Integrand size = 19, antiderivative size = 20 \[ \int \frac {\arctan (a x)^n}{c+a^2 c x^2} \, dx=\frac {\arctan (a x)^{1+n}}{a c (1+n)} \]
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Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {5004} \[ \int \frac {\arctan (a x)^n}{c+a^2 c x^2} \, dx=\frac {\arctan (a x)^{n+1}}{a c (n+1)} \]
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Rule 5004
Rubi steps \begin{align*} \text {integral}& = \frac {\arctan (a x)^{1+n}}{a c (1+n)} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\arctan (a x)^n}{c+a^2 c x^2} \, dx=\frac {\arctan (a x)^{1+n}}{a c (1+n)} \]
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Time = 0.34 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05
method | result | size |
default | \(\frac {\arctan \left (a x \right )^{1+n}}{a c \left (1+n \right )}\) | \(21\) |
parallelrisch | \(\frac {\arctan \left (a x \right )^{n} \arctan \left (a x \right )}{c a \left (1+n \right )}\) | \(23\) |
risch | \(\frac {i \left (\ln \left (-i a x +1\right )-\ln \left (i a x +1\right )\right ) \left (\frac {i \left (\ln \left (-i a x +1\right )-\ln \left (i a x +1\right )\right )}{2}\right )^{n}}{2 c a \left (1+n \right )}\) | \(58\) |
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none
Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int \frac {\arctan (a x)^n}{c+a^2 c x^2} \, dx=\frac {\arctan \left (a x\right )^{n} \arctan \left (a x\right )}{a c n + a c} \]
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\[ \int \frac {\arctan (a x)^n}{c+a^2 c x^2} \, dx=\frac {\int \frac {\operatorname {atan}^{n}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \]
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Exception generated. \[ \int \frac {\arctan (a x)^n}{c+a^2 c x^2} \, dx=\text {Exception raised: RuntimeError} \]
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none
Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\arctan (a x)^n}{c+a^2 c x^2} \, dx=\frac {\arctan \left (a x\right )^{n + 1}}{a c {\left (n + 1\right )}} \]
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Time = 0.44 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\arctan (a x)^n}{c+a^2 c x^2} \, dx=\frac {{\mathrm {atan}\left (a\,x\right )}^{n+1}}{a\,c\,\left (n+1\right )} \]
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