\(\int \frac {(a+b \arctan (c x^2))^2}{x^6} \, dx\) [85]

Optimal result

Integrand size = 16, antiderivative size = 1444 \[ \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{x^6} \, dx=-\frac {2 a b c}{15 x^3}+\frac {2 i a b c^2}{5 x}-\frac {8 b^2 c^2}{15 x}-\frac {4}{15} (-1)^{3/4} b^2 c^{5/2} \arctan \left ((-1)^{3/4} \sqrt {c} x\right )-\frac {1}{5} \sqrt [4]{-1} b^2 c^{5/2} \arctan \left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{5} \sqrt [4]{-1} a b c^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )+\frac {4}{15} (-1)^{3/4} b^2 c^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )+\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{5} (-1)^{3/4} b^2 c^{5/2} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {2}{5} (-1)^{3/4} b^2 c^{5/2} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1+\sqrt [4]{-1} \sqrt {c} x}\right )+\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \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 )-\frac {2}{5} (-1)^{3/4} b^2 c^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )+\frac {2}{5} (-1)^{3/4} b^2 c^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1+(-1)^{3/4} \sqrt {c} x}\right )-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \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 )-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \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 )+\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \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 )-\frac {i b^2 c \log \left (1-i c x^2\right )}{15 x^3}-\frac {b^2 c^2 \log \left (1-i c x^2\right )}{5 x}+\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )-\frac {b c \left (2 a+i b \log \left (1-i c x^2\right )\right )}{15 x^3}-\frac {i b c^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )}{5 x}+\frac {1}{5} \sqrt [4]{-1} b c^{5/2} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{20 x^5}+\frac {i a b \log \left (1+i c x^2\right )}{5 x^5}+\frac {2 i b^2 c \log \left (1+i c x^2\right )}{15 x^3}-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )-\frac {b^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{10 x^5}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{20 x^5}-\frac {1}{5} \sqrt [4]{-1} b^2 c^{5/2} \operatorname {PolyLog}\left (2,1-\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {1}{5} \sqrt [4]{-1} b^2 c^{5/2} \operatorname {PolyLog}\left (2,1-\frac {2}{1+\sqrt [4]{-1} \sqrt {c} x}\right )+\frac {1}{10} \sqrt [4]{-1} b^2 c^{5/2} \operatorname {PolyLog}\left (2,1-\frac {\sqrt {2} \left (\sqrt [4]{-1}+\sqrt {c} x\right )}{1+\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \operatorname {PolyLog}\left (2,1-\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \operatorname {PolyLog}\left (2,1-\frac {2}{1+(-1)^{3/4} \sqrt {c} x}\right )+\frac {1}{10} (-1)^{3/4} b^2 c^{5/2} \operatorname {PolyLog}\left (2,1+\frac {\sqrt {2} \left ((-1)^{3/4}+\sqrt {c} x\right )}{1+(-1)^{3/4} \sqrt {c} x}\right )+\frac {1}{10} (-1)^{3/4} b^2 c^{5/2} \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 )+\frac {1}{10} \sqrt [4]{-1} b^2 c^{5/2} \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 ) \]

[Out]

-8/15*b^2*c^2/x-1/20*(2*a+I*b*ln(1-I*c*x^2))^2/x^5-2/15*a*b*c/x^3-4/15*(-1)^(3/4)*b^2*c^(5/2)*arctan((-1)^(3/4
)*x*c^(1/2))-1/5*(-1)^(1/4)*b^2*c^(5/2)*arctan((-1)^(3/4)*x*c^(1/2))^2+4/15*(-1)^(3/4)*b^2*c^(5/2)*arctanh((-1
)^(3/4)*x*c^(1/2))+1/5*(-1)^(3/4)*b^2*c^(5/2)*arctanh((-1)^(3/4)*x*c^(1/2))^2-1/5*b^2*c^2*ln(1-I*c*x^2)/x-1/15
*b*c*(2*a+I*b*ln(1-I*c*x^2))/x^3-1/10*b^2*ln(1-I*c*x^2)*ln(1+I*c*x^2)/x^5-1/5*(-1)^(1/4)*b^2*c^(5/2)*polylog(2
,1-2/(1-(-1)^(1/4)*x*c^(1/2)))-1/5*(-1)^(1/4)*b^2*c^(5/2)*polylog(2,1-2/(1+(-1)^(1/4)*x*c^(1/2)))+1/10*(-1)^(1
/4)*b^2*c^(5/2)*polylog(2,1-2^(1/2)*((-1)^(1/4)+x*c^(1/2))/(1+(-1)^(1/4)*x*c^(1/2)))-1/5*(-1)^(3/4)*b^2*c^(5/2
)*polylog(2,1-2/(1-(-1)^(3/4)*x*c^(1/2)))-1/5*(-1)^(3/4)*b^2*c^(5/2)*polylog(2,1-2/(1+(-1)^(3/4)*x*c^(1/2)))+1
/10*(-1)^(3/4)*b^2*c^(5/2)*polylog(2,1+2^(1/2)*((-1)^(3/4)+x*c^(1/2))/(1+(-1)^(3/4)*x*c^(1/2)))+1/10*(-1)^(3/4
)*b^2*c^(5/2)*polylog(2,1-(1+I)*(1+(-1)^(1/4)*x*c^(1/2))/(1+(-1)^(3/4)*x*c^(1/2)))+1/10*(-1)^(1/4)*b^2*c^(5/2)
*polylog(2,1+(-1+I)*(1+(-1)^(3/4)*x*c^(1/2))/(1+(-1)^(1/4)*x*c^(1/2)))-1/5*(-1)^(3/4)*b^2*c^(5/2)*arctanh((-1)
^(3/4)*x*c^(1/2))*ln(-2^(1/2)*((-1)^(3/4)+x*c^(1/2))/(1+(-1)^(3/4)*x*c^(1/2)))-1/5*(-1)^(3/4)*b^2*c^(5/2)*arct
anh((-1)^(3/4)*x*c^(1/2))*ln((1+I)*(1+(-1)^(1/4)*x*c^(1/2))/(1+(-1)^(3/4)*x*c^(1/2)))+1/5*(-1)^(3/4)*b^2*c^(5/
2)*arctan((-1)^(3/4)*x*c^(1/2))*ln((1-I)*(1+(-1)^(3/4)*x*c^(1/2))/(1+(-1)^(1/4)*x*c^(1/2)))+1/5*(-1)^(3/4)*b^2
*c^(5/2)*arctanh((-1)^(3/4)*x*c^(1/2))*ln(1-I*c*x^2)+1/5*(-1)^(1/4)*b*c^(5/2)*arctan((-1)^(3/4)*x*c^(1/2))*(2*
a+I*b*ln(1-I*c*x^2))-1/5*(-1)^(3/4)*b^2*c^(5/2)*arctan((-1)^(3/4)*x*c^(1/2))*ln(1+I*c*x^2)-1/5*(-1)^(3/4)*b^2*
c^(5/2)*arctanh((-1)^(3/4)*x*c^(1/2))*ln(1+I*c*x^2)+2/5*(-1)^(3/4)*b^2*c^(5/2)*arctan((-1)^(3/4)*x*c^(1/2))*ln
(2/(1-(-1)^(1/4)*x*c^(1/2)))-2/5*(-1)^(3/4)*b^2*c^(5/2)*arctan((-1)^(3/4)*x*c^(1/2))*ln(2/(1+(-1)^(1/4)*x*c^(1
/2)))+1/5*(-1)^(3/4)*b^2*c^(5/2)*arctan((-1)^(3/4)*x*c^(1/2))*ln(2^(1/2)*((-1)^(1/4)+x*c^(1/2))/(1+(-1)^(1/4)*
x*c^(1/2)))-2/5*(-1)^(3/4)*b^2*c^(5/2)*arctanh((-1)^(3/4)*x*c^(1/2))*ln(2/(1-(-1)^(3/4)*x*c^(1/2)))+2/5*(-1)^(
3/4)*b^2*c^(5/2)*arctanh((-1)^(3/4)*x*c^(1/2))*ln(2/(1+(-1)^(3/4)*x*c^(1/2)))+2/5*(-1)^(1/4)*a*b*c^(5/2)*arcta
nh((-1)^(3/4)*x*c^(1/2))-1/15*I*b^2*c*ln(1-I*c*x^2)/x^3-1/5*I*b*c^2*(2*a+I*b*ln(1-I*c*x^2))/x+1/20*b^2*ln(1+I*
c*x^2)^2/x^5+1/5*I*a*b*ln(1+I*c*x^2)/x^5+2/15*I*b^2*c*ln(1+I*c*x^2)/x^3+2/5*I*a*b*c^2/x

Rubi [A] (verified)

Time = 1.68 (sec) , antiderivative size = 1444, normalized size of antiderivative = 1.00, number of steps used = 77, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.562, Rules used = {4950, 2507, 2526, 2505, 331, 209, 211, 2520, 12, 5040, 4964, 2449, 2352, 6874, 212, 30, 2637, 5048, 4966, 2497, 214, 6139, 6057, 6131, 6055} \[ \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{x^6} \, dx=-\frac {1}{5} \sqrt [4]{-1} b^2 \arctan \left ((-1)^{3/4} \sqrt {c} x\right )^2 c^{5/2}+\frac {1}{5} (-1)^{3/4} b^2 \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )^2 c^{5/2}-\frac {4}{15} (-1)^{3/4} b^2 \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) c^{5/2}+\frac {4}{15} (-1)^{3/4} b^2 \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) c^{5/2}+\frac {2}{5} \sqrt [4]{-1} a b \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) c^{5/2}+\frac {2}{5} (-1)^{3/4} b^2 \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right ) c^{5/2}-\frac {2}{5} (-1)^{3/4} b^2 \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{\sqrt [4]{-1} \sqrt {c} x+1}\right ) c^{5/2}+\frac {1}{5} (-1)^{3/4} b^2 \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 ) c^{5/2}-\frac {2}{5} (-1)^{3/4} b^2 \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right ) c^{5/2}+\frac {2}{5} (-1)^{3/4} b^2 \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{(-1)^{3/4} \sqrt {c} x+1}\right ) c^{5/2}-\frac {1}{5} (-1)^{3/4} b^2 \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 ) c^{5/2}-\frac {1}{5} (-1)^{3/4} b^2 \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 ) c^{5/2}+\frac {1}{5} (-1)^{3/4} b^2 \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 ) c^{5/2}+\frac {1}{5} (-1)^{3/4} b^2 \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right ) c^{5/2}+\frac {1}{5} \sqrt [4]{-1} b \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right ) c^{5/2}-\frac {1}{5} (-1)^{3/4} b^2 \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (i c x^2+1\right ) c^{5/2}-\frac {1}{5} (-1)^{3/4} b^2 \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (i c x^2+1\right ) c^{5/2}-\frac {1}{5} \sqrt [4]{-1} b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right ) c^{5/2}-\frac {1}{5} \sqrt [4]{-1} b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{\sqrt [4]{-1} \sqrt {c} x+1}\right ) c^{5/2}+\frac {1}{10} \sqrt [4]{-1} b^2 \operatorname {PolyLog}\left (2,1-\frac {\sqrt {2} \left (\sqrt {c} x+\sqrt [4]{-1}\right )}{\sqrt [4]{-1} \sqrt {c} x+1}\right ) c^{5/2}-\frac {1}{5} (-1)^{3/4} b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right ) c^{5/2}-\frac {1}{5} (-1)^{3/4} b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{(-1)^{3/4} \sqrt {c} x+1}\right ) c^{5/2}+\frac {1}{10} (-1)^{3/4} b^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {2} \left (\sqrt {c} x+(-1)^{3/4}\right )}{(-1)^{3/4} \sqrt {c} x+1}+1\right ) c^{5/2}+\frac {1}{10} (-1)^{3/4} b^2 \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 ) c^{5/2}+\frac {1}{10} \sqrt [4]{-1} b^2 \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 ) c^{5/2}-\frac {b^2 \log \left (1-i c x^2\right ) c^2}{5 x}-\frac {i b \left (2 a+i b \log \left (1-i c x^2\right )\right ) c^2}{5 x}-\frac {8 b^2 c^2}{15 x}+\frac {2 i a b c^2}{5 x}-\frac {i b^2 \log \left (1-i c x^2\right ) c}{15 x^3}-\frac {b \left (2 a+i b \log \left (1-i c x^2\right )\right ) c}{15 x^3}+\frac {2 i b^2 \log \left (i c x^2+1\right ) c}{15 x^3}-\frac {2 a b c}{15 x^3}-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{20 x^5}+\frac {b^2 \log ^2\left (i c x^2+1\right )}{20 x^5}-\frac {b^2 \log \left (1-i c x^2\right ) \log \left (i c x^2+1\right )}{10 x^5}+\frac {i a b \log \left (i c x^2+1\right )}{5 x^5} \]

[In]

Int[(a + b*ArcTan[c*x^2])^2/x^6,x]

[Out]

(-2*a*b*c)/(15*x^3) + (((2*I)/5)*a*b*c^2)/x - (8*b^2*c^2)/(15*x) - (4*(-1)^(3/4)*b^2*c^(5/2)*ArcTan[(-1)^(3/4)
*Sqrt[c]*x])/15 - ((-1)^(1/4)*b^2*c^(5/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x]^2)/5 + (2*(-1)^(1/4)*a*b*c^(5/2)*ArcTan
h[(-1)^(3/4)*Sqrt[c]*x])/5 + (4*(-1)^(3/4)*b^2*c^(5/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x])/15 + ((-1)^(3/4)*b^2*c^(
5/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]^2)/5 + (2*(-1)^(3/4)*b^2*c^(5/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 - (-
1)^(1/4)*Sqrt[c]*x)])/5 - (2*(-1)^(3/4)*b^2*c^(5/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 + (-1)^(1/4)*Sqrt[c]
*x)])/5 + ((-1)^(3/4)*b^2*c^(5/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[(Sqrt[2]*((-1)^(1/4) + Sqrt[c]*x))/(1 + (-1
)^(1/4)*Sqrt[c]*x)])/5 - (2*(-1)^(3/4)*b^2*c^(5/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 - (-1)^(3/4)*Sqrt[c]
*x)])/5 + (2*(-1)^(3/4)*b^2*c^(5/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 + (-1)^(3/4)*Sqrt[c]*x)])/5 - ((-1)
^(3/4)*b^2*c^(5/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[-((Sqrt[2]*((-1)^(3/4) + Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt
[c]*x))])/5 - ((-1)^(3/4)*b^2*c^(5/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[((1 + I)*(1 + (-1)^(1/4)*Sqrt[c]*x))/(
1 + (-1)^(3/4)*Sqrt[c]*x)])/5 + ((-1)^(3/4)*b^2*c^(5/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[((1 - I)*(1 + (-1)^(3
/4)*Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/5 - ((I/15)*b^2*c*Log[1 - I*c*x^2])/x^3 - (b^2*c^2*Log[1 - I*c*x^
2])/(5*x) + ((-1)^(3/4)*b^2*c^(5/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[1 - I*c*x^2])/5 - (b*c*(2*a + I*b*Log[1
- I*c*x^2]))/(15*x^3) - ((I/5)*b*c^2*(2*a + I*b*Log[1 - I*c*x^2]))/x + ((-1)^(1/4)*b*c^(5/2)*ArcTan[(-1)^(3/4)
*Sqrt[c]*x]*(2*a + I*b*Log[1 - I*c*x^2]))/5 - (2*a + I*b*Log[1 - I*c*x^2])^2/(20*x^5) + ((I/5)*a*b*Log[1 + I*c
*x^2])/x^5 + (((2*I)/15)*b^2*c*Log[1 + I*c*x^2])/x^3 - ((-1)^(3/4)*b^2*c^(5/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Lo
g[1 + I*c*x^2])/5 - ((-1)^(3/4)*b^2*c^(5/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[1 + I*c*x^2])/5 - (b^2*Log[1 - I
*c*x^2]*Log[1 + I*c*x^2])/(10*x^5) + (b^2*Log[1 + I*c*x^2]^2)/(20*x^5) - ((-1)^(1/4)*b^2*c^(5/2)*PolyLog[2, 1
- 2/(1 - (-1)^(1/4)*Sqrt[c]*x)])/5 - ((-1)^(1/4)*b^2*c^(5/2)*PolyLog[2, 1 - 2/(1 + (-1)^(1/4)*Sqrt[c]*x)])/5 +
((-1)^(1/4)*b^2*c^(5/2)*PolyLog[2, 1 - (Sqrt[2]*((-1)^(1/4) + Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/10 - (
(-1)^(3/4)*b^2*c^(5/2)*PolyLog[2, 1 - 2/(1 - (-1)^(3/4)*Sqrt[c]*x)])/5 - ((-1)^(3/4)*b^2*c^(5/2)*PolyLog[2, 1
- 2/(1 + (-1)^(3/4)*Sqrt[c]*x)])/5 + ((-1)^(3/4)*b^2*c^(5/2)*PolyLog[2, 1 + (Sqrt[2]*((-1)^(3/4) + Sqrt[c]*x))
/(1 + (-1)^(3/4)*Sqrt[c]*x)])/10 + ((-1)^(3/4)*b^2*c^(5/2)*PolyLog[2, 1 - ((1 + I)*(1 + (-1)^(1/4)*Sqrt[c]*x))
/(1 + (-1)^(3/4)*Sqrt[c]*x)])/10 + ((-1)^(1/4)*b^2*c^(5/2)*PolyLog[2, 1 - ((1 - I)*(1 + (-1)^(3/4)*Sqrt[c]*x))
/(1 + (-1)^(1/4)*Sqrt[c]*x)])/10

Rule 12

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

Rule 30

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

Rule 209

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

Rule 211

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

Rule 212

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

Rule 214

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

Rule 331

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

Rule 2352

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

Rule 2449

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

Rule 2497

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/D[u, x])]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rule 2505

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

Rule 2507

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

Rule 2520

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

Rule 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 2637

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

Rule 4950

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

Rule 4964

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

Rule 4966

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

Rule 5040

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

Rule 5048

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

Rule 6055

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

Rule 6057

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

Rule 6131

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

Rule 6139

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

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{4 x^6}+\frac {b \left (-2 i a+b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{2 x^6}-\frac {b^2 \log ^2\left (1+i c x^2\right )}{4 x^6}\right ) \, dx \\ & = \frac {1}{4} \int \frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{x^6} \, dx+\frac {1}{2} b \int \frac {\left (-2 i a+b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{x^6} \, dx-\frac {1}{4} b^2 \int \frac {\log ^2\left (1+i c x^2\right )}{x^6} \, dx \\ & = -\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{20 x^5}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{20 x^5}+\frac {1}{2} b \int \left (-\frac {2 i a \log \left (1+i c x^2\right )}{x^6}+\frac {b \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{x^6}\right ) \, dx+\frac {1}{5} (b c) \int \frac {2 a+i b \log \left (1-i c x^2\right )}{x^4 \left (1-i c x^2\right )} \, dx-\frac {1}{5} \left (i b^2 c\right ) \int \frac {\log \left (1+i c x^2\right )}{x^4 \left (1+i c x^2\right )} \, dx \\ & = -\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{20 x^5}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{20 x^5}-(i a b) \int \frac {\log \left (1+i c x^2\right )}{x^6} \, dx+\frac {1}{2} b^2 \int \frac {\log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{x^6} \, dx+\frac {1}{5} (b c) \int \left (\frac {2 a+i b \log \left (1-i c x^2\right )}{x^4}+\frac {i c \left (2 a+i b \log \left (1-i c x^2\right )\right )}{x^2}-\frac {i c^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )}{i+c x^2}\right ) \, dx-\frac {1}{5} \left (i b^2 c\right ) \int \left (\frac {\log \left (1+i c x^2\right )}{x^4}-\frac {i c \log \left (1+i c x^2\right )}{x^2}+\frac {i c^2 \log \left (1+i c x^2\right )}{-i+c x^2}\right ) \, dx \\ & = -\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{20 x^5}+\frac {i a b \log \left (1+i c x^2\right )}{5 x^5}-\frac {b^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{10 x^5}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{20 x^5}-\frac {1}{2} b^2 \int \frac {2 c \log \left (1-i c x^2\right )}{5 x^4 \left (i-c x^2\right )} \, dx-\frac {1}{2} b^2 \int \frac {2 c \log \left (1+i c x^2\right )}{5 x^4 \left (-i-c x^2\right )} \, dx+\frac {1}{5} (b c) \int \frac {2 a+i b \log \left (1-i c x^2\right )}{x^4} \, dx+\frac {1}{5} (2 a b c) \int \frac {1}{x^4 \left (1+i c x^2\right )} \, dx-\frac {1}{5} \left (i b^2 c\right ) \int \frac {\log \left (1+i c x^2\right )}{x^4} \, dx+\frac {1}{5} \left (i b c^2\right ) \int \frac {2 a+i b \log \left (1-i c x^2\right )}{x^2} \, dx-\frac {1}{5} \left (b^2 c^2\right ) \int \frac {\log \left (1+i c x^2\right )}{x^2} \, dx-\frac {1}{5} \left (i b c^3\right ) \int \frac {2 a+i b \log \left (1-i c x^2\right )}{i+c x^2} \, dx+\frac {1}{5} \left (b^2 c^3\right ) \int \frac {\log \left (1+i c x^2\right )}{-i+c x^2} \, dx \\ & = -\frac {2 a b c}{15 x^3}-\frac {b c \left (2 a+i b \log \left (1-i c x^2\right )\right )}{15 x^3}-\frac {i b c^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )}{5 x}+\frac {1}{5} \sqrt [4]{-1} b c^{5/2} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{20 x^5}+\frac {i a b \log \left (1+i c x^2\right )}{5 x^5}+\frac {i b^2 c \log \left (1+i c x^2\right )}{15 x^3}+\frac {b^2 c^2 \log \left (1+i c x^2\right )}{5 x}-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )-\frac {b^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{10 x^5}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{20 x^5}-\frac {1}{5} \left (b^2 c\right ) \int \frac {\log \left (1-i c x^2\right )}{x^4 \left (i-c x^2\right )} \, dx-\frac {1}{5} \left (b^2 c\right ) \int \frac {\log \left (1+i c x^2\right )}{x^4 \left (-i-c x^2\right )} \, dx-\frac {1}{5} \left (2 i a b c^2\right ) \int \frac {1}{x^2 \left (1+i c x^2\right )} \, dx+\frac {1}{15} \left (2 b^2 c^2\right ) \int \frac {1}{x^2 \left (1-i c x^2\right )} \, dx+\frac {1}{15} \left (2 b^2 c^2\right ) \int \frac {1}{x^2 \left (1+i c x^2\right )} \, dx+\frac {1}{5} \left (2 i b^2 c^3\right ) \int \frac {1}{1-i c x^2} \, dx-\frac {1}{5} \left (2 i b^2 c^3\right ) \int \frac {1}{1+i c x^2} \, dx+\frac {1}{5} \left (2 i b^2 c^4\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+\frac {1}{5} \left (2 i b^2 c^4\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 \\ & = -\frac {2 a b c}{15 x^3}+\frac {2 i a b c^2}{5 x}-\frac {4 b^2 c^2}{15 x}-\frac {2}{5} (-1)^{3/4} b^2 c^{5/2} \arctan \left ((-1)^{3/4} \sqrt {c} x\right )+\frac {2}{5} (-1)^{3/4} b^2 c^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )-\frac {b c \left (2 a+i b \log \left (1-i c x^2\right )\right )}{15 x^3}-\frac {i b c^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )}{5 x}+\frac {1}{5} \sqrt [4]{-1} b c^{5/2} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{20 x^5}+\frac {i a b \log \left (1+i c x^2\right )}{5 x^5}+\frac {i b^2 c \log \left (1+i c x^2\right )}{15 x^3}+\frac {b^2 c^2 \log \left (1+i c x^2\right )}{5 x}-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )-\frac {b^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{10 x^5}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{20 x^5}-\frac {1}{5} \left (b^2 c\right ) \int \left (-\frac {i \log \left (1-i c x^2\right )}{x^4}-\frac {c \log \left (1-i c x^2\right )}{x^2}+\frac {c^2 \log \left (1-i c x^2\right )}{-i+c x^2}\right ) \, dx-\frac {1}{5} \left (b^2 c\right ) \int \left (\frac {i \log \left (1+i c x^2\right )}{x^4}-\frac {c \log \left (1+i c x^2\right )}{x^2}+\frac {c^2 \log \left (1+i c x^2\right )}{i+c x^2}\right ) \, dx-\frac {1}{5} \left (2 a b c^3\right ) \int \frac {1}{1+i c x^2} \, dx+\frac {1}{15} \left (2 i b^2 c^3\right ) \int \frac {1}{1-i c x^2} \, dx-\frac {1}{15} \left (2 i b^2 c^3\right ) \int \frac {1}{1+i c x^2} \, dx-\frac {1}{5} \left (2 \sqrt [4]{-1} b^2 c^{7/2}\right ) \int \frac {x \arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{1-i c x^2} \, dx-\frac {1}{5} \left (2 \sqrt [4]{-1} b^2 c^{7/2}\right ) \int \frac {x \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{1+i c x^2} \, dx \\ & = -\frac {2 a b c}{15 x^3}+\frac {2 i a b c^2}{5 x}-\frac {4 b^2 c^2}{15 x}-\frac {8}{15} (-1)^{3/4} b^2 c^{5/2} \arctan \left ((-1)^{3/4} \sqrt {c} x\right )-\frac {1}{5} \sqrt [4]{-1} b^2 c^{5/2} \arctan \left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{5} \sqrt [4]{-1} a b c^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )+\frac {8}{15} (-1)^{3/4} b^2 c^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )+\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )^2-\frac {b c \left (2 a+i b \log \left (1-i c x^2\right )\right )}{15 x^3}-\frac {i b c^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )}{5 x}+\frac {1}{5} \sqrt [4]{-1} b c^{5/2} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{20 x^5}+\frac {i a b \log \left (1+i c x^2\right )}{5 x^5}+\frac {i b^2 c \log \left (1+i c x^2\right )}{15 x^3}+\frac {b^2 c^2 \log \left (1+i c x^2\right )}{5 x}-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )-\frac {b^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{10 x^5}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{20 x^5}+\frac {1}{5} \left (i b^2 c\right ) \int \frac {\log \left (1-i c x^2\right )}{x^4} \, dx-\frac {1}{5} \left (i b^2 c\right ) \int \frac {\log \left (1+i c x^2\right )}{x^4} \, dx+\frac {1}{5} \left (b^2 c^2\right ) \int \frac {\log \left (1-i c x^2\right )}{x^2} \, dx+\frac {1}{5} \left (b^2 c^2\right ) \int \frac {\log \left (1+i c x^2\right )}{x^2} \, dx-\frac {1}{5} \left (2 i b^2 c^3\right ) \int \frac {\arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{i-(-1)^{3/4} \sqrt {c} x} \, dx+\frac {1}{5} \left (2 i b^2 c^3\right ) \int \frac {\text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{1-(-1)^{3/4} \sqrt {c} x} \, dx-\frac {1}{5} \left (b^2 c^3\right ) \int \frac {\log \left (1-i c x^2\right )}{-i+c x^2} \, dx-\frac {1}{5} \left (b^2 c^3\right ) \int \frac {\log \left (1+i c x^2\right )}{i+c x^2} \, dx \\ & = -\frac {2 a b c}{15 x^3}+\frac {2 i a b c^2}{5 x}-\frac {4 b^2 c^2}{15 x}-\frac {8}{15} (-1)^{3/4} b^2 c^{5/2} \arctan \left ((-1)^{3/4} \sqrt {c} x\right )-\frac {1}{5} \sqrt [4]{-1} b^2 c^{5/2} \arctan \left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{5} \sqrt [4]{-1} a b c^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )+\frac {8}{15} (-1)^{3/4} b^2 c^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )+\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{5} (-1)^{3/4} b^2 c^{5/2} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {2}{5} (-1)^{3/4} b^2 c^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )-\frac {i b^2 c \log \left (1-i c x^2\right )}{15 x^3}-\frac {b^2 c^2 \log \left (1-i c x^2\right )}{5 x}+\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )-\frac {b c \left (2 a+i b \log \left (1-i c x^2\right )\right )}{15 x^3}-\frac {i b c^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )}{5 x}+\frac {1}{5} \sqrt [4]{-1} b c^{5/2} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{20 x^5}+\frac {i a b \log \left (1+i c x^2\right )}{5 x^5}+\frac {2 i b^2 c \log \left (1+i c x^2\right )}{15 x^3}-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )-\frac {b^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{10 x^5}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{20 x^5}+\frac {1}{15} \left (2 b^2 c^2\right ) \int \frac {1}{x^2 \left (1-i c x^2\right )} \, dx+\frac {1}{15} \left (2 b^2 c^2\right ) \int \frac {1}{x^2 \left (1+i c x^2\right )} \, dx-\frac {1}{5} \left (2 i b^2 c^3\right ) \int \frac {1}{1-i c x^2} \, dx+\frac {1}{5} \left (2 i b^2 c^3\right ) \int \frac {1}{1+i c x^2} \, dx+\frac {1}{5} \left (2 i b^2 c^3\right ) \int \frac {\log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )}{1-i c x^2} \, dx-\frac {1}{5} \left (2 i b^2 c^3\right ) \int \frac {\log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )}{1+i c x^2} \, dx+\frac {1}{5} \left (2 i b^2 c^4\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+\frac {1}{5} \left (2 i b^2 c^4\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 \\ & = -\frac {2 a b c}{15 x^3}+\frac {2 i a b c^2}{5 x}-\frac {8 b^2 c^2}{15 x}-\frac {2}{15} (-1)^{3/4} b^2 c^{5/2} \arctan \left ((-1)^{3/4} \sqrt {c} x\right )-\frac {1}{5} \sqrt [4]{-1} b^2 c^{5/2} \arctan \left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{5} \sqrt [4]{-1} a b c^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )+\frac {2}{15} (-1)^{3/4} b^2 c^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )+\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{5} (-1)^{3/4} b^2 c^{5/2} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {2}{5} (-1)^{3/4} b^2 c^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )-\frac {i b^2 c \log \left (1-i c x^2\right )}{15 x^3}-\frac {b^2 c^2 \log \left (1-i c x^2\right )}{5 x}+\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )-\frac {b c \left (2 a+i b \log \left (1-i c x^2\right )\right )}{15 x^3}-\frac {i b c^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )}{5 x}+\frac {1}{5} \sqrt [4]{-1} b c^{5/2} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{20 x^5}+\frac {i a b \log \left (1+i c x^2\right )}{5 x^5}+\frac {2 i b^2 c \log \left (1+i c x^2\right )}{15 x^3}-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )-\frac {b^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{10 x^5}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{20 x^5}-\frac {1}{5} \left (2 \sqrt [4]{-1} b^2 c^{5/2}\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {1}{5} \left (2 (-1)^{3/4} b^2 c^{5/2}\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-(-1)^{3/4} \sqrt {c} x}\right )+\frac {1}{15} \left (2 i b^2 c^3\right ) \int \frac {1}{1-i c x^2} \, dx-\frac {1}{15} \left (2 i b^2 c^3\right ) \int \frac {1}{1+i c x^2} \, dx-\frac {1}{5} \left (2 \sqrt [4]{-1} b^2 c^{7/2}\right ) \int \frac {x \arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{1+i c x^2} \, dx-\frac {1}{5} \left (2 \sqrt [4]{-1} b^2 c^{7/2}\right ) \int \frac {x \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{1-i c x^2} \, dx \\ & = -\frac {2 a b c}{15 x^3}+\frac {2 i a b c^2}{5 x}-\frac {8 b^2 c^2}{15 x}-\frac {4}{15} (-1)^{3/4} b^2 c^{5/2} \arctan \left ((-1)^{3/4} \sqrt {c} x\right )-\frac {1}{5} \sqrt [4]{-1} b^2 c^{5/2} \arctan \left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{5} \sqrt [4]{-1} a b c^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )+\frac {4}{15} (-1)^{3/4} b^2 c^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )+\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{5} (-1)^{3/4} b^2 c^{5/2} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {2}{5} (-1)^{3/4} b^2 c^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )-\frac {i b^2 c \log \left (1-i c x^2\right )}{15 x^3}-\frac {b^2 c^2 \log \left (1-i c x^2\right )}{5 x}+\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )-\frac {b c \left (2 a+i b \log \left (1-i c x^2\right )\right )}{15 x^3}-\frac {i b c^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )}{5 x}+\frac {1}{5} \sqrt [4]{-1} b c^{5/2} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{20 x^5}+\frac {i a b \log \left (1+i c x^2\right )}{5 x^5}+\frac {2 i b^2 c \log \left (1+i c x^2\right )}{15 x^3}-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )-\frac {b^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{10 x^5}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{20 x^5}-\frac {1}{5} \sqrt [4]{-1} b^2 c^{5/2} \operatorname {PolyLog}\left (2,1-\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \operatorname {PolyLog}\left (2,1-\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )-\frac {1}{5} \left (2 \sqrt [4]{-1} b^2 c^{7/2}\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-\frac {1}{5} \left (2 \sqrt [4]{-1} b^2 c^{7/2}\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 \\ & = -\frac {2 a b c}{15 x^3}+\frac {2 i a b c^2}{5 x}-\frac {8 b^2 c^2}{15 x}-\frac {4}{15} (-1)^{3/4} b^2 c^{5/2} \arctan \left ((-1)^{3/4} \sqrt {c} x\right )-\frac {1}{5} \sqrt [4]{-1} b^2 c^{5/2} \arctan \left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{5} \sqrt [4]{-1} a b c^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )+\frac {4}{15} (-1)^{3/4} b^2 c^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )+\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{5} (-1)^{3/4} b^2 c^{5/2} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {2}{5} (-1)^{3/4} b^2 c^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )-\frac {i b^2 c \log \left (1-i c x^2\right )}{15 x^3}-\frac {b^2 c^2 \log \left (1-i c x^2\right )}{5 x}+\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )-\frac {b c \left (2 a+i b \log \left (1-i c x^2\right )\right )}{15 x^3}-\frac {i b c^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )}{5 x}+\frac {1}{5} \sqrt [4]{-1} b c^{5/2} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{20 x^5}+\frac {i a b \log \left (1+i c x^2\right )}{5 x^5}+\frac {2 i b^2 c \log \left (1+i c x^2\right )}{15 x^3}-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )-\frac {b^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{10 x^5}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{20 x^5}-\frac {1}{5} \sqrt [4]{-1} b^2 c^{5/2} \operatorname {PolyLog}\left (2,1-\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \operatorname {PolyLog}\left (2,1-\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )-\frac {1}{5} \left ((-1)^{3/4} b^2 c^3\right ) \int \frac {\arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt [4]{-1}-\sqrt {c} x} \, dx+\frac {1}{5} \left ((-1)^{3/4} b^2 c^3\right ) \int \frac {\arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt [4]{-1}+\sqrt {c} x} \, dx+\frac {1}{5} \left ((-1)^{3/4} b^2 c^3\right ) \int \frac {\text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{-(-1)^{3/4}-\sqrt {c} x} \, dx-\frac {1}{5} \left ((-1)^{3/4} b^2 c^3\right ) \int \frac {\text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{-(-1)^{3/4}+\sqrt {c} x} \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [F]

\[ \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{x^6} \, dx=\int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{x^6} \, dx \]

[In]

Integrate[(a + b*ArcTan[c*x^2])^2/x^6,x]

[Out]

Integrate[(a + b*ArcTan[c*x^2])^2/x^6, x]

Maple [F]

\[\int \frac {{\left (a +b \arctan \left (c \,x^{2}\right )\right )}^{2}}{x^{6}}d x\]

[In]

int((a+b*arctan(c*x^2))^2/x^6,x)

[Out]

int((a+b*arctan(c*x^2))^2/x^6,x)

Fricas [F]

\[ \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{x^6} \, dx=\int { \frac {{\left (b \arctan \left (c x^{2}\right ) + a\right )}^{2}}{x^{6}} \,d x } \]

[In]

integrate((a+b*arctan(c*x^2))^2/x^6,x, algorithm="fricas")

[Out]

integral((b^2*arctan(c*x^2)^2 + 2*a*b*arctan(c*x^2) + a^2)/x^6, x)

Sympy [F]

\[ \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{x^6} \, dx=\int \frac {\left (a + b \operatorname {atan}{\left (c x^{2} \right )}\right )^{2}}{x^{6}}\, dx \]

[In]

integrate((a+b*atan(c*x**2))**2/x**6,x)

[Out]

Integral((a + b*atan(c*x**2))**2/x**6, x)

Maxima [F]

\[ \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{x^6} \, dx=\int { \frac {{\left (b \arctan \left (c x^{2}\right ) + a\right )}^{2}}{x^{6}} \,d x } \]

[In]

integrate((a+b*arctan(c*x^2))^2/x^6,x, algorithm="maxima")

[Out]

-1/30*((6*sqrt(2)*c^(3/2)*arctan(1/2*sqrt(2)*(2*c*x + sqrt(2)*sqrt(c))/sqrt(c)) + 6*sqrt(2)*c^(3/2)*arctan(1/2
*sqrt(2)*(2*c*x - sqrt(2)*sqrt(c))/sqrt(c)) + 3*sqrt(2)*c^(3/2)*log(c*x^2 + sqrt(2)*sqrt(c)*x + 1) - 3*sqrt(2)
*c^(3/2)*log(c*x^2 - sqrt(2)*sqrt(c)*x + 1) + 8/x^3)*c + 12*arctan(c*x^2)/x^5)*a*b + 1/80*(80*x^5*integrate(-1
/80*(8*c^2*x^4*log(c^2*x^4 + 1) - 16*c*x^2*arctan(c*x^2) - 60*(c^2*x^4 + 1)*arctan(c*x^2)^2 - 5*(c^2*x^4 + 1)*
log(c^2*x^4 + 1)^2)/(c^2*x^10 + x^6), x) - 4*arctan(c*x^2)^2 + log(c^2*x^4 + 1)^2)*b^2/x^5 - 1/5*a^2/x^5

Giac [F]

\[ \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{x^6} \, dx=\int { \frac {{\left (b \arctan \left (c x^{2}\right ) + a\right )}^{2}}{x^{6}} \,d x } \]

[In]

integrate((a+b*arctan(c*x^2))^2/x^6,x, algorithm="giac")

[Out]

integrate((b*arctan(c*x^2) + a)^2/x^6, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{x^6} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x^2\right )\right )}^2}{x^6} \,d x \]

[In]

int((a + b*atan(c*x^2))^2/x^6,x)

[Out]

int((a + b*atan(c*x^2))^2/x^6, x)