Integrand size = 24, antiderivative size = 24 \[ \int x^m \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=\text {Int}\left (x^m \sqrt {c+a^2 c x^2} \arctan (a x)^2,x\right ) \]
[Out]
Not integrable
Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int x^m \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=\int x^m \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \int x^m \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx \\ \end{align*}
Not integrable
Time = 0.16 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int x^m \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=\int x^m \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx \]
[In]
[Out]
Not integrable
Time = 1.12 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92
\[\int x^{m} \sqrt {a^{2} c \,x^{2}+c}\, \arctan \left (a x \right )^{2}d x\]
[In]
[Out]
Not integrable
Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int x^m \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=\int { \sqrt {a^{2} c x^{2} + c} x^{m} \arctan \left (a x\right )^{2} \,d x } \]
[In]
[Out]
Not integrable
Time = 29.93 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int x^m \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=\int x^{m} \sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}^{2}{\left (a x \right )}\, dx \]
[In]
[Out]
Not integrable
Time = 0.41 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int x^m \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=\int { \sqrt {a^{2} c x^{2} + c} x^{m} \arctan \left (a x\right )^{2} \,d x } \]
[In]
[Out]
Exception generated. \[ \int x^m \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Not integrable
Time = 0.43 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int x^m \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=\int x^m\,{\mathrm {atan}\left (a\,x\right )}^2\,\sqrt {c\,a^2\,x^2+c} \,d x \]
[In]
[Out]