Integrand size = 16, antiderivative size = 1360 \[ \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{x^4} \, dx =\text {Too large to display} \]
-1/12*(2*a+I*b*ln(1-I*c*x^2))^2/x^3-1/3*b*c*(2*a+I*b*ln(1-I*c*x^2))/x-1/6* b^2*ln(1-I*c*x^2)*ln(1+I*c*x^2)/x^3+1/3*(-1)^(3/4)*b^2*c^(3/2)*polylog(2,1 -2/(1-(-1)^(1/4)*x*c^(1/2)))+1/3*(-1)^(3/4)*b^2*c^(3/2)*polylog(2,1-2/(1+( -1)^(1/4)*x*c^(1/2)))-1/6*(-1)^(3/4)*b^2*c^(3/2)*polylog(2,1-2^(1/2)*((-1) ^(1/4)+x*c^(1/2))/(1+(-1)^(1/4)*x*c^(1/2)))+1/3*(-1)^(1/4)*b^2*c^(3/2)*pol ylog(2,1-2/(1-(-1)^(3/4)*x*c^(1/2)))+1/3*(-1)^(1/4)*b^2*c^(3/2)*polylog(2, 1-2/(1+(-1)^(3/4)*x*c^(1/2)))-1/6*(-1)^(1/4)*b^2*c^(3/2)*polylog(2,1+2^(1/ 2)*((-1)^(3/4)+x*c^(1/2))/(1+(-1)^(3/4)*x*c^(1/2)))-1/6*(-1)^(1/4)*b^2*c^( 3/2)*polylog(2,1-(1+I)*(1+(-1)^(1/4)*x*c^(1/2))/(1+(-1)^(3/4)*x*c^(1/2)))- 1/6*(-1)^(3/4)*b^2*c^(3/2)*polylog(2,1+(-1+I)*(1+(-1)^(3/4)*x*c^(1/2))/(1+ (-1)^(1/4)*x*c^(1/2)))-2/3*a*b*c/x-4/3*(-1)^(1/4)*b^2*c^(3/2)*arctan((-1)^ (3/4)*x*c^(1/2))+1/3*(-1)^(3/4)*b^2*c^(3/2)*arctan((-1)^(3/4)*x*c^(1/2))^2 -4/3*(-1)^(1/4)*b^2*c^(3/2)*arctanh((-1)^(3/4)*x*c^(1/2))-1/3*(-1)^(1/4)*b ^2*c^(3/2)*arctanh((-1)^(3/4)*x*c^(1/2))^2+1/12*b^2*ln(1+I*c*x^2)^2/x^3+1/ 3*I*a*b*ln(1+I*c*x^2)/x^3+2/3*I*b^2*c*ln(1+I*c*x^2)/x+1/3*(-1)^(1/4)*b^2*c ^(3/2)*arctan((-1)^(3/4)*x*c^(1/2))*ln(2^(1/2)*((-1)^(1/4)+x*c^(1/2))/(1+( -1)^(1/4)*x*c^(1/2)))+2/3*(-1)^(1/4)*b^2*c^(3/2)*arctanh((-1)^(3/4)*x*c^(1 /2))*ln(2/(1-(-1)^(3/4)*x*c^(1/2)))-2/3*(-1)^(1/4)*b^2*c^(3/2)*arctanh((-1 )^(3/4)*x*c^(1/2))*ln(2/(1+(-1)^(3/4)*x*c^(1/2)))+1/3*(-1)^(1/4)*b^2*c^(3/ 2)*arctanh((-1)^(3/4)*x*c^(1/2))*ln(-2^(1/2)*((-1)^(3/4)+x*c^(1/2))/(1+...
\[ \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{x^4} \, dx=\int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{x^4} \, dx \]
Time = 2.26 (sec) , antiderivative size = 1360, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5365, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{x^4} \, dx\) |
| \(\Big \downarrow \) 5365 |
| \(\displaystyle \int \left (\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{4 x^4}+\frac {b \log \left (1+i c x^2\right ) \left (b \log \left (1-i c x^2\right )-2 i a\right )}{2 x^4}-\frac {b^2 \log ^2\left (1+i c x^2\right )}{4 x^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} (-1)^{3/4} c^{3/2} \arctan \left ((-1)^{3/4} \sqrt {c} x\right )^2 b^2-\frac {1}{3} \sqrt [4]{-1} c^{3/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )^2 b^2+\frac {\log ^2\left (i c x^2+1\right ) b^2}{12 x^3}-\frac {4}{3} \sqrt [4]{-1} c^{3/2} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) b^2-\frac {4}{3} \sqrt [4]{-1} c^{3/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) b^2+\frac {2}{3} \sqrt [4]{-1} c^{3/2} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right ) b^2-\frac {2}{3} \sqrt [4]{-1} c^{3/2} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{\sqrt [4]{-1} \sqrt {c} x+1}\right ) b^2+\frac {1}{3} \sqrt [4]{-1} c^{3/2} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {\sqrt {2} \left (\sqrt {c} x+\sqrt [4]{-1}\right )}{\sqrt [4]{-1} \sqrt {c} x+1}\right ) b^2+\frac {2}{3} \sqrt [4]{-1} c^{3/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right ) b^2-\frac {2}{3} \sqrt [4]{-1} c^{3/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{(-1)^{3/4} \sqrt {c} x+1}\right ) b^2+\frac {1}{3} \sqrt [4]{-1} c^{3/2} \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 ) b^2+\frac {1}{3} \sqrt [4]{-1} c^{3/2} \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 ) b^2+\frac {1}{3} \sqrt [4]{-1} c^{3/2} \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 ) b^2-\frac {1}{3} \sqrt [4]{-1} c^{3/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right ) b^2-\frac {i c \log \left (1-i c x^2\right ) b^2}{3 x}-\frac {1}{3} \sqrt [4]{-1} c^{3/2} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (i c x^2+1\right ) b^2+\frac {1}{3} \sqrt [4]{-1} c^{3/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (i c x^2+1\right ) b^2-\frac {\log \left (1-i c x^2\right ) \log \left (i c x^2+1\right ) b^2}{6 x^3}+\frac {2 i c \log \left (i c x^2+1\right ) b^2}{3 x}+\frac {1}{3} (-1)^{3/4} c^{3/2} \operatorname {PolyLog}\left (2,1-\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right ) b^2+\frac {1}{3} (-1)^{3/4} c^{3/2} \operatorname {PolyLog}\left (2,1-\frac {2}{\sqrt [4]{-1} \sqrt {c} x+1}\right ) b^2-\frac {1}{6} (-1)^{3/4} c^{3/2} \operatorname {PolyLog}\left (2,1-\frac {\sqrt {2} \left (\sqrt {c} x+\sqrt [4]{-1}\right )}{\sqrt [4]{-1} \sqrt {c} x+1}\right ) b^2+\frac {1}{3} \sqrt [4]{-1} c^{3/2} \operatorname {PolyLog}\left (2,1-\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right ) b^2+\frac {1}{3} \sqrt [4]{-1} c^{3/2} \operatorname {PolyLog}\left (2,1-\frac {2}{(-1)^{3/4} \sqrt {c} x+1}\right ) b^2-\frac {1}{6} \sqrt [4]{-1} c^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {2} \left (\sqrt {c} x+(-1)^{3/4}\right )}{(-1)^{3/4} \sqrt {c} x+1}+1\right ) b^2-\frac {1}{6} \sqrt [4]{-1} c^{3/2} \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 ) b^2-\frac {1}{6} (-1)^{3/4} c^{3/2} \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 ) b^2+\frac {2}{3} (-1)^{3/4} a c^{3/2} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) b-\frac {1}{3} (-1)^{3/4} c^{3/2} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right ) b-\frac {c \left (2 a+i b \log \left (1-i c x^2\right )\right ) b}{3 x}+\frac {i a \log \left (i c x^2+1\right ) b}{3 x^3}-\frac {2 a c b}{3 x}-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{12 x^3}\) |
(-2*a*b*c)/(3*x) - (4*(-1)^(1/4)*b^2*c^(3/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x]) /3 + ((-1)^(3/4)*b^2*c^(3/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x]^2)/3 + (2*(-1)^( 3/4)*a*b*c^(3/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x])/3 - (4*(-1)^(1/4)*b^2*c^(3 /2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x])/3 - ((-1)^(1/4)*b^2*c^(3/2)*ArcTanh[(-1 )^(3/4)*Sqrt[c]*x]^2)/3 + (2*(-1)^(1/4)*b^2*c^(3/2)*ArcTan[(-1)^(3/4)*Sqrt [c]*x]*Log[2/(1 - (-1)^(1/4)*Sqrt[c]*x)])/3 - (2*(-1)^(1/4)*b^2*c^(3/2)*Ar cTan[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 + (-1)^(1/4)*Sqrt[c]*x)])/3 + ((-1)^(1 /4)*b^2*c^(3/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[(Sqrt[2]*((-1)^(1/4) + Sq rt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/3 + (2*(-1)^(1/4)*b^2*c^(3/2)*ArcTa nh[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 - (-1)^(3/4)*Sqrt[c]*x)])/3 - (2*(-1)^(1 /4)*b^2*c^(3/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 + (-1)^(3/4)*Sqrt[c ]*x)])/3 + ((-1)^(1/4)*b^2*c^(3/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[-((Sq rt[2]*((-1)^(3/4) + Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x))])/3 + ((-1)^(1 /4)*b^2*c^(3/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[((1 + I)*(1 + (-1)^(1/4) *Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x)])/3 + ((-1)^(1/4)*b^2*c^(3/2)*ArcT an[(-1)^(3/4)*Sqrt[c]*x]*Log[((1 - I)*(1 + (-1)^(3/4)*Sqrt[c]*x))/(1 + (-1 )^(1/4)*Sqrt[c]*x)])/3 - ((I/3)*b^2*c*Log[1 - I*c*x^2])/x - ((-1)^(1/4)*b^ 2*c^(3/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[1 - I*c*x^2])/3 - (b*c*(2*a + I*b*Log[1 - I*c*x^2]))/(3*x) - ((-1)^(3/4)*b*c^(3/2)*ArcTan[(-1)^(3/4)*Sqr t[c]*x]*(2*a + I*b*Log[1 - I*c*x^2]))/3 - (2*a + I*b*Log[1 - I*c*x^2])^...
3.1.84.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTan[(c_.)*(x_)^(n_)]*(b_.))^(p_)*(x_)^(m_.), x_Symbol] :> I nt[ExpandIntegrand[x^m*(a + (I*b*Log[1 - I*c*x^n])/2 - (I*b*Log[1 + I*c*x^n ])/2)^p, x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1] && IGtQ[n, 0] && Integ erQ[m]
\[\int \frac {{\left (a +b \arctan \left (c \,x^{2}\right )\right )}^{2}}{x^{4}}d x\]
\[ \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{x^4} \, dx=\int { \frac {{\left (b \arctan \left (c x^{2}\right ) + a\right )}^{2}}{x^{4}} \,d x } \]
\[ \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{x^4} \, dx=\int \frac {\left (a + b \operatorname {atan}{\left (c x^{2} \right )}\right )^{2}}{x^{4}}\, dx \]
\[ \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{x^4} \, dx=\int { \frac {{\left (b \arctan \left (c x^{2}\right ) + a\right )}^{2}}{x^{4}} \,d x } \]
-1/6*((c^2*(2*sqrt(2)*arctan(1/2*sqrt(2)*(2*c*x + sqrt(2)*sqrt(c))/sqrt(c) )/c^(3/2) + 2*sqrt(2)*arctan(1/2*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)*l og(c*x^2 - sqrt(2)*sqrt(c)*x + 1)/c^(3/2)) + 8/x)*c + 4*arctan(c*x^2)/x^3) *a*b + 1/48*(48*x^3*integrate(-1/48*(8*c^2*x^4*log(c^2*x^4 + 1) - 16*c*x^2 *arctan(c*x^2) - 36*(c^2*x^4 + 1)*arctan(c*x^2)^2 - 3*(c^2*x^4 + 1)*log(c^ 2*x^4 + 1)^2)/(c^2*x^8 + x^4), x) - 4*arctan(c*x^2)^2 + log(c^2*x^4 + 1)^2 )*b^2/x^3 - 1/3*a^2/x^3
\[ \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{x^4} \, dx=\int { \frac {{\left (b \arctan \left (c x^{2}\right ) + a\right )}^{2}}{x^{4}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x^2\right )\right )}^2}{x^4} \,d x \]