Integrand size = 21, antiderivative size = 18 \[ \int \frac {1}{\left (c+a^2 c x^2\right ) \arctan (a x)^{5/2}} \, dx=-\frac {2}{3 a c \arctan (a x)^{3/2}} \]
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Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {5004} \[ \int \frac {1}{\left (c+a^2 c x^2\right ) \arctan (a x)^{5/2}} \, dx=-\frac {2}{3 a c \arctan (a x)^{3/2}} \]
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Rule 5004
Rubi steps \begin{align*} \text {integral}& = -\frac {2}{3 a c \arctan (a x)^{3/2}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (c+a^2 c x^2\right ) \arctan (a x)^{5/2}} \, dx=-\frac {2}{3 a c \arctan (a x)^{3/2}} \]
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Time = 0.99 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83
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none
Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\left (c+a^2 c x^2\right ) \arctan (a x)^{5/2}} \, dx=-\frac {2}{3 \, a c \arctan \left (a x\right )^{\frac {3}{2}}} \]
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Time = 3.63 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\left (c+a^2 c x^2\right ) \arctan (a x)^{5/2}} \, dx=- \frac {2}{3 a c \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}} \]
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Exception generated. \[ \int \frac {1}{\left (c+a^2 c x^2\right ) \arctan (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]
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none
Time = 0.29 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\left (c+a^2 c x^2\right ) \arctan (a x)^{5/2}} \, dx=-\frac {2}{3 \, a c \arctan \left (a x\right )^{\frac {3}{2}}} \]
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Time = 0.38 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\left (c+a^2 c x^2\right ) \arctan (a x)^{5/2}} \, dx=-\frac {2}{3\,a\,c\,{\mathrm {atan}\left (a\,x\right )}^{3/2}} \]
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