Integrand size = 21, antiderivative size = 19 \[ \int \sec ^2(e+f x) \left (2-3 \sec ^2(e+f x)\right ) \, dx=-\frac {\sec ^2(e+f x) \tan (e+f x)}{f} \]
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Time = 0.03 (sec) , antiderivative size = 19, 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 = {4128} \[ \int \sec ^2(e+f x) \left (2-3 \sec ^2(e+f x)\right ) \, dx=-\frac {\tan (e+f x) \sec ^2(e+f x)}{f} \]
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Rule 4128
Rubi steps \begin{align*} \text {integral}& = -\frac {\sec ^2(e+f x) \tan (e+f x)}{f} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \sec ^2(e+f x) \left (2-3 \sec ^2(e+f x)\right ) \, dx=-\frac {\sec ^2(e+f x) \tan (e+f x)}{f} \]
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Time = 0.14 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.68
method | result | size |
parallelrisch | \(-\frac {4 \sin \left (f x +e \right )}{f \left (\cos \left (3 f x +3 e \right )+3 \cos \left (f x +e \right )\right )}\) | \(32\) |
derivativedivides | \(\frac {2 \tan \left (f x +e \right )+3 \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )}{f}\) | \(34\) |
default | \(\frac {2 \tan \left (f x +e \right )+3 \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )}{f}\) | \(34\) |
parts | \(\frac {2 \tan \left (f x +e \right )}{f}+\frac {3 \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )}{f}\) | \(36\) |
risch | \(\frac {4 i \left ({\mathrm e}^{4 i \left (f x +e \right )}-{\mathrm e}^{2 i \left (f x +e \right )}\right )}{f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{3}}\) | \(41\) |
norman | \(\frac {\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f}+\frac {4 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{f}+\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{3}}\) | \(64\) |
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none
Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \sec ^2(e+f x) \left (2-3 \sec ^2(e+f x)\right ) \, dx=-\frac {\sin \left (f x + e\right )}{f \cos \left (f x + e\right )^{3}} \]
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\[ \int \sec ^2(e+f x) \left (2-3 \sec ^2(e+f x)\right ) \, dx=- \int \left (- 2 \sec ^{2}{\left (e + f x \right )}\right )\, dx - \int 3 \sec ^{4}{\left (e + f x \right )}\, dx \]
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none
Time = 0.20 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \sec ^2(e+f x) \left (2-3 \sec ^2(e+f x)\right ) \, dx=-\frac {\tan \left (f x + e\right )^{3} + \tan \left (f x + e\right )}{f} \]
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Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \sec ^2(e+f x) \left (2-3 \sec ^2(e+f x)\right ) \, dx=-\frac {\tan \left (f x + e\right )^{3} + \tan \left (f x + e\right )}{f} \]
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Time = 15.14 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \sec ^2(e+f x) \left (2-3 \sec ^2(e+f x)\right ) \, dx=-\frac {\mathrm {tan}\left (e+f\,x\right )\,\left ({\mathrm {tan}\left (e+f\,x\right )}^2+1\right )}{f} \]
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