Integrand size = 19, antiderivative size = 244 \[ \int \sqrt {c+a^2 c x^2} \arctan (a x) \, dx=-\frac {\sqrt {c+a^2 c x^2}}{2 a}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {i c \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a \sqrt {c+a^2 c x^2}}+\frac {i c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {c+a^2 c x^2}}-\frac {i c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {c+a^2 c x^2}} \]
[Out]
Time = 0.08 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4998, 5010, 5006} \[ \int \sqrt {c+a^2 c x^2} \arctan (a x) \, dx=-\frac {i c \sqrt {a^2 x^2+1} \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right ) \arctan (a x)}{a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}+\frac {i c \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {a^2 c x^2+c}}-\frac {i c \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a} \]
[In]
[Out]
Rule 4998
Rule 5006
Rule 5010
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {c+a^2 c x^2}}{2 a}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{2} c \int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx \\ & = -\frac {\sqrt {c+a^2 c x^2}}{2 a}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {\left (c \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{\sqrt {1+a^2 x^2}} \, dx}{2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {\sqrt {c+a^2 c x^2}}{2 a}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {i c \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a \sqrt {c+a^2 c x^2}}+\frac {i c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {c+a^2 c x^2}}-\frac {i c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {c+a^2 c x^2}} \\ \end{align*}
Time = 0.58 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.58 \[ \int \sqrt {c+a^2 c x^2} \arctan (a x) \, dx=\frac {\sqrt {c \left (1+a^2 x^2\right )} \left (\sqrt {1+a^2 x^2} (-1+a x \arctan (a x))+\arctan (a x) \left (\log \left (1-i e^{i \arctan (a x)}\right )-\log \left (1+i e^{i \arctan (a x)}\right )\right )+i \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-i \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )\right )}{2 a \sqrt {1+a^2 x^2}} \]
[In]
[Out]
Time = 0.42 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.73
method | result | size |
default | \(\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (x \arctan \left (a x \right ) a -1\right )}{2 a}-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (\arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-\arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+i \operatorname {dilog}\left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{2 a \sqrt {a^{2} x^{2}+1}}\) | \(178\) |
[In]
[Out]
\[ \int \sqrt {c+a^2 c x^2} \arctan (a x) \, dx=\int { \sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right ) \,d x } \]
[In]
[Out]
\[ \int \sqrt {c+a^2 c x^2} \arctan (a x) \, dx=\int \sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}{\left (a x \right )}\, dx \]
[In]
[Out]
\[ \int \sqrt {c+a^2 c x^2} \arctan (a x) \, dx=\int { \sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right ) \,d x } \]
[In]
[Out]
Exception generated. \[ \int \sqrt {c+a^2 c x^2} \arctan (a x) \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int \sqrt {c+a^2 c x^2} \arctan (a x) \, dx=\int \mathrm {atan}\left (a\,x\right )\,\sqrt {c\,a^2\,x^2+c} \,d x \]
[In]
[Out]