sqrt[c]*x))
/(1 + (-1)^(1/4)*Sqrt[c]*x)] + 2*(-1)^(3/4)*b^2*Sqrt[c]*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 - (-1)^(3/4)*Sq
rt[c]*x)] - 2*(-1)^(3/4)*b^2*Sqrt[c]*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 + (-1)^(3/4)*Sqrt[c]*x)] + (-1)^(3
/4)*b^2*Sqrt[c]*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[-((Sqrt[2]*((-1)^(3/4) + Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]
*x))] + (-1)^(3/4)*b^2*Sqrt[c]*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[((1 + I)*(1 + (-1)^(1/4)*Sqrt[c]*x))/(1 + (-1
)^(3/4)*Sqrt[c]*x)] - (-1)^(3/4)*b^2*Sqrt[c]*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[((1 - I)*(1 + (-1)^(3/4)*Sqrt[c]
*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)] - (-1)^(3/4)*b^2*Sqrt[c]*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[1 - I*c*x^2] - (-1
)^(1/4)*b*Sqrt[c]*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*(2*a + I*b*Log[1 - I*c*x^2]) - (2*a + I*b*Log[1 - I*c*x^2])^2/(
4*x) + (I*a*b*Log[1 + I*c*x^2])/x + (-1)^(3/4)*b^2*Sqrt[c]*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[1 + I*c*x^2] + (-1
)^(3/4)*b^2*Sqrt[c]*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[1 + I*c*x^2] - (b^2*Log[1 - I*c*x^2]*Log[1 + I*c*x^2])/(
2*x) + (b^2*Log[1 + I*c*x^2]^2)/(4*x) + (-1)^(1/4)*b^2*Sqrt[c]*PolyLog[2, 1 - 2/(1 - (-1)^(1/4)*Sqrt[c]*x)] +
(-1)^(1/4)*b^2*Sqrt[c]*PolyLog[2, 1 - 2/(1 + (-1)^(1/4)*Sqrt[c]*x)] - ((-1)^(1/4)*b^2*Sqrt[c]*PolyLog[2, 1 - (
Sqrt[2]*((-1)^(1/4) + Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/2 + (-1)^(3/4)*b^2*Sqrt[c]*PolyLog[2, 1 - 2/(1
- (-1)^(3/4)*Sqrt[c]*x)] + (-1)^(3/4)*b^2*Sqrt[c]*PolyLog[2, 1 - 2/(1 + (-1)^(3/4)*Sqrt[c]*x)] - ((-1)^(3/4)*b
^2*Sqrt[c]*PolyLog[2, 1 + (Sqrt[2]*((-1)^(3/4) + Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x)])/2 - ((-1)^(3/4)*b^2*
Sqrt[c]*PolyLog[2, 1 - ((1 + I)*(1 + (-1)^(1/4)*Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x)])/2 - ((-1)^(1/4)*b^2*S
qrt[c]*PolyLog[2, 1 - ((1 - I)*(1 + (-1)^(3/4)*Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/2

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 209

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

Rule 211

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

Rule 212

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

Rule 214

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

Rule 2352

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

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2497

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 2505

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 2507

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 2520

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 2637

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 4950

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

Rule 4964

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

Rule 4966

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 5040

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

Rule 5048

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 6055

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 6057

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 6131

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 6139

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 6874

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^2} \, dx &=\int \left (\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{4 x^2}+\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^2}-\frac {b^2 \log ^2\left (1+i c x^2\right )}{4 x^2}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{x^2} \, 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^2} \, dx-\frac {1}{4} b^2 \int \frac {\log ^2\left (1+i c x^2\right )}{x^2} \, dx\\ &=-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{4 x}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{4 x}+\frac {1}{2} b \int \left (-\frac {2 i a \log \left (1+i c x^2\right )}{x^2}+\frac {b \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{x^2}\right ) \, dx+(b c) \int \frac {2 a+i b \log \left (1-i c x^2\right )}{1-i c x^2} \, dx-\left (i b^2 c\right ) \int \frac {\log \left (1+i c x^2\right )}{1+i c x^2} \, dx\\ &=-\sqrt [4]{-1} b \sqrt {c} \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}{4 x}+(-1)^{3/4} b^2 \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )+\frac {b^2 \log ^2\left (1+i c x^2\right )}{4 x}-(i a b) \int \frac {\log \left (1+i c x^2\right )}{x^2} \, 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^2} \, dx+\left (2 b^2 c^2\right ) \int \frac {\sqrt [4]{-1} x \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c} \left (1-i c x^2\right )} \, dx+\left (2 b^2 c^2\right ) \int \frac {\sqrt [4]{-1} x \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c} \left (1+i c x^2\right )} \, dx\\ &=-\sqrt [4]{-1} b \sqrt {c} \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}{4 x}+\frac {i a b \log \left (1+i c x^2\right )}{x}+(-1)^{3/4} b^2 \sqrt {c} \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 )}{2 x}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{4 x}-\frac {1}{2} b^2 \int \frac {2 c \log \left (1-i c x^2\right )}{i-c x^2} \, dx-\frac {1}{2} b^2 \int \frac {2 c \log \left (1+i c x^2\right )}{-i-c x^2} \, dx+(2 a b c) \int \frac {1}{1+i c x^2} \, dx+\left (2 \sqrt [4]{-1} b^2 c^{3/2}\right ) \int \frac {x \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{1-i c x^2} \, dx+\left (2 \sqrt [4]{-1} b^2 c^{3/2}\right ) \int \frac {x \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{1+i c x^2} \, dx\\ &=\sqrt [4]{-1} b^2 \sqrt {c} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2-2 \sqrt [4]{-1} a b \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )-(-1)^{3/4} b^2 \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2-\sqrt [4]{-1} b \sqrt {c} \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}{4 x}+\frac {i a b \log \left (1+i c x^2\right )}{x}+(-1)^{3/4} b^2 \sqrt {c} \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 )}{2 x}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{4 x}+\left (2 i b^2 c\right ) \int \frac {\tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{i-(-1)^{3/4} \sqrt {c} x} \, dx-\left (2 i b^2 c\right ) \int \frac {\tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{1-(-1)^{3/4} \sqrt {c} x} \, dx-\left (b^2 c\right ) \int \frac {\log \left (1-i c x^2\right )}{i-c x^2} \, dx-\left (b^2 c\right ) \int \frac {\log \left (1+i c x^2\right )}{-i-c x^2} \, dx\\ &=\sqrt [4]{-1} b^2 \sqrt {c} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2-2 \sqrt [4]{-1} a b \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )-(-1)^{3/4} b^2 \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2-2 (-1)^{3/4} b^2 \sqrt {c} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )+2 (-1)^{3/4} b^2 \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )-(-1)^{3/4} b^2 \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )-\sqrt [4]{-1} b \sqrt {c} \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}{4 x}+\frac {i a b \log \left (1+i c x^2\right )}{x}+(-1)^{3/4} b^2 \sqrt {c} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )+(-1)^{3/4} b^2 \sqrt {c} \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 )}{2 x}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{4 x}-\left (2 i b^2 c\right ) \int \frac {\log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )}{1-i c x^2} \, dx+\left (2 i b^2 c\right ) \int \frac {\log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )}{1+i c x^2} \, dx-\left (2 i b^2 c^2\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-\left (2 i b^2 c^2\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\\ &=\sqrt [4]{-1} b^2 \sqrt {c} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2-2 \sqrt [4]{-1} a b \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )-(-1)^{3/4} b^2 \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2-2 (-1)^{3/4} b^2 \sqrt {c} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )+2 (-1)^{3/4} b^2 \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )-(-1)^{3/4} b^2 \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )-\sqrt [4]{-1} b \sqrt {c} \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}{4 x}+\frac {i a b \log \left (1+i c x^2\right )}{x}+(-1)^{3/4} b^2 \sqrt {c} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )+(-1)^{3/4} b^2 \sqrt {c} \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 )}{2 x}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{4 x}+\left (2 \sqrt [4]{-1} b^2 \sqrt {c}\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\sqrt [4]{-1} \sqrt {c} x}\right )+\left (2 (-1)^{3/4} b^2 \sqrt {c}\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-(-1)^{3/4} \sqrt {c} x}\right )+\left (2 \sqrt [4]{-1} b^2 c^{3/2}\right ) \int \frac {x \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{1+i c x^2} \, dx+\left (2 \sqrt [4]{-1} b^2 c^{3/2}\right ) \int \frac {x \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{1-i c x^2} \, dx\\ &=\sqrt [4]{-1} b^2 \sqrt {c} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2-2 \sqrt [4]{-1} a b \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )-(-1)^{3/4} b^2 \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2-2 (-1)^{3/4} b^2 \sqrt {c} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )+2 (-1)^{3/4} b^2 \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )-(-1)^{3/4} b^2 \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )-\sqrt [4]{-1} b \sqrt {c} \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}{4 x}+\frac {i a b \log \left (1+i c x^2\right )}{x}+(-1)^{3/4} b^2 \sqrt {c} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )+(-1)^{3/4} b^2 \sqrt {c} \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 )}{2 x}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{4 x}+\sqrt [4]{-1} b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )+(-1)^{3/4} b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )+\left (2 \sqrt [4]{-1} b^2 c^{3/2}\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+\left (2 \sqrt [4]{-1} b^2 c^{3/2}\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\\ &=\sqrt [4]{-1} b^2 \sqrt {c} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2-2 \sqrt [4]{-1} a b \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )-(-1)^{3/4} b^2 \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2-2 (-1)^{3/4} b^2 \sqrt {c} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )+2 (-1)^{3/4} b^2 \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )-(-1)^{3/4} b^2 \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )-\sqrt [4]{-1} b \sqrt {c} \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}{4 x}+\frac {i a b \log \left (1+i c x^2\right )}{x}+(-1)^{3/4} b^2 \sqrt {c} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )+(-1)^{3/4} b^2 \sqrt {c} \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 )}{2 x}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{4 x}+\sqrt [4]{-1} b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )+(-1)^{3/4} b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )+\left ((-1)^{3/4} b^2 c\right ) \int \frac {\tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt [4]{-1}-\sqrt {c} x} \, dx-\left ((-1)^{3/4} b^2 c\right ) \int \frac {\tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt [4]{-1}+\sqrt {c} x} \, dx-\left ((-1)^{3/4} b^2 c\right ) \int \frac {\tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{-(-1)^{3/4}-\sqrt {c} x} \, dx+\left ((-1)^{3/4} b^2 c\right ) \int \frac {\tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{-(-1)^{3/4}+\sqrt {c} x} \, dx\\ &=\sqrt [4]{-1} b^2 \sqrt {c} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2-2 \sqrt [4]{-1} a b \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )-(-1)^{3/4} b^2 \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2-2 (-1)^{3/4} b^2 \sqrt {c} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )+2 (-1)^{3/4} b^2 \sqrt {c} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1+\sqrt [4]{-1} \sqrt {c} x}\right )-(-1)^{3/4} b^2 \sqrt {c} \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 )+2 (-1)^{3/4} b^2 \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )-2 (-1)^{3/4} b^2 \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1+(-1)^{3/4} \sqrt {c} x}\right )+(-1)^{3/4} b^2 \sqrt {c} \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 )+(-1)^{3/4} b^2 \sqrt {c} \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 )-(-1)^{3/4} b^2 \sqrt {c} \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 )-(-1)^{3/4} b^2 \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )-\sqrt [4]{-1} b \sqrt {c} \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}{4 x}+\frac {i a b \log \left (1+i c x^2\right )}{x}+(-1)^{3/4} b^2 \sqrt {c} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )+(-1)^{3/4} b^2 \sqrt {c} \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 )}{2 x}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{4 x}+\sqrt [4]{-1} b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )+(-1)^{3/4} b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )+2 \left (\left (i b^2 c\right ) \int \frac {\log \left (\frac {2}{1+\sqrt [4]{-1} \sqrt {c} x}\right )}{1-i c x^2} \, dx\right )-\left (i b^2 c\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-\left (i b^2 c\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 (\left (i b^2 c\right ) \int \frac {\log \left (\frac {2}{1+(-1)^{3/4} \sqrt {c} x}\right )}{1+i c x^2} \, dx\right )+\left (i b^2 c\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+\left (i b^2 c\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\\ &=\sqrt [4]{-1} b^2 \sqrt {c} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2-2 \sqrt [4]{-1} a b \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )-(-1)^{3/4} b^2 \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2-2 (-1)^{3/4} b^2 \sqrt {c} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )+2 (-1)^{3/4} b^2 \sqrt {c} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1+\sqrt [4]{-1} \sqrt {c} x}\right )-(-1)^{3/4} b^2 \sqrt {c} \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 )+2 (-1)^{3/4} b^2 \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )-2 (-1)^{3/4} b^2 \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1+(-1)^{3/4} \sqrt {c} x}\right )+(-1)^{3/4} b^2 \sqrt {c} \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 )+(-1)^{3/4} b^2 \sqrt {c} \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 )-(-1)^{3/4} b^2 \sqrt {c} \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 )-(-1)^{3/4} b^2 \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )-\sqrt [4]{-1} b \sqrt {c} \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}{4 x}+\frac {i a b \log \left (1+i c x^2\right )}{x}+(-1)^{3/4} b^2 \sqrt {c} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )+(-1)^{3/4} b^2 \sqrt {c} \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 )}{2 x}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{4 x}+\sqrt [4]{-1} b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {1}{2} \sqrt [4]{-1} b^2 \sqrt {c} \text {Li}_2\left (1-\frac {\sqrt {2} \left (\sqrt [4]{-1}+\sqrt {c} x\right )}{1+\sqrt [4]{-1} \sqrt {c} x}\right )+(-1)^{3/4} b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )-\frac {1}{2} (-1)^{3/4} b^2 \sqrt {c} \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}{2} (-1)^{3/4} b^2 \sqrt {c} \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}{2} \sqrt [4]{-1} b^2 \sqrt {c} \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 (\left (\sqrt [4]{-1} b^2 \sqrt {c}\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\sqrt [4]{-1} \sqrt {c} x}\right )\right )+2 \left (\left ((-1)^{3/4} b^2 \sqrt {c}\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+(-1)^{3/4} \sqrt {c} x}\right )\right )\\ &=\sqrt [4]{-1} b^2 \sqrt {c} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2-2 \sqrt [4]{-1} a b \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )-(-1)^{3/4} b^2 \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2-2 (-1)^{3/4} b^2 \sqrt {c} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )+2 (-1)^{3/4} b^2 \sqrt {c} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1+\sqrt [4]{-1} \sqrt {c} x}\right )-(-1)^{3/4} b^2 \sqrt {c} \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 )+2 (-1)^{3/4} b^2 \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )-2 (-1)^{3/4} b^2 \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1+(-1)^{3/4} \sqrt {c} x}\right )+(-1)^{3/4} b^2 \sqrt {c} \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 )+(-1)^{3/4} b^2 \sqrt {c} \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 )-(-1)^{3/4} b^2 \sqrt {c} \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 )-(-1)^{3/4} b^2 \sqrt {c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )-\sqrt [4]{-1} b \sqrt {c} \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}{4 x}+\frac {i a b \log \left (1+i c x^2\right )}{x}+(-1)^{3/4} b^2 \sqrt {c} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )+(-1)^{3/4} b^2 \sqrt {c} \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 )}{2 x}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{4 x}+\sqrt [4]{-1} b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )+\sqrt [4]{-1} b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2}{1+\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {1}{2} \sqrt [4]{-1} b^2 \sqrt {c} \text {Li}_2\left (1-\frac {\sqrt {2} \left (\sqrt [4]{-1}+\sqrt {c} x\right )}{1+\sqrt [4]{-1} \sqrt {c} x}\right )+(-1)^{3/4} b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )+(-1)^{3/4} b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2}{1+(-1)^{3/4} \sqrt {c} x}\right )-\frac {1}{2} (-1)^{3/4} b^2 \sqrt {c} \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}{2} (-1)^{3/4} b^2 \sqrt {c} \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}{2} \sqrt [4]{-1} b^2 \sqrt {c} \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 [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(5293\) vs. \(2(1164)=2328\).
time = 30.55, size = 5293, normalized size = 4.55 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arctan \left (c \,x^{2}\right )\right )^{2}}{x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(c*(2*sqrt(2)*arctan(1/2*sqrt(2)*(2*c*x + sqrt(2)*sqrt(c))/sqrt(c))/sqrt(c) + 2*sqrt(2)*arctan(1/2*sqrt(2)
*(2*c*x - sqrt(2)*sqrt(c))/sqrt(c))/sqrt(c) + sqrt(2)*log(c*x^2 + sqrt(2)*sqrt(c)*x + 1)/sqrt(c) - sqrt(2)*log
(c*x^2 - sqrt(2)*sqrt(c)*x + 1)/sqrt(c)) - 4*arctan(c*x^2)/x)*a*b - 1/16*(4*arctan(c*x^2)^2 - 16*x*integrate(-
1/16*(8*c^2*x^4*log(c^2*x^4 + 1) - 16*c*x^2*arctan(c*x^2) - 12*(c^2*x^4 + 1)*arctan(c*x^2)^2 - (c^2*x^4 + 1)*l
og(c^2*x^4 + 1)^2)/(c^2*x^6 + x^2), x) - log(c^2*x^4 + 1)^2)*b^2/x - a^2/x

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atan(c*x^2))^2/x^2,x)

[Out]

int((a + b*atan(c*x^2))^2/x^2, x)

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