Integrand size = 19, antiderivative size = 17 \[ \int \sec (e+f x) \left (1-2 \sec ^2(e+f x)\right ) \, dx=-\frac {\sec (e+f x) \tan (e+f x)}{f} \]
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Time = 0.02 (sec) , antiderivative size = 17, 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 = {4128} \[ \int \sec (e+f x) \left (1-2 \sec ^2(e+f x)\right ) \, dx=-\frac {\tan (e+f x) \sec (e+f x)}{f} \]
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Rule 4128
Rubi steps \begin{align*} \text {integral}& = -\frac {\sec (e+f x) \tan (e+f x)}{f} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \sec (e+f x) \left (1-2 \sec ^2(e+f x)\right ) \, dx=-\frac {\sec (e+f x) \tan (e+f x)}{f} \]
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Time = 0.11 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06
method | result | size |
derivativedivides | \(-\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{f}\) | \(18\) |
default | \(-\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{f}\) | \(18\) |
parts | \(-\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{f}\) | \(18\) |
parallelrisch | \(-\frac {2 \sin \left (f x +e \right )}{f \left (1+\cos \left (2 f x +2 e \right )\right )}\) | \(25\) |
risch | \(\frac {2 i \left ({\mathrm e}^{3 i \left (f x +e \right )}-{\mathrm e}^{i \left (f x +e \right )}\right )}{f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2}}\) | \(41\) |
norman | \(\frac {-\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f}-\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{2}}\) | \(48\) |
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none
Time = 0.23 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \sec (e+f x) \left (1-2 \sec ^2(e+f x)\right ) \, dx=-\frac {\sin \left (f x + e\right )}{f \cos \left (f x + e\right )^{2}} \]
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\[ \int \sec (e+f x) \left (1-2 \sec ^2(e+f x)\right ) \, dx=- \int \left (- \sec {\left (e + f x \right )}\right )\, dx - \int 2 \sec ^{3}{\left (e + f x \right )}\, dx \]
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none
Time = 0.19 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.29 \[ \int \sec (e+f x) \left (1-2 \sec ^2(e+f x)\right ) \, dx=\frac {\sin \left (f x + e\right )}{{\left (\sin \left (f x + e\right )^{2} - 1\right )} f} \]
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none
Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.41 \[ \int \sec (e+f x) \left (1-2 \sec ^2(e+f x)\right ) \, dx=-\frac {1}{f {\left (\frac {1}{\sin \left (f x + e\right )} - \sin \left (f x + e\right )\right )}} \]
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Time = 15.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.29 \[ \int \sec (e+f x) \left (1-2 \sec ^2(e+f x)\right ) \, dx=\frac {\sin \left (e+f\,x\right )}{f\,\left ({\sin \left (e+f\,x\right )}^2-1\right )} \]
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