Integrand size = 21, antiderivative size = 18 \[ \int \frac {\arctan (a x)^{5/2}}{c+a^2 c x^2} \, dx=\frac {2 \arctan (a x)^{7/2}}{7 a c} \]
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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} \[ \int \frac {\arctan (a x)^{5/2}}{c+a^2 c x^2} \, dx=\frac {2 \arctan (a x)^{7/2}}{7 a c} \]
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Rule 5004
Rubi steps \begin{align*} \text {integral}& = \frac {2 \arctan (a x)^{7/2}}{7 a c} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {\arctan (a x)^{5/2}}{c+a^2 c x^2} \, dx=\frac {2 \arctan (a x)^{7/2}}{7 a c} \]
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Time = 2.54 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83
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none
Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {\arctan (a x)^{5/2}}{c+a^2 c x^2} \, dx=\frac {2 \, \arctan \left (a x\right )^{\frac {7}{2}}}{7 \, a c} \]
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Time = 7.28 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {\arctan (a x)^{5/2}}{c+a^2 c x^2} \, dx=\begin {cases} \frac {2 \operatorname {atan}^{\frac {7}{2}}{\left (a x \right )}}{7 a c} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int \frac {\arctan (a x)^{5/2}}{c+a^2 c x^2} \, dx=\text {Exception raised: RuntimeError} \]
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none
Time = 0.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {\arctan (a x)^{5/2}}{c+a^2 c x^2} \, dx=\frac {2 \, \arctan \left (a x\right )^{\frac {7}{2}}}{7 \, a c} \]
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Time = 0.44 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {\arctan (a x)^{5/2}}{c+a^2 c x^2} \, dx=\frac {2\,{\mathrm {atan}\left (a\,x\right )}^{7/2}}{7\,a\,c} \]
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