Integrand size = 23, antiderivative size = 23 \[ \int \frac {1}{\sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}} \, dx=\text {Int}\left (\frac {1}{\sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}},x\right ) \]
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Not integrable
Time = 0.02 (sec) , antiderivative size = 23, 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 = {} \[ \int \frac {1}{\sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}} \, dx=\int \frac {1}{\sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}} \, dx \\ \end{align*}
Not integrable
Time = 0.58 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}} \, dx=\int \frac {1}{\sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}} \, dx \]
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Not integrable
Time = 3.53 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83
\[\int \frac {1}{\arctan \left (a x \right )^{\frac {3}{2}} \sqrt {a^{2} c \,x^{2}+c}}d x\]
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Exception generated. \[ \int \frac {1}{\sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 8.46 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {1}{\sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}} \, dx=\int \frac {1}{\sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]
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Exception generated. \[ \int \frac {1}{\sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]
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Not integrable
Time = 206.07 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.13 \[ \int \frac {1}{\sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}} \, dx=\int { \frac {1}{\sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right )^{\frac {3}{2}}} \,d x } \]
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Not integrable
Time = 0.35 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}} \, dx=\int \frac {1}{{\mathrm {atan}\left (a\,x\right )}^{3/2}\,\sqrt {c\,a^2\,x^2+c}} \,d x \]
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