Integrand size = 14, antiderivative size = 132 \[ \int \frac {(a+b \arctan (c x))^2}{x} \, dx=2 (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )-i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )+i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )-\frac {1}{2} b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )+\frac {1}{2} b^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right ) \]
[Out]
Time = 0.21 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {4942, 5108, 5004, 5114, 6745} \[ \int \frac {(a+b \arctan (c x))^2}{x} \, dx=2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2-i b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))+i b \operatorname {PolyLog}\left (2,\frac {2}{i c x+1}-1\right ) (a+b \arctan (c x))-\frac {1}{2} b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right )+\frac {1}{2} b^2 \operatorname {PolyLog}\left (3,\frac {2}{i c x+1}-1\right ) \]
[In]
[Out]
Rule 4942
Rule 5004
Rule 5108
Rule 5114
Rule 6745
Rubi steps \begin{align*} \text {integral}& = 2 (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )-(4 b c) \int \frac {(a+b \arctan (c x)) \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx \\ & = 2 (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )+(2 b c) \int \frac {(a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx-(2 b c) \int \frac {(a+b \arctan (c x)) \log \left (2-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx \\ & = 2 (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )-i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )+i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )+\left (i b^2 c\right ) \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx-\left (i b^2 c\right ) \int \frac {\operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx \\ & = 2 (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )-i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )+i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )-\frac {1}{2} b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )+\frac {1}{2} b^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right ) \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.36 \[ \int \frac {(a+b \arctan (c x))^2}{x} \, dx=a^2 \log (c x)+i a b (\operatorname {PolyLog}(2,-i c x)-\operatorname {PolyLog}(2,i c x))+b^2 \left (-\frac {i \pi ^3}{24}+\frac {2}{3} i \arctan (c x)^3+\arctan (c x)^2 \log \left (1-e^{-2 i \arctan (c x)}\right )-\arctan (c x)^2 \log \left (1+e^{2 i \arctan (c x)}\right )+i \arctan (c x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (c x)}\right )+i \arctan (c x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,e^{-2 i \arctan (c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )\right ) \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.74 (sec) , antiderivative size = 1002, normalized size of antiderivative = 7.59
\[\text {Expression too large to display}\]
[In]
[Out]
\[ \int \frac {(a+b \arctan (c x))^2}{x} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
[In]
[Out]
\[ \int \frac {(a+b \arctan (c x))^2}{x} \, dx=\int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right )^{2}}{x}\, dx \]
[In]
[Out]
\[ \int \frac {(a+b \arctan (c x))^2}{x} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
[In]
[Out]
\[ \int \frac {(a+b \arctan (c x))^2}{x} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(a+b \arctan (c x))^2}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{x} \,d x \]
[In]
[Out]