3.1.82 \(\int (a+b \text {ArcTan}(c x^2))^2 \, dx\) [82]

Optimal. Leaf size=1191 \[ a^2 x-\frac {2 (-1)^{3/4} a b \text {ArcTan}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}+\frac {(-1)^{3/4} b^2 \text {ArcTan}\left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}+\frac {2 (-1)^{3/4} a b \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}-\frac {\sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}+\frac {2 \sqrt [4]{-1} b^2 \text {ArcTan}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )}{\sqrt {c}}-\frac {2 \sqrt [4]{-1} b^2 \text {ArcTan}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1+\sqrt [4]{-1} \sqrt {c} x}\right )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^2 \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 )}{\sqrt {c}}+\frac {2 \sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )}{\sqrt {c}}-\frac {2 \sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1+(-1)^{3/4} \sqrt {c} x}\right )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^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 )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^2 \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 )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^2 \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 )}{\sqrt {c}}+i a b x \log \left (1-i c x^2\right )+\frac {\sqrt [4]{-1} b^2 \text {ArcTan}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )}{\sqrt {c}}-\frac {\sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )}{\sqrt {c}}-\frac {1}{4} b^2 x \log ^2\left (1-i c x^2\right )-i a b x \log \left (1+i c x^2\right )-\frac {\sqrt [4]{-1} b^2 \text {ArcTan}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )}{\sqrt {c}}+\frac {1}{2} b^2 x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac {1}{4} b^2 x \log ^2\left (1+i c x^2\right )+\frac {(-1)^{3/4} b^2 \text {PolyLog}\left (2,1-\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )}{\sqrt {c}}+\frac {(-1)^{3/4} b^2 \text {PolyLog}\left (2,1-\frac {2}{1+\sqrt [4]{-1} \sqrt {c} x}\right )}{\sqrt {c}}-\frac {(-1)^{3/4} b^2 \text {PolyLog}\left (2,1-\frac {\sqrt {2} \left (\sqrt [4]{-1}+\sqrt {c} x\right )}{1+\sqrt [4]{-1} \sqrt {c} x}\right )}{2 \sqrt {c}}+\frac {\sqrt [4]{-1} b^2 \text {PolyLog}\left (2,1-\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^2 \text {PolyLog}\left (2,1-\frac {2}{1+(-1)^{3/4} \sqrt {c} x}\right )}{\sqrt {c}}-\frac {\sqrt [4]{-1} b^2 \text {PolyLog}\left (2,1+\frac {\sqrt {2} \left ((-1)^{3/4}+\sqrt {c} x\right )}{1+(-1)^{3/4} \sqrt {c} x}\right )}{2 \sqrt {c}}-\frac {\sqrt [4]{-1} b^2 \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 )}{2 \sqrt {c}}-\frac {(-1)^{3/4} b^2 \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 )}{2 \sqrt {c}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 1.23, antiderivative size = 1191, normalized size of antiderivative = 1.00, number of steps used = 69, number of rules used = 23, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.917, Rules used = {4932, 2498, 327, 209, 2500, 2526, 2520, 12, 5040, 4964, 2449, 2352, 212, 2636, 211, 5048, 4966, 2497, 214, 6139, 6057, 6131, 6055} \begin {gather*} x a^2-\frac {2 (-1)^{3/4} b \text {ArcTan}\left ((-1)^{3/4} \sqrt {c} x\right ) a}{\sqrt {c}}+\frac {2 (-1)^{3/4} b \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) a}{\sqrt {c}}+i b x \log \left (1-i c x^2\right ) a-i b x \log \left (i c x^2+1\right ) a+\frac {(-1)^{3/4} b^2 \text {ArcTan}\left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}-\frac {\sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}-\frac {1}{4} b^2 x \log ^2\left (1-i c x^2\right )-\frac {1}{4} b^2 x \log ^2\left (i c x^2+1\right )+\frac {2 \sqrt [4]{-1} b^2 \text {ArcTan}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )}{\sqrt {c}}-\frac {2 \sqrt [4]{-1} b^2 \text {ArcTan}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{\sqrt [4]{-1} \sqrt {c} x+1}\right )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^2 \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 )}{\sqrt {c}}+\frac {2 \sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )}{\sqrt {c}}-\frac {2 \sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{(-1)^{3/4} \sqrt {c} x+1}\right )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^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 )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^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 )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^2 \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 )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^2 \text {ArcTan}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )}{\sqrt {c}}-\frac {\sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )}{\sqrt {c}}-\frac {\sqrt [4]{-1} b^2 \text {ArcTan}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (i c x^2+1\right )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (i c x^2+1\right )}{\sqrt {c}}+\frac {1}{2} b^2 x \log \left (1-i c x^2\right ) \log \left (i c x^2+1\right )+\frac {(-1)^{3/4} b^2 \text {Li}_2\left (1-\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )}{\sqrt {c}}+\frac {(-1)^{3/4} b^2 \text {Li}_2\left (1-\frac {2}{\sqrt [4]{-1} \sqrt {c} x+1}\right )}{\sqrt {c}}-\frac {(-1)^{3/4} b^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 )}{2 \sqrt {c}}+\frac {\sqrt [4]{-1} b^2 \text {Li}_2\left (1-\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^2 \text {Li}_2\left (1-\frac {2}{(-1)^{3/4} \sqrt {c} x+1}\right )}{\sqrt {c}}-\frac {\sqrt [4]{-1} b^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 )}{2 \sqrt {c}}-\frac {\sqrt [4]{-1} b^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 )}{2 \sqrt {c}}-\frac {(-1)^{3/4} b^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 )}{2 \sqrt {c}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTan[c*x^2])^2,x]

[Out]

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

Rule 12

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

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 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

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 2498

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

Rule 2500

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

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 2526

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 2636

Int[Log[v_]*Log[w_], x_Symbol] :> Simp[x*Log[v]*Log[w], x] + (-Int[SimplifyIntegrand[x*Log[w]*(D[v, x]/v), x],
x] - Int[SimplifyIntegrand[x*Log[v]*(D[w, x]/w), x], x]) /; InverseFunctionFreeQ[v, x] && InverseFunctionFree
Q[w, x]

Rule 4932

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_)]*(b_.))^(p_), x_Symbol] :> Int[ExpandIntegrand[(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}, x] && IGtQ[p, 1] && IGtQ[n, 0]

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])

Rubi steps

\begin {align*} \int \left (a+b \tan ^{-1}\left (c x^2\right )\right )^2 \, dx &=\int \left (a^2+i a b \log \left (1-i c x^2\right )-\frac {1}{4} b^2 \log ^2\left (1-i c x^2\right )-i a b \log \left (1+i c x^2\right )+\frac {1}{2} b^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac {1}{4} b^2 \log ^2\left (1+i c x^2\right )\right ) \, dx\\ &=a^2 x+(i a b) \int \log \left (1-i c x^2\right ) \, dx-(i a b) \int \log \left (1+i c x^2\right ) \, dx-\frac {1}{4} b^2 \int \log ^2\left (1-i c x^2\right ) \, dx-\frac {1}{4} b^2 \int \log ^2\left (1+i c x^2\right ) \, dx+\frac {1}{2} b^2 \int \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right ) \, dx\\ &=a^2 x+i a b x \log \left (1-i c x^2\right )-\frac {1}{4} b^2 x \log ^2\left (1-i c x^2\right )-i a b x \log \left (1+i c x^2\right )+\frac {1}{2} b^2 x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac {1}{4} b^2 x \log ^2\left (1+i c x^2\right )-\frac {1}{2} b^2 \int \frac {2 c x^2 \log \left (1-i c x^2\right )}{-i+c x^2} \, dx-\frac {1}{2} b^2 \int \frac {2 c x^2 \log \left (1+i c x^2\right )}{i+c x^2} \, dx-(2 a b c) \int \frac {x^2}{1-i c x^2} \, dx-(2 a b c) \int \frac {x^2}{1+i c x^2} \, dx-\left (i b^2 c\right ) \int \frac {x^2 \log \left (1-i c x^2\right )}{1-i c x^2} \, dx+\left (i b^2 c\right ) \int \frac {x^2 \log \left (1+i c x^2\right )}{1+i c x^2} \, dx\\ &=a^2 x+i a b x \log \left (1-i c x^2\right )-\frac {1}{4} b^2 x \log ^2\left (1-i c x^2\right )-i a b x \log \left (1+i c x^2\right )+\frac {1}{2} b^2 x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac {1}{4} b^2 x \log ^2\left (1+i c x^2\right )+(2 i a b) \int \frac {1}{1-i c x^2} \, dx-(2 i a b) \int \frac {1}{1+i c x^2} \, dx-\left (i b^2 c\right ) \int \left (\frac {i \log \left (1-i c x^2\right )}{c}-\frac {i \log \left (1-i c x^2\right )}{c \left (1-i c x^2\right )}\right ) \, dx+\left (i b^2 c\right ) \int \left (-\frac {i \log \left (1+i c x^2\right )}{c}+\frac {i \log \left (1+i c x^2\right )}{c \left (1+i c x^2\right )}\right ) \, dx-\left (b^2 c\right ) \int \frac {x^2 \log \left (1-i c x^2\right )}{-i+c x^2} \, dx-\left (b^2 c\right ) \int \frac {x^2 \log \left (1+i c x^2\right )}{i+c x^2} \, dx\\ &=a^2 x-\frac {2 (-1)^{3/4} a b \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}+\frac {2 (-1)^{3/4} a b \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}+i a b x \log \left (1-i c x^2\right )-\frac {1}{4} b^2 x \log ^2\left (1-i c x^2\right )-i a b x \log \left (1+i c x^2\right )+\frac {1}{2} b^2 x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac {1}{4} b^2 x \log ^2\left (1+i c x^2\right )+b^2 \int \log \left (1-i c x^2\right ) \, dx-b^2 \int \frac {\log \left (1-i c x^2\right )}{1-i c x^2} \, dx+b^2 \int \log \left (1+i c x^2\right ) \, dx-b^2 \int \frac {\log \left (1+i c x^2\right )}{1+i c x^2} \, dx-\left (b^2 c\right ) \int \left (\frac {\log \left (1-i c x^2\right )}{c}+\frac {i \log \left (1-i c x^2\right )}{c \left (-i+c x^2\right )}\right ) \, dx-\left (b^2 c\right ) \int \left (\frac {\log \left (1+i c x^2\right )}{c}-\frac {i \log \left (1+i c x^2\right )}{c \left (i+c x^2\right )}\right ) \, dx\\ &=a^2 x-\frac {2 (-1)^{3/4} a b \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}+\frac {2 (-1)^{3/4} a b \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}+i a b x \log \left (1-i c x^2\right )+b^2 x \log \left (1-i c x^2\right )+\frac {\sqrt [4]{-1} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )}{\sqrt {c}}-\frac {1}{4} b^2 x \log ^2\left (1-i c x^2\right )-i a b x \log \left (1+i c x^2\right )+b^2 x \log \left (1+i c x^2\right )+\frac {\sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )}{\sqrt {c}}+\frac {1}{2} b^2 x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac {1}{4} b^2 x \log ^2\left (1+i c x^2\right )-\left (i b^2\right ) \int \frac {\log \left (1-i c x^2\right )}{-i+c x^2} \, dx+\left (i b^2\right ) \int \frac {\log \left (1+i c x^2\right )}{i+c x^2} \, dx-b^2 \int \log \left (1-i c x^2\right ) \, dx-b^2 \int \log \left (1+i c x^2\right ) \, dx+\left (2 i b^2 c\right ) \int \frac {x^2}{1-i c x^2} \, dx-\left (2 i b^2 c\right ) \int \frac {x^2}{1+i c x^2} \, dx+\left (2 i b^2 c\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 i b^2 c\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\\ &=a^2 x-4 b^2 x-\frac {2 (-1)^{3/4} a b \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}+\frac {2 (-1)^{3/4} a b \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}+i a b x \log \left (1-i c x^2\right )+\frac {\sqrt [4]{-1} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )}{\sqrt {c}}-\frac {\sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )}{\sqrt {c}}-\frac {1}{4} b^2 x \log ^2\left (1-i c x^2\right )-i a b x \log \left (1+i c x^2\right )-\frac {\sqrt [4]{-1} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )}{\sqrt {c}}+\frac {1}{2} b^2 x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac {1}{4} b^2 x \log ^2\left (1+i c x^2\right )+\left (2 b^2\right ) \int \frac {1}{1-i c x^2} \, dx+\left (2 b^2\right ) \int \frac {1}{1+i c x^2} \, dx+\left (2 (-1)^{3/4} b^2 \sqrt {c}\right ) \int \frac {x \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{1-i c x^2} \, dx-\left (2 (-1)^{3/4} b^2 \sqrt {c}\right ) \int \frac {x \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{1+i c x^2} \, dx-\left (2 i b^2 c\right ) \int \frac {x^2}{1-i c x^2} \, dx+\left (2 i b^2 c\right ) \int \frac {x^2}{1+i c x^2} \, dx+\left (2 b^2 c\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 b^2 c\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\\ &=a^2 x-\frac {2 (-1)^{3/4} a b \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}-\frac {2 \sqrt [4]{-1} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}+\frac {(-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}+\frac {2 (-1)^{3/4} a b \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}-\frac {2 \sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}-\frac {\sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}+i a b x \log \left (1-i c x^2\right )+\frac {\sqrt [4]{-1} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )}{\sqrt {c}}-\frac {\sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )}{\sqrt {c}}-\frac {1}{4} b^2 x \log ^2\left (1-i c x^2\right )-i a b x \log \left (1+i c x^2\right )-\frac {\sqrt [4]{-1} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )}{\sqrt {c}}+\frac {1}{2} b^2 x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac {1}{4} b^2 x \log ^2\left (1+i c x^2\right )-\left (2 b^2\right ) \int \frac {1}{1-i c x^2} \, dx-\left (2 b^2\right ) \int \frac {1}{1+i c x^2} \, dx-\left (2 b^2\right ) \int \frac {\tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{i-(-1)^{3/4} \sqrt {c} x} \, dx-\left (2 b^2\right ) \int \frac {\tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{1-(-1)^{3/4} \sqrt {c} x} \, dx+\left (2 (-1)^{3/4} b^2 \sqrt {c}\right ) \int \frac {x \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{1+i c x^2} \, dx-\left (2 (-1)^{3/4} b^2 \sqrt {c}\right ) \int \frac {x \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{1-i c x^2} \, dx\\ &=a^2 x-\frac {2 (-1)^{3/4} a b \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}+\frac {(-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}+\frac {2 (-1)^{3/4} a b \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}-\frac {\sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}+\frac {2 \sqrt [4]{-1} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )}{\sqrt {c}}+\frac {2 \sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )}{\sqrt {c}}+i a b x \log \left (1-i c x^2\right )+\frac {\sqrt [4]{-1} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )}{\sqrt {c}}-\frac {\sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )}{\sqrt {c}}-\frac {1}{4} b^2 x \log ^2\left (1-i c x^2\right )-i a b x \log \left (1+i c x^2\right )-\frac {\sqrt [4]{-1} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )}{\sqrt {c}}+\frac {1}{2} b^2 x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac {1}{4} b^2 x \log ^2\left (1+i c x^2\right )+\left (2 b^2\right ) \int \frac {\log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )}{1-i c x^2} \, dx+\left (2 b^2\right ) \int \frac {\log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )}{1+i c x^2} \, dx+\left (2 (-1)^{3/4} b^2 \sqrt {c}\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 (-1)^{3/4} b^2 \sqrt {c}\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\\ &=a^2 x-\frac {2 (-1)^{3/4} a b \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}+\frac {(-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}+\frac {2 (-1)^{3/4} a b \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}-\frac {\sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}+\frac {2 \sqrt [4]{-1} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )}{\sqrt {c}}+\frac {2 \sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )}{\sqrt {c}}+i a b x \log \left (1-i c x^2\right )+\frac {\sqrt [4]{-1} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )}{\sqrt {c}}-\frac {\sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )}{\sqrt {c}}-\frac {1}{4} b^2 x \log ^2\left (1-i c x^2\right )-i a b x \log \left (1+i c x^2\right )-\frac {\sqrt [4]{-1} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )}{\sqrt {c}}+\frac {1}{2} b^2 x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac {1}{4} b^2 x \log ^2\left (1+i c x^2\right )-\left (\sqrt [4]{-1} b^2\right ) \int \frac {\tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt [4]{-1}-\sqrt {c} x} \, dx+\left (\sqrt [4]{-1} b^2\right ) \int \frac {\tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt [4]{-1}+\sqrt {c} x} \, dx-\left (\sqrt [4]{-1} b^2\right ) \int \frac {\tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{-(-1)^{3/4}-\sqrt {c} x} \, dx+\left (\sqrt [4]{-1} b^2\right ) \int \frac {\tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{-(-1)^{3/4}+\sqrt {c} x} \, dx+\frac {\left (2 \sqrt [4]{-1} b^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-(-1)^{3/4} \sqrt {c} x}\right )}{\sqrt {c}}+\frac {\left (2 (-1)^{3/4} b^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\sqrt [4]{-1} \sqrt {c} x}\right )}{\sqrt {c}}\\ &=a^2 x-\frac {2 (-1)^{3/4} a b \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}+\frac {(-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}+\frac {2 (-1)^{3/4} a b \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}-\frac {\sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}+\frac {2 \sqrt [4]{-1} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )}{\sqrt {c}}-\frac {2 \sqrt [4]{-1} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1+\sqrt [4]{-1} \sqrt {c} x}\right )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^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 )}{\sqrt {c}}+\frac {2 \sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )}{\sqrt {c}}-\frac {2 \sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1+(-1)^{3/4} \sqrt {c} x}\right )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^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 )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^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 )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^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 )}{\sqrt {c}}+i a b x \log \left (1-i c x^2\right )+\frac {\sqrt [4]{-1} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )}{\sqrt {c}}-\frac {\sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )}{\sqrt {c}}-\frac {1}{4} b^2 x \log ^2\left (1-i c x^2\right )-i a b x \log \left (1+i c x^2\right )-\frac {\sqrt [4]{-1} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )}{\sqrt {c}}+\frac {1}{2} b^2 x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac {1}{4} b^2 x \log ^2\left (1+i c x^2\right )+\frac {(-1)^{3/4} b^2 \text {Li}_2\left (1-\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^2 \text {Li}_2\left (1-\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )}{\sqrt {c}}-2 \left (b^2 \int \frac {\log \left (\frac {2}{1+\sqrt [4]{-1} \sqrt {c} x}\right )}{1-i c x^2} \, dx\right )+b^2 \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+b^2 \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 (b^2 \int \frac {\log \left (\frac {2}{1+(-1)^{3/4} \sqrt {c} x}\right )}{1+i c x^2} \, dx\right )+b^2 \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+b^2 \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\\ &=a^2 x-\frac {2 (-1)^{3/4} a b \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}+\frac {(-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}+\frac {2 (-1)^{3/4} a b \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}-\frac {\sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}+\frac {2 \sqrt [4]{-1} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )}{\sqrt {c}}-\frac {2 \sqrt [4]{-1} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1+\sqrt [4]{-1} \sqrt {c} x}\right )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^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 )}{\sqrt {c}}+\frac {2 \sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )}{\sqrt {c}}-\frac {2 \sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1+(-1)^{3/4} \sqrt {c} x}\right )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^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 )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^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 )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^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 )}{\sqrt {c}}+i a b x \log \left (1-i c x^2\right )+\frac {\sqrt [4]{-1} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )}{\sqrt {c}}-\frac {\sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )}{\sqrt {c}}-\frac {1}{4} b^2 x \log ^2\left (1-i c x^2\right )-i a b x \log \left (1+i c x^2\right )-\frac {\sqrt [4]{-1} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )}{\sqrt {c}}+\frac {1}{2} b^2 x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac {1}{4} b^2 x \log ^2\left (1+i c x^2\right )+\frac {(-1)^{3/4} b^2 \text {Li}_2\left (1-\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )}{\sqrt {c}}-\frac {(-1)^{3/4} b^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 )}{2 \sqrt {c}}+\frac {\sqrt [4]{-1} b^2 \text {Li}_2\left (1-\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )}{\sqrt {c}}-\frac {\sqrt [4]{-1} b^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 )}{2 \sqrt {c}}-\frac {\sqrt [4]{-1} b^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 )}{2 \sqrt {c}}-\frac {(-1)^{3/4} b^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 \sqrt {c}}+2 \frac {\left (\sqrt [4]{-1} b^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+(-1)^{3/4} \sqrt {c} x}\right )}{\sqrt {c}}+2 \frac {\left ((-1)^{3/4} b^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\sqrt [4]{-1} \sqrt {c} x}\right )}{\sqrt {c}}\\ &=a^2 x-\frac {2 (-1)^{3/4} a b \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}+\frac {(-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}+\frac {2 (-1)^{3/4} a b \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}-\frac {\sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}+\frac {2 \sqrt [4]{-1} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )}{\sqrt {c}}-\frac {2 \sqrt [4]{-1} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1+\sqrt [4]{-1} \sqrt {c} x}\right )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^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 )}{\sqrt {c}}+\frac {2 \sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )}{\sqrt {c}}-\frac {2 \sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1+(-1)^{3/4} \sqrt {c} x}\right )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^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 )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^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 )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^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 )}{\sqrt {c}}+i a b x \log \left (1-i c x^2\right )+\frac {\sqrt [4]{-1} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )}{\sqrt {c}}-\frac {\sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )}{\sqrt {c}}-\frac {1}{4} b^2 x \log ^2\left (1-i c x^2\right )-i a b x \log \left (1+i c x^2\right )-\frac {\sqrt [4]{-1} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )}{\sqrt {c}}+\frac {1}{2} b^2 x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac {1}{4} b^2 x \log ^2\left (1+i c x^2\right )+\frac {(-1)^{3/4} b^2 \text {Li}_2\left (1-\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )}{\sqrt {c}}+\frac {(-1)^{3/4} b^2 \text {Li}_2\left (1-\frac {2}{1+\sqrt [4]{-1} \sqrt {c} x}\right )}{\sqrt {c}}-\frac {(-1)^{3/4} b^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 )}{2 \sqrt {c}}+\frac {\sqrt [4]{-1} b^2 \text {Li}_2\left (1-\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^2 \text {Li}_2\left (1-\frac {2}{1+(-1)^{3/4} \sqrt {c} x}\right )}{\sqrt {c}}-\frac {\sqrt [4]{-1} b^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 )}{2 \sqrt {c}}-\frac {\sqrt [4]{-1} b^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 )}{2 \sqrt {c}}-\frac {(-1)^{3/4} b^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 \sqrt {c}}\\ \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. \(2(1191)=2382\).
time = 30.76, size = 5293, normalized size = 4.44 \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]

[Out]

Result too large to show

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

-1/2*(c*(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*sqrt(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)*lo
g(c*x^2 - sqrt(2)*sqrt(c)*x + 1)/c^(3/2)) - 4*x*arctan(c*x^2))*a*b + 1/16*(4*x*arctan(c*x^2)^2 - x*log(c^2*x^4
+ 1)^2 + 16*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)*log(c^2*x^4 + 1)^2)/(c^2*x^4 + 1), x))*b^2 + 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, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*atan(c*x**2))**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, algorithm="giac")

[Out]

integrate((b*arctan(c*x^2) + a)^2, x)

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

[Out]

int((a + b*atan(c*x^2))^2, x)

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