Integrand size = 12, antiderivative size = 1191 \[ \int \left (a+b \arctan \left (c x^2\right )\right )^2 \, dx=a^2 x-\frac {2 (-1)^{3/4} a b \arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}+\frac {(-1)^{3/4} b^2 \arctan \left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}+\frac {2 (-1)^{3/4} a b \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}-\frac {\sqrt [4]{-1} b^2 \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}+\frac {2 \sqrt [4]{-1} b^2 \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 \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 \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 \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 \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 \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 \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 \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {(1-i) \left (1+(-1)^{3/4} \sqrt {c} x\right )}{1+\sqrt [4]{-1} \sqrt {c} x}\right )}{\sqrt {c}}+i a b x \log \left (1-i c x^2\right )+\frac {\sqrt [4]{-1} b^2 \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 \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 x \log ^2\left (1-i c x^2\right )-i a b x \log \left (1+i c x^2\right )-\frac {\sqrt [4]{-1} b^2 \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 \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 x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac {1}{4} b^2 x \log ^2\left (1+i c x^2\right )+\frac {(-1)^{3/4} b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )}{\sqrt {c}}+\frac {(-1)^{3/4} b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+\sqrt [4]{-1} \sqrt {c} x}\right )}{\sqrt {c}}-\frac {(-1)^{3/4} b^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 )}{2 \sqrt {c}}+\frac {\sqrt [4]{-1} b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+(-1)^{3/4} \sqrt {c} x}\right )}{\sqrt {c}}-\frac {\sqrt [4]{-1} b^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 )}{2 \sqrt {c}}-\frac {\sqrt [4]{-1} b^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 )}{2 \sqrt {c}}-\frac {(-1)^{3/4} b^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 )}{2 \sqrt {c}} \]
[Out]
Time = 1.26 (sec) , antiderivative size = 1191, normalized size of antiderivative = 1.00, number of steps used = 69, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.917, Rules used = {4932, 2498, 327, 209, 2500, 2526, 2520, 12, 5040, 4964, 2449, 2352, 212, 2636, 211, 5048, 4966, 2497, 214, 6139, 6057, 6131, 6055} \[ \int \left (a+b \arctan \left (c x^2\right )\right )^2 \, dx=x a^2-\frac {2 (-1)^{3/4} b \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) a}{\sqrt {c}}+\frac {2 (-1)^{3/4} b \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) a}{\sqrt {c}}+i b x \log \left (1-i c x^2\right ) a-i b x \log \left (i c x^2+1\right ) a+\frac {(-1)^{3/4} b^2 \arctan \left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}-\frac {\sqrt [4]{-1} b^2 \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}-\frac {1}{4} b^2 x \log ^2\left (1-i c x^2\right )-\frac {1}{4} b^2 x \log ^2\left (i c x^2+1\right )+\frac {2 \sqrt [4]{-1} b^2 \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 \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 \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 \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 \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 \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 \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 \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 \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 \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )}{\sqrt {c}}-\frac {\sqrt [4]{-1} b^2 \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 \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 x \log \left (1-i c x^2\right ) \log \left (i c x^2+1\right )+\frac {(-1)^{3/4} b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )}{\sqrt {c}}+\frac {(-1)^{3/4} b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{\sqrt [4]{-1} \sqrt {c} x+1}\right )}{\sqrt {c}}-\frac {(-1)^{3/4} 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 )}{2 \sqrt {c}}+\frac {\sqrt [4]{-1} b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )}{\sqrt {c}}+\frac {\sqrt [4]{-1} b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{(-1)^{3/4} \sqrt {c} x+1}\right )}{\sqrt {c}}-\frac {\sqrt [4]{-1} 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 )}{2 \sqrt {c}}-\frac {\sqrt [4]{-1} 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 )}{2 \sqrt {c}}-\frac {(-1)^{3/4} 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 )}{2 \sqrt {c}} \]
[In]
[Out]
Rule 12
Rule 209
Rule 211
Rule 212
Rule 214
Rule 327
Rule 2352
Rule 2449
Rule 2497
Rule 2498
Rule 2500
Rule 2520
Rule 2526
Rule 2636
Rule 4932
Rule 4964
Rule 4966
Rule 5040
Rule 5048
Rule 6055
Rule 6057
Rule 6131
Rule 6139
Rubi steps \begin{align*} \text {integral}& = \int \left (a^2+i a b \log \left (1-i c x^2\right )-\frac {1}{4} b^2 \log ^2\left (1-i c x^2\right )-i a b \log \left (1+i c x^2\right )+\frac {1}{2} b^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac {1}{4} b^2 \log ^2\left (1+i c x^2\right )\right ) \, dx \\ & = a^2 x+(i a b) \int \log \left (1-i c x^2\right ) \, dx-(i a b) \int \log \left (1+i c x^2\right ) \, dx-\frac {1}{4} b^2 \int \log ^2\left (1-i c x^2\right ) \, dx-\frac {1}{4} b^2 \int \log ^2\left (1+i c x^2\right ) \, dx+\frac {1}{2} b^2 \int \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right ) \, dx \\ & = a^2 x+i a b x \log \left (1-i c x^2\right )-\frac {1}{4} b^2 x \log ^2\left (1-i c x^2\right )-i a b x \log \left (1+i c x^2\right )+\frac {1}{2} b^2 x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac {1}{4} b^2 x \log ^2\left (1+i c x^2\right )-\frac {1}{2} b^2 \int \frac {2 c x^2 \log \left (1-i c x^2\right )}{-i+c x^2} \, dx-\frac {1}{2} b^2 \int \frac {2 c x^2 \log \left (1+i c x^2\right )}{i+c x^2} \, dx-(2 a b c) \int \frac {x^2}{1-i c x^2} \, dx-(2 a b c) \int \frac {x^2}{1+i c x^2} \, dx-\left (i b^2 c\right ) \int \frac {x^2 \log \left (1-i c x^2\right )}{1-i c x^2} \, dx+\left (i b^2 c\right ) \int \frac {x^2 \log \left (1+i c x^2\right )}{1+i c x^2} \, dx \\ & = a^2 x+i a b x \log \left (1-i c x^2\right )-\frac {1}{4} b^2 x \log ^2\left (1-i c x^2\right )-i a b x \log \left (1+i c x^2\right )+\frac {1}{2} b^2 x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac {1}{4} b^2 x \log ^2\left (1+i c x^2\right )+(2 i a b) \int \frac {1}{1-i c x^2} \, dx-(2 i a b) \int \frac {1}{1+i c x^2} \, dx-\left (i b^2 c\right ) \int \left (\frac {i \log \left (1-i c x^2\right )}{c}-\frac {i \log \left (1-i c x^2\right )}{c \left (1-i c x^2\right )}\right ) \, dx+\left (i b^2 c\right ) \int \left (-\frac {i \log \left (1+i c x^2\right )}{c}+\frac {i \log \left (1+i c x^2\right )}{c \left (1+i c x^2\right )}\right ) \, dx-\left (b^2 c\right ) \int \frac {x^2 \log \left (1-i c x^2\right )}{-i+c x^2} \, dx-\left (b^2 c\right ) \int \frac {x^2 \log \left (1+i c x^2\right )}{i+c x^2} \, dx \\ & = a^2 x-\frac {2 (-1)^{3/4} a b \arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}+\frac {2 (-1)^{3/4} a b \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}+i a b x \log \left (1-i c x^2\right )-\frac {1}{4} b^2 x \log ^2\left (1-i c x^2\right )-i a b x \log \left (1+i c x^2\right )+\frac {1}{2} b^2 x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac {1}{4} b^2 x \log ^2\left (1+i c x^2\right )+b^2 \int \log \left (1-i c x^2\right ) \, dx-b^2 \int \frac {\log \left (1-i c x^2\right )}{1-i c x^2} \, dx+b^2 \int \log \left (1+i c x^2\right ) \, dx-b^2 \int \frac {\log \left (1+i c x^2\right )}{1+i c x^2} \, dx-\left (b^2 c\right ) \int \left (\frac {\log \left (1-i c x^2\right )}{c}+\frac {i \log \left (1-i c x^2\right )}{c \left (-i+c x^2\right )}\right ) \, dx-\left (b^2 c\right ) \int \left (\frac {\log \left (1+i c x^2\right )}{c}-\frac {i \log \left (1+i c x^2\right )}{c \left (i+c x^2\right )}\right ) \, dx \\ & = a^2 x-\frac {2 (-1)^{3/4} a b \arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}+\frac {2 (-1)^{3/4} a b \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}+i a b x \log \left (1-i c x^2\right )+b^2 x \log \left (1-i c x^2\right )+\frac {\sqrt [4]{-1} b^2 \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )}{\sqrt {c}}-\frac {1}{4} b^2 x \log ^2\left (1-i c x^2\right )-i a b x \log \left (1+i c x^2\right )+b^2 x \log \left (1+i c x^2\right )+\frac {\sqrt [4]{-1} b^2 \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 x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac {1}{4} b^2 x \log ^2\left (1+i c x^2\right )-\left (i b^2\right ) \int \frac {\log \left (1-i c x^2\right )}{-i+c x^2} \, dx+\left (i b^2\right ) \int \frac {\log \left (1+i c x^2\right )}{i+c x^2} \, dx-b^2 \int \log \left (1-i c x^2\right ) \, dx-b^2 \int \log \left (1+i c x^2\right ) \, dx+\left (2 i b^2 c\right ) \int \frac {x^2}{1-i c x^2} \, dx-\left (2 i b^2 c\right ) \int \frac {x^2}{1+i c x^2} \, dx+\left (2 i b^2 c\right ) \int \frac {\sqrt [4]{-1} x \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\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 x-4 b^2 x-\frac {2 (-1)^{3/4} a b \arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}+\frac {2 (-1)^{3/4} a b \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}+i a b x \log \left (1-i c x^2\right )+\frac {\sqrt [4]{-1} b^2 \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 \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 x \log ^2\left (1-i c x^2\right )-i a b x \log \left (1+i c x^2\right )-\frac {\sqrt [4]{-1} b^2 \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 \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 x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac {1}{4} b^2 x \log ^2\left (1+i c x^2\right )+\left (2 b^2\right ) \int \frac {1}{1-i c x^2} \, dx+\left (2 b^2\right ) \int \frac {1}{1+i c x^2} \, dx+\left (2 (-1)^{3/4} b^2 \sqrt {c}\right ) \int \frac {x \arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{1-i c x^2} \, dx-\left (2 (-1)^{3/4} b^2 \sqrt {c}\right ) \int \frac {x \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{1+i c x^2} \, dx-\left (2 i b^2 c\right ) \int \frac {x^2}{1-i c x^2} \, dx+\left (2 i b^2 c\right ) \int \frac {x^2}{1+i c x^2} \, dx+\left (2 b^2 c\right ) \int \frac {(-1)^{3/4} x \arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c} \left (1+i c x^2\right )} \, dx-\left (2 b^2 c\right ) \int \frac {(-1)^{3/4} x \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c} \left (1-i c x^2\right )} \, dx \\ & = a^2 x-\frac {2 (-1)^{3/4} a b \arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}-\frac {2 \sqrt [4]{-1} b^2 \arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}+\frac {(-1)^{3/4} b^2 \arctan \left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}+\frac {2 (-1)^{3/4} a b \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}-\frac {2 \sqrt [4]{-1} b^2 \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}-\frac {\sqrt [4]{-1} b^2 \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}+i a b x \log \left (1-i c x^2\right )+\frac {\sqrt [4]{-1} b^2 \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 \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 x \log ^2\left (1-i c x^2\right )-i a b x \log \left (1+i c x^2\right )-\frac {\sqrt [4]{-1} b^2 \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 \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 x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac {1}{4} b^2 x \log ^2\left (1+i c x^2\right )-\left (2 b^2\right ) \int \frac {1}{1-i c x^2} \, dx-\left (2 b^2\right ) \int \frac {1}{1+i c x^2} \, dx-\left (2 b^2\right ) \int \frac {\arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{i-(-1)^{3/4} \sqrt {c} x} \, dx-\left (2 b^2\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}\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}\right ) \int \frac {x \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{1-i c x^2} \, dx \\ & = a^2 x-\frac {2 (-1)^{3/4} a b \arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}+\frac {(-1)^{3/4} b^2 \arctan \left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}+\frac {2 (-1)^{3/4} a b \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}-\frac {\sqrt [4]{-1} b^2 \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}+\frac {2 \sqrt [4]{-1} b^2 \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )}{\sqrt {c}}+\frac {2 \sqrt [4]{-1} b^2 \text {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 x \log \left (1-i c x^2\right )+\frac {\sqrt [4]{-1} b^2 \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 \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 x \log ^2\left (1-i c x^2\right )-i a b x \log \left (1+i c x^2\right )-\frac {\sqrt [4]{-1} b^2 \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 \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 x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac {1}{4} b^2 x \log ^2\left (1+i c x^2\right )+\left (2 b^2\right ) \int \frac {\log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )}{1-i c x^2} \, dx+\left (2 b^2\right ) \int \frac {\log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )}{1+i c x^2} \, dx+\left (2 (-1)^{3/4} b^2 \sqrt {c}\right ) \int \left (\frac {i \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}\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 x-\frac {2 (-1)^{3/4} a b \arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}+\frac {(-1)^{3/4} b^2 \arctan \left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}+\frac {2 (-1)^{3/4} a b \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt {c}}-\frac {\sqrt [4]{-1} b^2 \text {arctanh}\left ((-1)^{3/4} \sqrt {c} x\right )^2}{\sqrt {c}}+\frac {2 \sqrt [4]{-1} b^2 \arctan \left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )}{\sqrt {c}}+\frac {2 \sqrt [4]{-1} b^2 \text {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 x \log \left (1-i c x^2\right )+\frac {\sqrt [4]{-1} b^2 \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 \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 x \log ^2\left (1-i c x^2\right )-i a b x \log \left (1+i c x^2\right )-\frac {\sqrt [4]{-1} b^2 \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 \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 x \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac {1}{4} b^2 x \log ^2\left (1+i c x^2\right )-\left (\sqrt [4]{-1} b^2\right ) \int \frac {\arctan \left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt [4]{-1}-\sqrt {c} x} \, dx+\left (\sqrt [4]{-1} b^2\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\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\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\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-(-1)^{3/4} \sqrt {c} x}\right )}{\sqrt {c}}+\frac {\left (2 (-1)^{3/4} b^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\sqrt [4]{-1} \sqrt {c} x}\right )}{\sqrt {c}} \\ & = \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. \(2(1191)=2382\).
Time = 38.10 (sec) , antiderivative size = 4697, normalized size of antiderivative = 3.94 \[ \int \left (a+b \arctan \left (c x^2\right )\right )^2 \, dx=\text {Result too large to show} \]
[In]
[Out]
\[\int {\left (a +b \arctan \left (c \,x^{2}\right )\right )}^{2}d x\]
[In]
[Out]
\[ \int \left (a+b \arctan \left (c x^2\right )\right )^2 \, dx=\int { {\left (b \arctan \left (c x^{2}\right ) + a\right )}^{2} \,d x } \]
[In]
[Out]
\[ \int \left (a+b \arctan \left (c x^2\right )\right )^2 \, dx=\int \left (a + b \operatorname {atan}{\left (c x^{2} \right )}\right )^{2}\, dx \]
[In]
[Out]
\[ \int \left (a+b \arctan \left (c x^2\right )\right )^2 \, dx=\int { {\left (b \arctan \left (c x^{2}\right ) + a\right )}^{2} \,d x } \]
[In]
[Out]
\[ \int \left (a+b \arctan \left (c x^2\right )\right )^2 \, dx=\int { {\left (b \arctan \left (c x^{2}\right ) + a\right )}^{2} \,d x } \]
[In]
[Out]
Timed out. \[ \int \left (a+b \arctan \left (c x^2\right )\right )^2 \, dx=\int {\left (a+b\,\mathrm {atan}\left (c\,x^2\right )\right )}^2 \,d x \]
[In]
[Out]