Integrand size = 19, antiderivative size = 16 \[ \int \frac {\arctan (a x)^3}{c+a^2 c x^2} \, dx=\frac {\arctan (a x)^4}{4 a c} \]
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Time = 0.02 (sec) , antiderivative size = 16, 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} \[ \int \frac {\arctan (a x)^3}{c+a^2 c x^2} \, dx=\frac {\arctan (a x)^4}{4 a c} \]
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Rule 5004
Rubi steps \begin{align*} \text {integral}& = \frac {\arctan (a x)^4}{4 a c} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\arctan (a x)^3}{c+a^2 c x^2} \, dx=\frac {\arctan (a x)^4}{4 a c} \]
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Time = 0.66 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(\frac {\arctan \left (a x \right )^{4}}{4 a c}\) | \(15\) |
default | \(\frac {\arctan \left (a x \right )^{4}}{4 a c}\) | \(15\) |
parallelrisch | \(\frac {\arctan \left (a x \right )^{4}}{4 a c}\) | \(15\) |
parts | \(\frac {\arctan \left (a x \right )^{4}}{4 a c}\) | \(15\) |
risch | \(\frac {\ln \left (i a x +1\right )^{4}}{64 c a}-\frac {\ln \left (-i a x +1\right ) \ln \left (i a x +1\right )^{3}}{16 c a}+\frac {3 \ln \left (-i a x +1\right )^{2} \ln \left (i a x +1\right )^{2}}{32 c a}-\frac {\ln \left (-i a x +1\right )^{3} \ln \left (i a x +1\right )}{16 c a}+\frac {\ln \left (-i a x +1\right )^{4}}{64 c a}\) | \(118\) |
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none
Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {\arctan (a x)^3}{c+a^2 c x^2} \, dx=\frac {\arctan \left (a x\right )^{4}}{4 \, a c} \]
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\[ \int \frac {\arctan (a x)^3}{c+a^2 c x^2} \, dx=\frac {\int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \]
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none
Time = 0.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {\arctan (a x)^3}{c+a^2 c x^2} \, dx=\frac {\arctan \left (a x\right )^{4}}{4 \, a c} \]
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\[ \int \frac {\arctan (a x)^3}{c+a^2 c x^2} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{a^{2} c x^{2} + c} \,d x } \]
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Time = 0.13 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {\arctan (a x)^3}{c+a^2 c x^2} \, dx=\frac {{\mathrm {atan}\left (a\,x\right )}^4}{4\,a\,c} \]
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