\(\int (d+e x) (a+b \arctan (c x^2))^2 \, dx\) [25]

Optimal result

Integrand size = 18, antiderivative size = 1325 \[ \int (d+e x) \left (a+b \arctan \left (c x^2\right )\right )^2 \, dx=a^2 d x-\frac {2 (-1)^{3/4} a b d \arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}+\frac {(-1)^{3/4} b^2 d \arctan \left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}+\frac {i e \left (a+b \arctan \left (c x^2\right )\right )^2}{2 c}+\frac {1}{2} e x^2 \left (a+b \arctan \left (c x^2\right )\right )^2+\frac {2 (-1)^{3/4} a b d \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}-\frac {\sqrt [4]{-1} b^2 d \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}+\frac {2 \sqrt [4]{-1} b^2 d \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 d \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 d \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 d \text {arctanh}\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 d \text {arctanh}\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 d \text {arctanh}\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 d \text {arctanh}\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 d \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 d x \log \left (1-i c x^2\right )+\frac {\sqrt [4]{-1} b^2 d \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 d \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )}{\sqrt {c}}-\frac {1}{4} b^2 d x \log ^2\left (1-i c x^2\right )+\frac {b e \left (a+b \arctan \left (c x^2\right )\right ) \log \left (\frac {2}{1+i c x^2}\right )}{c}-i a b d x \log \left (1+i c x^2\right )-\frac {\sqrt [4]{-1} b^2 d \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 d \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )}{\sqrt {c}}+\frac {1}{2} b^2 d x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac {1}{4} b^2 d x \log ^2\left (1+i c x^2\right )+\frac {(-1)^{3/4} b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )}{\sqrt {c}}+\frac {(-1)^{3/4} b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1+\sqrt [4]{-1} \sqrt {c} x}\right )}{\sqrt {c}}-\frac {(-1)^{3/4} b^2 d \operatorname {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 d \operatorname {PolyLog}\left (2,1-\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1+(-1)^{3/4} \sqrt {c} x}\right )}{\sqrt {c}}-\frac {\sqrt [4]{-1} b^2 d \operatorname {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 d \operatorname {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 d \operatorname {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}}+\frac {i b^2 e \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x^2}\right )}{2 c} \]

[Out]

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

Rubi [A] (verified)

Time = 1.58 (sec) , antiderivative size = 1325, normalized size of antiderivative = 1.00, number of steps used = 77, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.444, Rules used = {4982, 4932, 2498, 327, 209, 2500, 2526, 2520, 12, 5040, 4964, 2449, 2352, 212, 2636, 211, 5048, 4966, 2497, 214, 6139, 6057, 6131, 6055, 4948, 4930} \[ \int (d+e x) \left (a+b \arctan \left (c x^2\right )\right )^2 \, dx=d x a^2-\frac {2 (-1)^{3/4} b d \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) a}{\sqrt {c}}+\frac {2 (-1)^{3/4} b d \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) a}{\sqrt {c}}+i b d x \log \left (1-i c x^2\right ) a-i b d x \log \left (i c x^2+1\right ) a+\frac {(-1)^{3/4} b^2 d \arctan \left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}+\frac {1}{2} e x^2 \left (a+b \arctan \left (c x^2\right )\right )^2+\frac {i e \left (a+b \arctan \left (c x^2\right )\right )^2}{2 c}-\frac {\sqrt [4]{-1} b^2 d \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}-\frac {1}{4} b^2 d x \log ^2\left (1-i c x^2\right )-\frac {1}{4} b^2 d x \log ^2\left (i c x^2+1\right )+\frac {2 \sqrt [4]{-1} b^2 d \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 d \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 d \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 d \text {arctanh}\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 d \text {arctanh}\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 d \text {arctanh}\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 d \text {arctanh}\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 d \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 d \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 d \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )}{\sqrt {c}}+\frac {b e \left (a+b \arctan \left (c x^2\right )\right ) \log \left (\frac {2}{i c x^2+1}\right )}{c}-\frac {\sqrt [4]{-1} b^2 d \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 d \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (i c x^2+1\right )}{\sqrt {c}}+\frac {1}{2} b^2 d x \log \left (1-i c x^2\right ) \log \left (i c x^2+1\right )+\frac {(-1)^{3/4} b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )}{\sqrt {c}}+\frac {(-1)^{3/4} b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{\sqrt [4]{-1} \sqrt {c} x+1}\right )}{\sqrt {c}}-\frac {(-1)^{3/4} b^2 d \operatorname {PolyLog}\left (2,1-\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 d \operatorname {PolyLog}\left (2,1-\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{(-1)^{3/4} \sqrt {c} x+1}\right )}{\sqrt {c}}-\frac {\sqrt [4]{-1} b^2 d \operatorname {PolyLog}\left (2,\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 d \operatorname {PolyLog}\left (2,1-\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 d \operatorname {PolyLog}\left (2,1-\frac {(1-i) \left ((-1)^{3/4} \sqrt {c} x+1\right )}{\sqrt [4]{-1} \sqrt {c} x+1}\right )}{2 \sqrt {c}}+\frac {i b^2 e \operatorname {PolyLog}\left (2,1-\frac {2}{i c x^2+1}\right )}{2 c} \]

[In]

Int[(d + e*x)*(a + b*ArcTan[c*x^2])^2,x]

[Out]

a^2*d*x - (2*(-1)^(3/4)*a*b*d*ArcTan[(-1)^(3/4)*Sqrt[c]*x])/Sqrt[c] + ((-1)^(3/4)*b^2*d*ArcTan[(-1)^(3/4)*Sqrt
[c]*x]^2)/Sqrt[c] + ((I/2)*e*(a + b*ArcTan[c*x^2])^2)/c + (e*x^2*(a + b*ArcTan[c*x^2])^2)/2 + (2*(-1)^(3/4)*a*
b*d*ArcTanh[(-1)^(3/4)*Sqrt[c]*x])/Sqrt[c] - ((-1)^(1/4)*b^2*d*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]^2)/Sqrt[c] + (2*(
-1)^(1/4)*b^2*d*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 - (-1)^(1/4)*Sqrt[c]*x)])/Sqrt[c] - (2*(-1)^(1/4)*b^2*d*
ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 + (-1)^(1/4)*Sqrt[c]*x)])/Sqrt[c] + ((-1)^(1/4)*b^2*d*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*d*A
rcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 - (-1)^(3/4)*Sqrt[c]*x)])/Sqrt[c] - (2*(-1)^(1/4)*b^2*d*ArcTanh[(-1)^(3/
4)*Sqrt[c]*x]*Log[2/(1 + (-1)^(3/4)*Sqrt[c]*x)])/Sqrt[c] + ((-1)^(1/4)*b^2*d*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*d*ArcTanh[(-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*d*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*d*x*Log[1 - I*c*x^2] + ((-1)^(1/4)*b^2*d*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[1 - I*c*x^2])/Sqrt[c] - ((-
1)^(1/4)*b^2*d*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[1 - I*c*x^2])/Sqrt[c] - (b^2*d*x*Log[1 - I*c*x^2]^2)/4 + (b*e
*(a + b*ArcTan[c*x^2])*Log[2/(1 + I*c*x^2)])/c - I*a*b*d*x*Log[1 + I*c*x^2] - ((-1)^(1/4)*b^2*d*ArcTan[(-1)^(3
/4)*Sqrt[c]*x]*Log[1 + I*c*x^2])/Sqrt[c] + ((-1)^(1/4)*b^2*d*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[1 + I*c*x^2])/S
qrt[c] + (b^2*d*x*Log[1 - I*c*x^2]*Log[1 + I*c*x^2])/2 - (b^2*d*x*Log[1 + I*c*x^2]^2)/4 + ((-1)^(3/4)*b^2*d*Po
lyLog[2, 1 - 2/(1 - (-1)^(1/4)*Sqrt[c]*x)])/Sqrt[c] + ((-1)^(3/4)*b^2*d*PolyLog[2, 1 - 2/(1 + (-1)^(1/4)*Sqrt[
c]*x)])/Sqrt[c] - ((-1)^(3/4)*b^2*d*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*d*PolyLog[2, 1 - 2/(1 - (-1)^(3/4)*Sqrt[c]*x)])/Sqrt[c] + ((-1)^(1/4)*b^2*d
*PolyLog[2, 1 - 2/(1 + (-1)^(3/4)*Sqrt[c]*x)])/Sqrt[c] - ((-1)^(1/4)*b^2*d*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*d*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*d*PolyLog[2, 1 - ((1 - I)*(1 + (-
1)^(3/4)*Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/(2*Sqrt[c]) + ((I/2)*b^2*e*PolyLog[2, 1 - 2/(1 + I*c*x^2)])/
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 4930

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

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 4948

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

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 4982

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(
a + b*ArcTan[c*x^n])^p, (d + e*x)^m, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 1] && IGtQ[m, 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*} \text {integral}& = \int \left (d \left (a+b \arctan \left (c x^2\right )\right )^2+e x \left (a+b \arctan \left (c x^2\right )\right )^2\right ) \, dx \\ & = d \int \left (a+b \arctan \left (c x^2\right )\right )^2 \, dx+e \int x \left (a+b \arctan \left (c x^2\right )\right )^2 \, dx \\ & = d \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+\frac {1}{2} e \text {Subst}\left (\int (a+b \arctan (c x))^2 \, dx,x,x^2\right ) \\ & = a^2 d x+\frac {1}{2} e x^2 \left (a+b \arctan \left (c x^2\right )\right )^2+(i a b d) \int \log \left (1-i c x^2\right ) \, dx-(i a b d) \int \log \left (1+i c x^2\right ) \, dx-\frac {1}{4} \left (b^2 d\right ) \int \log ^2\left (1-i c x^2\right ) \, dx-\frac {1}{4} \left (b^2 d\right ) \int \log ^2\left (1+i c x^2\right ) \, dx+\frac {1}{2} \left (b^2 d\right ) \int \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right ) \, dx-(b c e) \text {Subst}\left (\int \frac {x (a+b \arctan (c x))}{1+c^2 x^2} \, dx,x,x^2\right ) \\ & = a^2 d x+\frac {i e \left (a+b \arctan \left (c x^2\right )\right )^2}{2 c}+\frac {1}{2} e x^2 \left (a+b \arctan \left (c x^2\right )\right )^2+i a b d x \log \left (1-i c x^2\right )-\frac {1}{4} b^2 d x \log ^2\left (1-i c x^2\right )-i a b d x \log \left (1+i c x^2\right )+\frac {1}{2} b^2 d x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac {1}{4} b^2 d x \log ^2\left (1+i c x^2\right )-\frac {1}{2} \left (b^2 d\right ) \int \frac {2 c x^2 \log \left (1-i c x^2\right )}{-i+c x^2} \, dx-\frac {1}{2} \left (b^2 d\right ) \int \frac {2 c x^2 \log \left (1+i c x^2\right )}{i+c x^2} \, dx-(2 a b c d) \int \frac {x^2}{1-i c x^2} \, dx-(2 a b c d) \int \frac {x^2}{1+i c x^2} \, dx-\left (i b^2 c d\right ) \int \frac {x^2 \log \left (1-i c x^2\right )}{1-i c x^2} \, dx+\left (i b^2 c d\right ) \int \frac {x^2 \log \left (1+i c x^2\right )}{1+i c x^2} \, dx+(b e) \text {Subst}\left (\int \frac {a+b \arctan (c x)}{i-c x} \, dx,x,x^2\right ) \\ & = a^2 d x+\frac {i e \left (a+b \arctan \left (c x^2\right )\right )^2}{2 c}+\frac {1}{2} e x^2 \left (a+b \arctan \left (c x^2\right )\right )^2+i a b d x \log \left (1-i c x^2\right )-\frac {1}{4} b^2 d x \log ^2\left (1-i c x^2\right )+\frac {b e \left (a+b \arctan \left (c x^2\right )\right ) \log \left (\frac {2}{1+i c x^2}\right )}{c}-i a b d x \log \left (1+i c x^2\right )+\frac {1}{2} b^2 d x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac {1}{4} b^2 d x \log ^2\left (1+i c x^2\right )+(2 i a b d) \int \frac {1}{1-i c x^2} \, dx-(2 i a b d) \int \frac {1}{1+i c x^2} \, dx-\left (i b^2 c d\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 d\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 d\right ) \int \frac {x^2 \log \left (1-i c x^2\right )}{-i+c x^2} \, dx-\left (b^2 c d\right ) \int \frac {x^2 \log \left (1+i c x^2\right )}{i+c x^2} \, dx-\left (b^2 e\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx,x,x^2\right ) \\ & = a^2 d x-\frac {2 (-1)^{3/4} a b d \arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}+\frac {i e \left (a+b \arctan \left (c x^2\right )\right )^2}{2 c}+\frac {1}{2} e x^2 \left (a+b \arctan \left (c x^2\right )\right )^2+\frac {2 (-1)^{3/4} a b d \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}+i a b d x \log \left (1-i c x^2\right )-\frac {1}{4} b^2 d x \log ^2\left (1-i c x^2\right )+\frac {b e \left (a+b \arctan \left (c x^2\right )\right ) \log \left (\frac {2}{1+i c x^2}\right )}{c}-i a b d x \log \left (1+i c x^2\right )+\frac {1}{2} b^2 d x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac {1}{4} b^2 d x \log ^2\left (1+i c x^2\right )+\left (b^2 d\right ) \int \log \left (1-i c x^2\right ) \, dx-\left (b^2 d\right ) \int \frac {\log \left (1-i c x^2\right )}{1-i c x^2} \, dx+\left (b^2 d\right ) \int \log \left (1+i c x^2\right ) \, dx-\left (b^2 d\right ) \int \frac {\log \left (1+i c x^2\right )}{1+i c x^2} \, dx-\left (b^2 c d\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 d\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+\frac {\left (i b^2 e\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x^2}\right )}{c} \\ & = a^2 d x-\frac {2 (-1)^{3/4} a b d \arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}+\frac {i e \left (a+b \arctan \left (c x^2\right )\right )^2}{2 c}+\frac {1}{2} e x^2 \left (a+b \arctan \left (c x^2\right )\right )^2+\frac {2 (-1)^{3/4} a b d \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}+i a b d x \log \left (1-i c x^2\right )+b^2 d x \log \left (1-i c x^2\right )+\frac {\sqrt [4]{-1} b^2 d \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )}{\sqrt {c}}-\frac {1}{4} b^2 d x \log ^2\left (1-i c x^2\right )+\frac {b e \left (a+b \arctan \left (c x^2\right )\right ) \log \left (\frac {2}{1+i c x^2}\right )}{c}-i a b d x \log \left (1+i c x^2\right )+b^2 d x \log \left (1+i c x^2\right )+\frac {\sqrt [4]{-1} b^2 d \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )}{\sqrt {c}}+\frac {1}{2} b^2 d x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac {1}{4} b^2 d x \log ^2\left (1+i c x^2\right )+\frac {i b^2 e \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x^2}\right )}{2 c}-\left (i b^2 d\right ) \int \frac {\log \left (1-i c x^2\right )}{-i+c x^2} \, dx+\left (i b^2 d\right ) \int \frac {\log \left (1+i c x^2\right )}{i+c x^2} \, dx-\left (b^2 d\right ) \int \log \left (1-i c x^2\right ) \, dx-\left (b^2 d\right ) \int \log \left (1+i c x^2\right ) \, dx+\left (2 i b^2 c d\right ) \int \frac {x^2}{1-i c x^2} \, dx-\left (2 i b^2 c d\right ) \int \frac {x^2}{1+i c x^2} \, dx+\left (2 i b^2 c d\right ) \int \frac {\sqrt [4]{-1} x \arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c} \left (1-i c x^2\right )} \, dx-\left (2 i b^2 c d\right ) \int \frac {\sqrt [4]{-1} x \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c} \left (1+i c x^2\right )} \, dx \\ & = a^2 d x-4 b^2 d x-\frac {2 (-1)^{3/4} a b d \arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}+\frac {i e \left (a+b \arctan \left (c x^2\right )\right )^2}{2 c}+\frac {1}{2} e x^2 \left (a+b \arctan \left (c x^2\right )\right )^2+\frac {2 (-1)^{3/4} a b d \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}+i a b d x \log \left (1-i c x^2\right )+\frac {\sqrt [4]{-1} b^2 d \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 d \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )}{\sqrt {c}}-\frac {1}{4} b^2 d x \log ^2\left (1-i c x^2\right )+\frac {b e \left (a+b \arctan \left (c x^2\right )\right ) \log \left (\frac {2}{1+i c x^2}\right )}{c}-i a b d x \log \left (1+i c x^2\right )-\frac {\sqrt [4]{-1} b^2 d \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 d \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )}{\sqrt {c}}+\frac {1}{2} b^2 d x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac {1}{4} b^2 d x \log ^2\left (1+i c x^2\right )+\frac {i b^2 e \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x^2}\right )}{2 c}+\left (2 b^2 d\right ) \int \frac {1}{1-i c x^2} \, dx+\left (2 b^2 d\right ) \int \frac {1}{1+i c x^2} \, dx+\left (2 (-1)^{3/4} b^2 \sqrt {c} d\right ) \int \frac {x \arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{1-i c x^2} \, dx-\left (2 (-1)^{3/4} b^2 \sqrt {c} d\right ) \int \frac {x \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{1+i c x^2} \, dx-\left (2 i b^2 c d\right ) \int \frac {x^2}{1-i c x^2} \, dx+\left (2 i b^2 c d\right ) \int \frac {x^2}{1+i c x^2} \, dx+\left (2 b^2 c d\right ) \int \frac {(-1)^{3/4} x \arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c} \left (1+i c x^2\right )} \, dx-\left (2 b^2 c d\right ) \int \frac {(-1)^{3/4} x \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c} \left (1-i c x^2\right )} \, dx \\ & = a^2 d x-\frac {2 (-1)^{3/4} a b d \arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}-\frac {2 \sqrt [4]{-1} b^2 d \arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}+\frac {(-1)^{3/4} b^2 d \arctan \left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}+\frac {i e \left (a+b \arctan \left (c x^2\right )\right )^2}{2 c}+\frac {1}{2} e x^2 \left (a+b \arctan \left (c x^2\right )\right )^2+\frac {2 (-1)^{3/4} a b d \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}-\frac {2 \sqrt [4]{-1} b^2 d \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}-\frac {\sqrt [4]{-1} b^2 d \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}+i a b d x \log \left (1-i c x^2\right )+\frac {\sqrt [4]{-1} b^2 d \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 d \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )}{\sqrt {c}}-\frac {1}{4} b^2 d x \log ^2\left (1-i c x^2\right )+\frac {b e \left (a+b \arctan \left (c x^2\right )\right ) \log \left (\frac {2}{1+i c x^2}\right )}{c}-i a b d x \log \left (1+i c x^2\right )-\frac {\sqrt [4]{-1} b^2 d \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 d \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )}{\sqrt {c}}+\frac {1}{2} b^2 d x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac {1}{4} b^2 d x \log ^2\left (1+i c x^2\right )+\frac {i b^2 e \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x^2}\right )}{2 c}-\left (2 b^2 d\right ) \int \frac {1}{1-i c x^2} \, dx-\left (2 b^2 d\right ) \int \frac {1}{1+i c x^2} \, dx-\left (2 b^2 d\right ) \int \frac {\arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{i-(-1)^{3/4} \sqrt {c} x} \, dx-\left (2 b^2 d\right ) \int \frac {\text {arctanh}\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} d\right ) \int \frac {x \arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{1+i c x^2} \, dx-\left (2 (-1)^{3/4} b^2 \sqrt {c} d\right ) \int \frac {x \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{1-i c x^2} \, dx \\ & = a^2 d x-\frac {2 (-1)^{3/4} a b d \arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}+\frac {(-1)^{3/4} b^2 d \arctan \left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}+\frac {i e \left (a+b \arctan \left (c x^2\right )\right )^2}{2 c}+\frac {1}{2} e x^2 \left (a+b \arctan \left (c x^2\right )\right )^2+\frac {2 (-1)^{3/4} a b d \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}-\frac {\sqrt [4]{-1} b^2 d \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}+\frac {2 \sqrt [4]{-1} b^2 d \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 d \text {arctanh}\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 d x \log \left (1-i c x^2\right )+\frac {\sqrt [4]{-1} b^2 d \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 d \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )}{\sqrt {c}}-\frac {1}{4} b^2 d x \log ^2\left (1-i c x^2\right )+\frac {b e \left (a+b \arctan \left (c x^2\right )\right ) \log \left (\frac {2}{1+i c x^2}\right )}{c}-i a b d x \log \left (1+i c x^2\right )-\frac {\sqrt [4]{-1} b^2 d \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 d \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )}{\sqrt {c}}+\frac {1}{2} b^2 d x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac {1}{4} b^2 d x \log ^2\left (1+i c x^2\right )+\frac {i b^2 e \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x^2}\right )}{2 c}+\left (2 b^2 d\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 d\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} d\right ) \int \left (\frac {i \arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{2 \sqrt {c} \left (\sqrt [4]{-1}-\sqrt {c} x\right )}-\frac {i \arctan \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} d\right ) \int \left (-\frac {i \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{2 \sqrt {c} \left (-(-1)^{3/4}-\sqrt {c} x\right )}+\frac {i \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{2 \sqrt {c} \left (-(-1)^{3/4}+\sqrt {c} x\right )}\right ) \, dx \\ & = a^2 d x-\frac {2 (-1)^{3/4} a b d \arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}+\frac {(-1)^{3/4} b^2 d \arctan \left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}+\frac {i e \left (a+b \arctan \left (c x^2\right )\right )^2}{2 c}+\frac {1}{2} e x^2 \left (a+b \arctan \left (c x^2\right )\right )^2+\frac {2 (-1)^{3/4} a b d \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}-\frac {\sqrt [4]{-1} b^2 d \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}+\frac {2 \sqrt [4]{-1} b^2 d \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 d \text {arctanh}\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 d x \log \left (1-i c x^2\right )+\frac {\sqrt [4]{-1} b^2 d \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 d \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )}{\sqrt {c}}-\frac {1}{4} b^2 d x \log ^2\left (1-i c x^2\right )+\frac {b e \left (a+b \arctan \left (c x^2\right )\right ) \log \left (\frac {2}{1+i c x^2}\right )}{c}-i a b d x \log \left (1+i c x^2\right )-\frac {\sqrt [4]{-1} b^2 d \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 d \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )}{\sqrt {c}}+\frac {1}{2} b^2 d x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac {1}{4} b^2 d x \log ^2\left (1+i c x^2\right )+\frac {i b^2 e \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x^2}\right )}{2 c}-\left (\sqrt [4]{-1} b^2 d\right ) \int \frac {\arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt [4]{-1}-\sqrt {c} x} \, dx+\left (\sqrt [4]{-1} b^2 d\right ) \int \frac {\arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt [4]{-1}+\sqrt {c} x} \, dx-\left (\sqrt [4]{-1} b^2 d\right ) \int \frac {\text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{-(-1)^{3/4}-\sqrt {c} x} \, dx+\left (\sqrt [4]{-1} b^2 d\right ) \int \frac {\text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{-(-1)^{3/4}+\sqrt {c} x} \, dx+\frac {\left (2 \sqrt [4]{-1} b^2 d\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 d\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}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(4824\) vs. \(2(1325)=2650\).

Time = 36.63 (sec) , antiderivative size = 4824, normalized size of antiderivative = 3.64 \[ \int (d+e x) \left (a+b \arctan \left (c x^2\right )\right )^2 \, dx=\text {Result too large to show} \]

[In]

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

[Out]

a^2*d*x + (a^2*e*x^2)/2 + (a*b*d*Sqrt[c*x^2]*(2*Sqrt[c*x^2]*ArcTan[c*x^2] - Sqrt[2]*(ArcTan[(-1 + c*x^2)/(Sqrt
[2]*Sqrt[c*x^2])] - ArcTanh[(Sqrt[2]*Sqrt[c*x^2])/(1 + c*x^2)])))/(c*x) + (a*b*e*(c*x^2*ArcTan[c*x^2] + Log[1/
Sqrt[1 + c^2*x^4]]))/c + (b^2*e*((-I)*ArcTan[c*x^2]^2 + c*x^2*ArcTan[c*x^2]^2 + 2*ArcTan[c*x^2]*Log[1 + E^((2*
I)*ArcTan[c*x^2])] - I*PolyLog[2, -E^((2*I)*ArcTan[c*x^2])]))/(2*c) + (b^2*d*Sqrt[c*x^2]*(2*Sqrt[c*x^2]*ArcTan
[c*x^2]^2 - 4*((ArcTan[c*x^2]*(-2*ArcTan[1 - Sqrt[2]*Sqrt[c*x^2]] + 2*ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]] + Log[1
+ c*x^2 - Sqrt[2]*Sqrt[c*x^2]] - Log[1 + c*x^2 + Sqrt[2]*Sqrt[c*x^2]]))/(2*Sqrt[2]) - (-((ArcTan[1 - Sqrt[2]*S
qrt[c*x^2]] + ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]])*Log[1 + c*x^2 - Sqrt[2]*Sqrt[c*x^2]]) + (ArcTan[1 - Sqrt[2]*Sqr
t[c*x^2]] + ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]])*Log[1 + c*x^2 + Sqrt[2]*Sqrt[c*x^2]] - (Sqrt[c*x^2]*(1 + (1 - Sqr
t[2]*Sqrt[c*x^2])^2)^(3/2)*(2*(-5*ArcTan[2 + I]*ArcTan[1 - Sqrt[2]*Sqrt[c*x^2]] + 4*ArcTan[1 - Sqrt[2]*Sqrt[c*
x^2]]^2 + ((1 + 2*I)*Sqrt[1 + I]*ArcTan[1 - Sqrt[2]*Sqrt[c*x^2]]^2)/E^(I*ArcTan[2 + I]) + ((1 - 2*I)*Sqrt[1 -
I]*ArcTan[1 - Sqrt[2]*Sqrt[c*x^2]]^2)/E^ArcTanh[1 + 2*I] - (5*I)*ArcTan[1 - Sqrt[2]*Sqrt[c*x^2]]*ArcTanh[1 + 2
*I] + (5*I)*(-ArcTan[2 + I] + ArcTan[1 - Sqrt[2]*Sqrt[c*x^2]])*Log[1 - E^((2*I)*(-ArcTan[2 + I] + ArcTan[1 - S
qrt[2]*Sqrt[c*x^2]]))] + 5*((-I)*ArcTan[1 - Sqrt[2]*Sqrt[c*x^2]] + ArcTanh[1 + 2*I])*Log[1 - E^((2*I)*ArcTan[1
- Sqrt[2]*Sqrt[c*x^2]] - 2*ArcTanh[1 + 2*I])] + (5*I)*ArcTan[2 + I]*Log[-Sin[ArcTan[2 + I] - ArcTan[1 - Sqrt[
2]*Sqrt[c*x^2]]]] - 5*ArcTanh[1 + 2*I]*Log[Sin[ArcTan[1 - Sqrt[2]*Sqrt[c*x^2]] + I*ArcTanh[1 + 2*I]]]) + 5*Pol
yLog[2, E^((2*I)*(-ArcTan[2 + I] + ArcTan[1 - Sqrt[2]*Sqrt[c*x^2]]))] - 5*PolyLog[2, E^((2*I)*ArcTan[1 - Sqrt[
2]*Sqrt[c*x^2]] - 2*ArcTanh[1 + 2*I])])*(3 + 2*Cos[2*ArcTan[1 - Sqrt[2]*Sqrt[c*x^2]]] - 2*Sin[2*ArcTan[1 - Sqr
t[2]*Sqrt[c*x^2]]]))/(20*Sqrt[2]*(-1 - c*x^2 + Sqrt[2]*Sqrt[c*x^2])*(1 + c*x^2 + Sqrt[2]*Sqrt[c*x^2])*(1/Sqrt[
1 + (1 - Sqrt[2]*Sqrt[c*x^2])^2] - (1 - Sqrt[2]*Sqrt[c*x^2])/Sqrt[1 + (1 - Sqrt[2]*Sqrt[c*x^2])^2])) + ((1/40
+ I/40)*c*x^2*(1 + (1 - Sqrt[2]*Sqrt[c*x^2])^2)*((5 + 5*I)*Pi*ArcTan[1 - Sqrt[2]*Sqrt[c*x^2]] + 10*ArcTan[2 +
I]*ArcTan[1 - Sqrt[2]*Sqrt[c*x^2]] + (4 - 4*I)*ArcTan[1 - Sqrt[2]*Sqrt[c*x^2]]^2 - ((2 + 4*I)*Sqrt[1 + I]*ArcT
an[1 - Sqrt[2]*Sqrt[c*x^2]]^2)/E^(I*ArcTan[2 + I]) + ((4 + 2*I)*Sqrt[1 - I]*ArcTan[1 - Sqrt[2]*Sqrt[c*x^2]]^2)
/E^ArcTanh[1 + 2*I] + 10*ArcTan[1 - Sqrt[2]*Sqrt[c*x^2]]*ArcTanh[1 + 2*I] + (5 - 5*I)*Pi*Log[1 + E^((-2*I)*Arc
Tan[1 - Sqrt[2]*Sqrt[c*x^2]])] + (10*I)*ArcTan[2 + I]*Log[1 - E^((2*I)*(-ArcTan[2 + I] + ArcTan[1 - Sqrt[2]*Sq
rt[c*x^2]]))] - (10*I)*ArcTan[1 - Sqrt[2]*Sqrt[c*x^2]]*Log[1 - E^((2*I)*(-ArcTan[2 + I] + ArcTan[1 - Sqrt[2]*S
qrt[c*x^2]]))] + 10*ArcTan[1 - Sqrt[2]*Sqrt[c*x^2]]*Log[1 - E^((2*I)*ArcTan[1 - Sqrt[2]*Sqrt[c*x^2]] - 2*ArcTa
nh[1 + 2*I])] + (10*I)*ArcTanh[1 + 2*I]*Log[1 - E^((2*I)*ArcTan[1 - Sqrt[2]*Sqrt[c*x^2]] - 2*ArcTanh[1 + 2*I])
] - (5 - 5*I)*Pi*Log[1/Sqrt[1 + (1 - Sqrt[2]*Sqrt[c*x^2])^2]] - (10*I)*ArcTan[2 + I]*Log[-Sin[ArcTan[2 + I] -
ArcTan[1 - Sqrt[2]*Sqrt[c*x^2]]]] - (10*I)*ArcTanh[1 + 2*I]*Log[Sin[ArcTan[1 - Sqrt[2]*Sqrt[c*x^2]] + I*ArcTan
h[1 + 2*I]]] - 5*PolyLog[2, E^((2*I)*(-ArcTan[2 + I] + ArcTan[1 - Sqrt[2]*Sqrt[c*x^2]]))] - (5*I)*PolyLog[2, E
^((2*I)*ArcTan[1 - Sqrt[2]*Sqrt[c*x^2]] - 2*ArcTanh[1 + 2*I])])*(3 + 2*Cos[2*ArcTan[1 - Sqrt[2]*Sqrt[c*x^2]]]
- 2*Sin[2*ArcTan[1 - Sqrt[2]*Sqrt[c*x^2]]]))/((-1 - c*x^2 + Sqrt[2]*Sqrt[c*x^2])*(1 + c*x^2 + Sqrt[2]*Sqrt[c*x
^2])*(1/Sqrt[1 + (1 - Sqrt[2]*Sqrt[c*x^2])^2] - (1 - Sqrt[2]*Sqrt[c*x^2])/Sqrt[1 + (1 - Sqrt[2]*Sqrt[c*x^2])^2
])^2) + ((1/80 + I/80)*(2 + 2*c*x^2 - 2*Sqrt[2]*Sqrt[c*x^2])^2*((-5 - 5*I)*Pi*ArcTan[1 - Sqrt[2]*Sqrt[c*x^2]]
- (10*I)*ArcTan[2 + I]*ArcTan[1 - Sqrt[2]*Sqrt[c*x^2]] + (8 - 8*I)*ArcTan[1 - Sqrt[2]*Sqrt[c*x^2]]^2 - ((4 - 2
*I)*Sqrt[1 + I]*ArcTan[1 - Sqrt[2]*Sqrt[c*x^2]]^2)/E^(I*ArcTan[2 + I]) - ((2 - 4*I)*Sqrt[1 - I]*ArcTan[1 - Sqr
t[2]*Sqrt[c*x^2]]^2)/E^ArcTanh[1 + 2*I] + (10*I)*ArcTan[1 - Sqrt[2]*Sqrt[c*x^2]]*ArcTanh[1 + 2*I] - (5 - 5*I)*
Pi*Log[1 + E^((-2*I)*ArcTan[1 - Sqrt[2]*Sqrt[c*x^2]])] + 10*ArcTan[2 + I]*Log[1 - E^((2*I)*(-ArcTan[2 + I] + A
rcTan[1 - Sqrt[2]*Sqrt[c*x^2]]))] - 10*ArcTan[1 - Sqrt[2]*Sqrt[c*x^2]]*Log[1 - E^((2*I)*(-ArcTan[2 + I] + ArcT
an[1 - Sqrt[2]*Sqrt[c*x^2]]))] + (10*I)*ArcTan[1 - Sqrt[2]*Sqrt[c*x^2]]*Log[1 - E^((2*I)*ArcTan[1 - Sqrt[2]*Sq
rt[c*x^2]] - 2*ArcTanh[1 + 2*I])] - 10*ArcTanh[1 + 2*I]*Log[1 - E^((2*I)*ArcTan[1 - Sqrt[2]*Sqrt[c*x^2]] - 2*A
rcTanh[1 + 2*I])] + (5 - 5*I)*Pi*Log[1/Sqrt[2 + 2*c*x^2 - 2*Sqrt[2]*Sqrt[c*x^2]]] - 10*ArcTan[2 + I]*Log[-Sin[
ArcTan[2 + I] - ArcTan[1 - Sqrt[2]*Sqrt[c*x^2]]]] + 10*ArcTanh[1 + 2*I]*Log[Sin[ArcTan[1 - Sqrt[2]*Sqrt[c*x^2]
] + I*ArcTanh[1 + 2*I]]] + (5*I)*PolyLog[2, E^((2*I)*(-ArcTan[2 + I] + ArcTan[1 - Sqrt[2]*Sqrt[c*x^2]]))] + 5*
PolyLog[2, E^((2*I)*ArcTan[1 - Sqrt[2]*Sqrt[c*x^2]] - 2*ArcTanh[1 + 2*I])])*(3 + 2*Cos[2*ArcTan[1 - Sqrt[2]*Sq
rt[c*x^2]]] - 2*Sin[2*ArcTan[1 - Sqrt[2]*Sqrt[c*x^2]]]))/(1 + c^2*x^4) - (Sqrt[c*x^2]*(1 + (1 + Sqrt[2]*Sqrt[c
*x^2])^2)^(3/2)*(2*(-5*ArcTan[2 + I]*ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]] + 4*ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]]^2 + (
(1 + 2*I)*Sqrt[1 + I]*ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]]^2)/E^(I*ArcTan[2 + I]) + ((1 - 2*I)*Sqrt[1 - I]*ArcTan[1
+ Sqrt[2]*Sqrt[c*x^2]]^2)/E^ArcTanh[1 + 2*I] - (5*I)*ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]]*ArcTanh[1 + 2*I] + (5*I)
*(-ArcTan[2 + I] + ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]])*Log[1 - E^((2*I)*(-ArcTan[2 + I] + ArcTan[1 + Sqrt[2]*Sqrt
[c*x^2]]))] + 5*((-I)*ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]] + ArcTanh[1 + 2*I])*Log[1 - E^((2*I)*ArcTan[1 + Sqrt[2]*
Sqrt[c*x^2]] - 2*ArcTanh[1 + 2*I])] + (5*I)*ArcTan[2 + I]*Log[-Sin[ArcTan[2 + I] - ArcTan[1 + Sqrt[2]*Sqrt[c*x
^2]]]] - 5*ArcTanh[1 + 2*I]*Log[Sin[ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]] + I*ArcTanh[1 + 2*I]]]) + 5*PolyLog[2, E^(
(2*I)*(-ArcTan[2 + I] + ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]]))] - 5*PolyLog[2, E^((2*I)*ArcTan[1 + Sqrt[2]*Sqrt[c*x
^2]] - 2*ArcTanh[1 + 2*I])])*(3 + 2*Cos[2*ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]]] - 2*Sin[2*ArcTan[1 + Sqrt[2]*Sqrt[c
*x^2]]]))/(20*Sqrt[2]*(-1 - c*x^2 + Sqrt[2]*Sqrt[c*x^2])*(1 + c*x^2 + Sqrt[2]*Sqrt[c*x^2])*(1/Sqrt[1 + (1 + Sq
rt[2]*Sqrt[c*x^2])^2] - (1 + Sqrt[2]*Sqrt[c*x^2])/Sqrt[1 + (1 + Sqrt[2]*Sqrt[c*x^2])^2])) - ((1/40 + I/40)*c*x
^2*(1 + (1 + Sqrt[2]*Sqrt[c*x^2])^2)*((5 + 5*I)*Pi*ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]] + 10*ArcTan[2 + I]*ArcTan[1
+ Sqrt[2]*Sqrt[c*x^2]] + (4 - 4*I)*ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]]^2 - ((2 + 4*I)*Sqrt[1 + I]*ArcTan[1 + Sqrt
[2]*Sqrt[c*x^2]]^2)/E^(I*ArcTan[2 + I]) + ((4 + 2*I)*Sqrt[1 - I]*ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]]^2)/E^ArcTanh[
1 + 2*I] + 10*ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]]*ArcTanh[1 + 2*I] + (5 - 5*I)*Pi*Log[1 + E^((-2*I)*ArcTan[1 + Sqr
t[2]*Sqrt[c*x^2]])] + (10*I)*ArcTan[2 + I]*Log[1 - E^((2*I)*(-ArcTan[2 + I] + ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]])
)] - (10*I)*ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]]*Log[1 - E^((2*I)*(-ArcTan[2 + I] + ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]]
))] + 10*ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]]*Log[1 - E^((2*I)*ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]] - 2*ArcTanh[1 + 2*I]
)] + (10*I)*ArcTanh[1 + 2*I]*Log[1 - E^((2*I)*ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]] - 2*ArcTanh[1 + 2*I])] - (5 - 5*
I)*Pi*Log[1/Sqrt[1 + (1 + Sqrt[2]*Sqrt[c*x^2])^2]] - (10*I)*ArcTan[2 + I]*Log[-Sin[ArcTan[2 + I] - ArcTan[1 +
Sqrt[2]*Sqrt[c*x^2]]]] - (10*I)*ArcTanh[1 + 2*I]*Log[Sin[ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]] + I*ArcTanh[1 + 2*I]]
] - 5*PolyLog[2, E^((2*I)*(-ArcTan[2 + I] + ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]]))] - (5*I)*PolyLog[2, E^((2*I)*Arc
Tan[1 + Sqrt[2]*Sqrt[c*x^2]] - 2*ArcTanh[1 + 2*I])])*(3 + 2*Cos[2*ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]]] - 2*Sin[2*A
rcTan[1 + Sqrt[2]*Sqrt[c*x^2]]]))/((-1 - c*x^2 + Sqrt[2]*Sqrt[c*x^2])*(1 + c*x^2 + Sqrt[2]*Sqrt[c*x^2])*(1/Sqr
t[1 + (1 + Sqrt[2]*Sqrt[c*x^2])^2] - (1 + Sqrt[2]*Sqrt[c*x^2])/Sqrt[1 + (1 + Sqrt[2]*Sqrt[c*x^2])^2])^2) + ((1
/80 + I/80)*(1 + (1 + Sqrt[2]*Sqrt[c*x^2])^2)^2*((-5 - 5*I)*Pi*ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]] - (10*I)*ArcTan
[2 + I]*ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]] + (8 - 8*I)*ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]]^2 - ((4 - 2*I)*Sqrt[1 + I]
*ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]]^2)/E^(I*ArcTan[2 + I]) - ((2 - 4*I)*Sqrt[1 - I]*ArcTan[1 + Sqrt[2]*Sqrt[c*x^2
]]^2)/E^ArcTanh[1 + 2*I] + (10*I)*ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]]*ArcTanh[1 + 2*I] - (5 - 5*I)*Pi*Log[1 + E^((
-2*I)*ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]])] + 10*ArcTan[2 + I]*Log[1 - E^((2*I)*(-ArcTan[2 + I] + ArcTan[1 + Sqrt[
2]*Sqrt[c*x^2]]))] - 10*ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]]*Log[1 - E^((2*I)*(-ArcTan[2 + I] + ArcTan[1 + Sqrt[2]*
Sqrt[c*x^2]]))] + (10*I)*ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]]*Log[1 - E^((2*I)*ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]] - 2*
ArcTanh[1 + 2*I])] - 10*ArcTanh[1 + 2*I]*Log[1 - E^((2*I)*ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]] - 2*ArcTanh[1 + 2*I]
)] + (5 - 5*I)*Pi*Log[1/Sqrt[1 + (1 + Sqrt[2]*Sqrt[c*x^2])^2]] - 10*ArcTan[2 + I]*Log[-Sin[ArcTan[2 + I] - Arc
Tan[1 + Sqrt[2]*Sqrt[c*x^2]]]] + 10*ArcTanh[1 + 2*I]*Log[Sin[ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]] + I*ArcTanh[1 + 2
*I]]] + (5*I)*PolyLog[2, E^((2*I)*(-ArcTan[2 + I] + ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]]))] + 5*PolyLog[2, E^((2*I)
*ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]] - 2*ArcTanh[1 + 2*I])])*(3 + 2*Cos[2*ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]]] - 2*Sin
[2*ArcTan[1 + Sqrt[2]*Sqrt[c*x^2]]]))/((-1 - c*x^2 + Sqrt[2]*Sqrt[c*x^2])*(1 + c*x^2 + Sqrt[2]*Sqrt[c*x^2])))/
(2*Sqrt[2]))))/(2*c*x)

Maple [F]

\[\int \left (e x +d \right ) {\left (a +b \arctan \left (c \,x^{2}\right )\right )}^{2}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int (d+e x) \left (a+b \arctan \left (c x^2\right )\right )^2 \, dx=\int { {\left (e x + d\right )} {\left (b \arctan \left (c x^{2}\right ) + a\right )}^{2} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int (d+e x) \left (a+b \arctan \left (c x^2\right )\right )^2 \, dx=\int \left (a + b \operatorname {atan}{\left (c x^{2} \right )}\right )^{2} \left (d + e x\right )\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int (d+e x) \left (a+b \arctan \left (c x^2\right )\right )^2 \, dx=\int { {\left (e x + d\right )} {\left (b \arctan \left (c x^{2}\right ) + a\right )}^{2} \,d x } \]

[In]

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

[Out]

12*b^2*c^2*e*integrate(1/16*x^5*arctan(c*x^2)^2/(c^2*x^4 + 1), x) + b^2*c^2*e*integrate(1/16*x^5*log(c^2*x^4 +
1)^2/(c^2*x^4 + 1), x) + 12*b^2*c^2*d*integrate(1/16*x^4*arctan(c*x^2)^2/(c^2*x^4 + 1), x) + 4*b^2*c^2*e*inte
grate(1/16*x^5*log(c^2*x^4 + 1)/(c^2*x^4 + 1), x) + b^2*c^2*d*integrate(1/16*x^4*log(c^2*x^4 + 1)^2/(c^2*x^4 +
1), x) + 8*b^2*c^2*d*integrate(1/16*x^4*log(c^2*x^4 + 1)/(c^2*x^4 + 1), x) + 1/2*a^2*e*x^2 + 1/8*b^2*e*arctan
(c*x^2)^3/c - 8*b^2*c*e*integrate(1/16*x^3*arctan(c*x^2)/(c^2*x^4 + 1), x) - 16*b^2*c*d*integrate(1/16*x^2*arc
tan(c*x^2)/(c^2*x^4 + 1), x) - 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)*log(c*x^2 - sqrt(2)*sqrt(c)*x + 1)/c^(3/2)) - 4*x*arctan(c*x^2))*a*b*d + a^2*d*x +
b^2*e*integrate(1/16*x*log(c^2*x^4 + 1)^2/(c^2*x^4 + 1), x) + 12*b^2*d*integrate(1/16*arctan(c*x^2)^2/(c^2*x^
4 + 1), x) + b^2*d*integrate(1/16*log(c^2*x^4 + 1)^2/(c^2*x^4 + 1), x) + 1/2*(2*c*x^2*arctan(c*x^2) - log(c^2*
x^4 + 1))*a*b*e/c + 1/8*(b^2*e*x^2 + 2*b^2*d*x)*arctan(c*x^2)^2 - 1/32*(b^2*e*x^2 + 2*b^2*d*x)*log(c^2*x^4 + 1
)^2

Giac [F]

\[ \int (d+e x) \left (a+b \arctan \left (c x^2\right )\right )^2 \, dx=\int { {\left (e x + d\right )} {\left (b \arctan \left (c x^{2}\right ) + a\right )}^{2} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int (d+e x) \left (a+b \arctan \left (c x^2\right )\right )^2 \, dx=\int {\left (a+b\,\mathrm {atan}\left (c\,x^2\right )\right )}^2\,\left (d+e\,x\right ) \,d x \]

[In]

int((a + b*atan(c*x^2))^2*(d + e*x),x)

[Out]

int((a + b*atan(c*x^2))^2*(d + e*x), x)