Integrand size = 16, antiderivative size = 1164 \[ \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{x^2} \, dx=\sqrt [4]{-1} b^2 \sqrt {c} \arctan \left ((-1)^{3/4} \sqrt {c} x\right )^2-2 \sqrt [4]{-1} a b \sqrt {c} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )-(-1)^{3/4} b^2 \sqrt {c} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )^2-2 (-1)^{3/4} b^2 \sqrt {c} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )+2 (-1)^{3/4} b^2 \sqrt {c} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1+\sqrt [4]{-1} \sqrt {c} x}\right )-(-1)^{3/4} b^2 \sqrt {c} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {\sqrt {2} \left (\sqrt [4]{-1}+\sqrt {c} x\right )}{1+\sqrt [4]{-1} \sqrt {c} x}\right )+2 (-1)^{3/4} b^2 \sqrt {c} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )-2 (-1)^{3/4} b^2 \sqrt {c} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1+(-1)^{3/4} \sqrt {c} x}\right )+(-1)^{3/4} b^2 \sqrt {c} \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 )+(-1)^{3/4} b^2 \sqrt {c} \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 )-(-1)^{3/4} b^2 \sqrt {c} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {(1-i) \left (1+(-1)^{3/4} \sqrt {c} x\right )}{1+\sqrt [4]{-1} \sqrt {c} x}\right )-(-1)^{3/4} b^2 \sqrt {c} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )-\sqrt [4]{-1} b \sqrt {c} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{4 x}+\frac {i a b \log \left (1+i c x^2\right )}{x}+(-1)^{3/4} b^2 \sqrt {c} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )+(-1)^{3/4} b^2 \sqrt {c} \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 )}{2 x}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{4 x}+\sqrt [4]{-1} b^2 \sqrt {c} \operatorname {PolyLog}\left (2,1-\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )+\sqrt [4]{-1} b^2 \sqrt {c} \operatorname {PolyLog}\left (2,1-\frac {2}{1+\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {1}{2} \sqrt [4]{-1} b^2 \sqrt {c} \operatorname {PolyLog}\left (2,1-\frac {\sqrt {2} \left (\sqrt [4]{-1}+\sqrt {c} x\right )}{1+\sqrt [4]{-1} \sqrt {c} x}\right )+(-1)^{3/4} b^2 \sqrt {c} \operatorname {PolyLog}\left (2,1-\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )+(-1)^{3/4} b^2 \sqrt {c} \operatorname {PolyLog}\left (2,1-\frac {2}{1+(-1)^{3/4} \sqrt {c} x}\right )-\frac {1}{2} (-1)^{3/4} b^2 \sqrt {c} \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}{2} (-1)^{3/4} b^2 \sqrt {c} \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}{2} \sqrt [4]{-1} b^2 \sqrt {c} \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]
Time = 1.20 (sec) , antiderivative size = 1164, normalized size of antiderivative = 1.00, number of steps used = 47, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.438, Rules used = {4950, 2507, 209, 2520, 12, 5040, 4964, 2449, 2352, 2505, 6874, 212, 30, 2637, 211, 5048, 4966, 2497, 214, 6139, 6057, 6131, 6055} \[ \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{x^2} \, dx=\sqrt [4]{-1} \sqrt {c} \arctan \left ((-1)^{3/4} \sqrt {c} x\right )^2 b^2-(-1)^{3/4} \sqrt {c} \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}{4 x}-2 (-1)^{3/4} \sqrt {c} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right ) b^2+2 (-1)^{3/4} \sqrt {c} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{\sqrt [4]{-1} \sqrt {c} x+1}\right ) b^2-(-1)^{3/4} \sqrt {c} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {\sqrt {2} \left (\sqrt {c} x+\sqrt [4]{-1}\right )}{\sqrt [4]{-1} \sqrt {c} x+1}\right ) b^2+2 (-1)^{3/4} \sqrt {c} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right ) b^2-2 (-1)^{3/4} \sqrt {c} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{(-1)^{3/4} \sqrt {c} x+1}\right ) b^2+(-1)^{3/4} \sqrt {c} \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+(-1)^{3/4} \sqrt {c} \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-(-1)^{3/4} \sqrt {c} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {(1-i) \left ((-1)^{3/4} \sqrt {c} x+1\right )}{\sqrt [4]{-1} \sqrt {c} x+1}\right ) b^2-(-1)^{3/4} \sqrt {c} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right ) b^2+(-1)^{3/4} \sqrt {c} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (i c x^2+1\right ) b^2+(-1)^{3/4} \sqrt {c} \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}{2 x}+\sqrt [4]{-1} \sqrt {c} \operatorname {PolyLog}\left (2,1-\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right ) b^2+\sqrt [4]{-1} \sqrt {c} \operatorname {PolyLog}\left (2,1-\frac {2}{\sqrt [4]{-1} \sqrt {c} x+1}\right ) b^2-\frac {1}{2} \sqrt [4]{-1} \sqrt {c} \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+(-1)^{3/4} \sqrt {c} \operatorname {PolyLog}\left (2,1-\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right ) b^2+(-1)^{3/4} \sqrt {c} \operatorname {PolyLog}\left (2,1-\frac {2}{(-1)^{3/4} \sqrt {c} x+1}\right ) b^2-\frac {1}{2} (-1)^{3/4} \sqrt {c} \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}{2} (-1)^{3/4} \sqrt {c} \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}{2} \sqrt [4]{-1} \sqrt {c} \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^2-2 \sqrt [4]{-1} a \sqrt {c} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) b-\sqrt [4]{-1} \sqrt {c} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right ) b+\frac {i a \log \left (i c x^2+1\right ) b}{x}-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{4 x} \]
[In]
[Out]
Rule 12
Rule 30
Rule 209
Rule 211
Rule 212
Rule 214
Rule 2352
Rule 2449
Rule 2497
Rule 2505
Rule 2507
Rule 2520
Rule 2637
Rule 4950
Rule 4964
Rule 4966
Rule 5040
Rule 5048
Rule 6055
Rule 6057
Rule 6131
Rule 6139
Rule 6874
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^2}+\frac {b \left (-2 i a+b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{2 x^2}-\frac {b^2 \log ^2\left (1+i c x^2\right )}{4 x^2}\right ) \, dx \\ & = \frac {1}{4} \int \frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{x^2} \, dx+\frac {1}{2} b \int \frac {\left (-2 i a+b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{x^2} \, dx-\frac {1}{4} b^2 \int \frac {\log ^2\left (1+i c x^2\right )}{x^2} \, dx \\ & = -\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{4 x}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{4 x}+\frac {1}{2} b \int \left (-\frac {2 i a \log \left (1+i c x^2\right )}{x^2}+\frac {b \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{x^2}\right ) \, dx+(b c) \int \frac {2 a+i b \log \left (1-i c x^2\right )}{1-i c x^2} \, dx-\left (i b^2 c\right ) \int \frac {\log \left (1+i c x^2\right )}{1+i c x^2} \, dx \\ & = -\sqrt [4]{-1} b \sqrt {c} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{4 x}+(-1)^{3/4} b^2 \sqrt {c} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )+\frac {b^2 \log ^2\left (1+i c x^2\right )}{4 x}-(i a b) \int \frac {\log \left (1+i c x^2\right )}{x^2} \, dx+\frac {1}{2} b^2 \int \frac {\log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{x^2} \, dx+\left (2 b^2 c^2\right ) \int \frac {\sqrt [4]{-1} x \arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c} \left (1-i c x^2\right )} \, dx+\left (2 b^2 c^2\right ) \int \frac {\sqrt [4]{-1} x \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c} \left (1+i c x^2\right )} \, dx \\ & = -\sqrt [4]{-1} b \sqrt {c} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{4 x}+\frac {i a b \log \left (1+i c x^2\right )}{x}+(-1)^{3/4} b^2 \sqrt {c} \text {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 )}{2 x}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{4 x}-\frac {1}{2} b^2 \int \frac {2 c \log \left (1-i c x^2\right )}{i-c x^2} \, dx-\frac {1}{2} b^2 \int \frac {2 c \log \left (1+i c x^2\right )}{-i-c x^2} \, dx+(2 a b c) \int \frac {1}{1+i c x^2} \, dx+\left (2 \sqrt [4]{-1} b^2 c^{3/2}\right ) \int \frac {x \arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{1-i c x^2} \, dx+\left (2 \sqrt [4]{-1} b^2 c^{3/2}\right ) \int \frac {x \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{1+i c x^2} \, dx \\ & = \sqrt [4]{-1} b^2 \sqrt {c} \arctan \left ((-1)^{3/4} \sqrt {c} x\right )^2-2 \sqrt [4]{-1} a b \sqrt {c} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )-(-1)^{3/4} b^2 \sqrt {c} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )^2-\sqrt [4]{-1} b \sqrt {c} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{4 x}+\frac {i a b \log \left (1+i c x^2\right )}{x}+(-1)^{3/4} b^2 \sqrt {c} \text {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 )}{2 x}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{4 x}+\left (2 i b^2 c\right ) \int \frac {\arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{i-(-1)^{3/4} \sqrt {c} x} \, dx-\left (2 i b^2 c\right ) \int \frac {\text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{1-(-1)^{3/4} \sqrt {c} x} \, dx-\left (b^2 c\right ) \int \frac {\log \left (1-i c x^2\right )}{i-c x^2} \, dx-\left (b^2 c\right ) \int \frac {\log \left (1+i c x^2\right )}{-i-c x^2} \, dx \\ & = \sqrt [4]{-1} b^2 \sqrt {c} \arctan \left ((-1)^{3/4} \sqrt {c} x\right )^2-2 \sqrt [4]{-1} a b \sqrt {c} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )-(-1)^{3/4} b^2 \sqrt {c} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )^2-2 (-1)^{3/4} b^2 \sqrt {c} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )+2 (-1)^{3/4} b^2 \sqrt {c} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )-(-1)^{3/4} b^2 \sqrt {c} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )-\sqrt [4]{-1} b \sqrt {c} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{4 x}+\frac {i a b \log \left (1+i c x^2\right )}{x}+(-1)^{3/4} b^2 \sqrt {c} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )+(-1)^{3/4} b^2 \sqrt {c} \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 )}{2 x}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{4 x}-\left (2 i b^2 c\right ) \int \frac {\log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )}{1-i c x^2} \, dx+\left (2 i b^2 c\right ) \int \frac {\log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )}{1+i c x^2} \, dx-\left (2 i b^2 c^2\right ) \int \frac {(-1)^{3/4} x \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^2\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 \\ & = \sqrt [4]{-1} b^2 \sqrt {c} \arctan \left ((-1)^{3/4} \sqrt {c} x\right )^2-2 \sqrt [4]{-1} a b \sqrt {c} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )-(-1)^{3/4} b^2 \sqrt {c} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )^2-2 (-1)^{3/4} b^2 \sqrt {c} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )+2 (-1)^{3/4} b^2 \sqrt {c} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )-(-1)^{3/4} b^2 \sqrt {c} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )-\sqrt [4]{-1} b \sqrt {c} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{4 x}+\frac {i a b \log \left (1+i c x^2\right )}{x}+(-1)^{3/4} b^2 \sqrt {c} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )+(-1)^{3/4} b^2 \sqrt {c} \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 )}{2 x}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{4 x}+\left (2 \sqrt [4]{-1} b^2 \sqrt {c}\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\sqrt [4]{-1} \sqrt {c} x}\right )+\left (2 (-1)^{3/4} b^2 \sqrt {c}\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-(-1)^{3/4} \sqrt {c} x}\right )+\left (2 \sqrt [4]{-1} b^2 c^{3/2}\right ) \int \frac {x \arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{1+i c x^2} \, dx+\left (2 \sqrt [4]{-1} b^2 c^{3/2}\right ) \int \frac {x \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{1-i c x^2} \, dx \\ & = \sqrt [4]{-1} b^2 \sqrt {c} \arctan \left ((-1)^{3/4} \sqrt {c} x\right )^2-2 \sqrt [4]{-1} a b \sqrt {c} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )-(-1)^{3/4} b^2 \sqrt {c} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )^2-2 (-1)^{3/4} b^2 \sqrt {c} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )+2 (-1)^{3/4} b^2 \sqrt {c} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )-(-1)^{3/4} b^2 \sqrt {c} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )-\sqrt [4]{-1} b \sqrt {c} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{4 x}+\frac {i a b \log \left (1+i c x^2\right )}{x}+(-1)^{3/4} b^2 \sqrt {c} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )+(-1)^{3/4} b^2 \sqrt {c} \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 )}{2 x}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{4 x}+\sqrt [4]{-1} b^2 \sqrt {c} \operatorname {PolyLog}\left (2,1-\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )+(-1)^{3/4} b^2 \sqrt {c} \operatorname {PolyLog}\left (2,1-\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )+\left (2 \sqrt [4]{-1} b^2 c^{3/2}\right ) \int \left (\frac {i \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 \sqrt [4]{-1} b^2 c^{3/2}\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 \\ & = \sqrt [4]{-1} b^2 \sqrt {c} \arctan \left ((-1)^{3/4} \sqrt {c} x\right )^2-2 \sqrt [4]{-1} a b \sqrt {c} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )-(-1)^{3/4} b^2 \sqrt {c} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )^2-2 (-1)^{3/4} b^2 \sqrt {c} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )+2 (-1)^{3/4} b^2 \sqrt {c} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )-(-1)^{3/4} b^2 \sqrt {c} \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )-\sqrt [4]{-1} b \sqrt {c} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{4 x}+\frac {i a b \log \left (1+i c x^2\right )}{x}+(-1)^{3/4} b^2 \sqrt {c} \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )+(-1)^{3/4} b^2 \sqrt {c} \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 )}{2 x}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{4 x}+\sqrt [4]{-1} b^2 \sqrt {c} \operatorname {PolyLog}\left (2,1-\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )+(-1)^{3/4} b^2 \sqrt {c} \operatorname {PolyLog}\left (2,1-\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )+\left ((-1)^{3/4} b^2 c\right ) \int \frac {\arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt [4]{-1}-\sqrt {c} x} \, dx-\left ((-1)^{3/4} b^2 c\right ) \int \frac {\arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt [4]{-1}+\sqrt {c} x} \, dx-\left ((-1)^{3/4} b^2 c\right ) \int \frac {\text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{-(-1)^{3/4}-\sqrt {c} x} \, dx+\left ((-1)^{3/4} b^2 c\right ) \int \frac {\text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{-(-1)^{3/4}+\sqrt {c} x} \, dx \\ & = \text {Too large to display} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(4697\) vs. \(2(1164)=2328\).
Time = 35.92 (sec) , antiderivative size = 4697, normalized size of antiderivative = 4.04 \[ \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{x^2} \, dx=\text {Result too large to show} \]
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\[\int \frac {{\left (a +b \arctan \left (c \,x^{2}\right )\right )}^{2}}{x^{2}}d x\]
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\[ \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{x^2} \, dx=\int { \frac {{\left (b \arctan \left (c x^{2}\right ) + a\right )}^{2}}{x^{2}} \,d x } \]
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\[ \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{x^2} \, dx=\int \frac {\left (a + b \operatorname {atan}{\left (c x^{2} \right )}\right )^{2}}{x^{2}}\, dx \]
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\[ \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{x^2} \, dx=\int { \frac {{\left (b \arctan \left (c x^{2}\right ) + a\right )}^{2}}{x^{2}} \,d x } \]
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\[ \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{x^2} \, dx=\int { \frac {{\left (b \arctan \left (c x^{2}\right ) + a\right )}^{2}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \arctan \left (c x^2\right )\right )^2}{x^2} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x^2\right )\right )}^2}{x^2} \,d x \]
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