Integrand size = 21, antiderivative size = 19 \[ \int \sec ^3(e+f x) \left (3-4 \sec ^2(e+f x)\right ) \, dx=-\frac {\sec ^3(e+f x) \tan (e+f x)}{f} \]
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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} \[ \int \sec ^3(e+f x) \left (3-4 \sec ^2(e+f x)\right ) \, dx=-\frac {\tan (e+f x) \sec ^3(e+f x)}{f} \]
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Rule 4128
Rubi steps \begin{align*} \text {integral}& = -\frac {\sec ^3(e+f x) \tan (e+f x)}{f} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \sec ^3(e+f x) \left (3-4 \sec ^2(e+f x)\right ) \, dx=-\frac {\sec ^3(e+f x) \tan (e+f x)}{f} \]
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Time = 0.19 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.89
method | result | size |
parallelrisch | \(-\frac {8 \sin \left (f x +e \right )}{f \left (\cos \left (4 f x +4 e \right )+4 \cos \left (2 f x +2 e \right )+3\right )}\) | \(36\) |
risch | \(\frac {8 i \left ({\mathrm e}^{5 i \left (f x +e \right )}-{\mathrm e}^{3 i \left (f x +e \right )}\right )}{f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{4}}\) | \(41\) |
derivativedivides | \(\frac {\frac {3 \sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+4 \left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )}{f}\) | \(47\) |
default | \(\frac {\frac {3 \sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+4 \left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )}{f}\) | \(47\) |
norman | \(\frac {-\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f}-\frac {6 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{f}-\frac {6 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{f}-\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{4}}\) | \(80\) |
parts | \(\frac {3 \sec \left (f x +e \right ) \tan \left (f x +e \right )}{2 f}+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2 f}-\frac {4 \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )}{f}\) | \(87\) |
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Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \sec ^3(e+f x) \left (3-4 \sec ^2(e+f x)\right ) \, dx=-\frac {\sin \left (f x + e\right )}{f \cos \left (f x + e\right )^{4}} \]
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\[ \int \sec ^3(e+f x) \left (3-4 \sec ^2(e+f x)\right ) \, dx=- \int \left (- 3 \sec ^{3}{\left (e + f x \right )}\right )\, dx - \int 4 \sec ^{5}{\left (e + f x \right )}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.74 \[ \int \sec ^3(e+f x) \left (3-4 \sec ^2(e+f x)\right ) \, dx=-\frac {\sin \left (f x + e\right )}{{\left (\sin \left (f x + e\right )^{4} - 2 \, \sin \left (f x + e\right )^{2} + 1\right )} f} \]
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Time = 0.30 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int \sec ^3(e+f x) \left (3-4 \sec ^2(e+f x)\right ) \, dx=-\frac {\sin \left (f x + e\right )}{{\left (\sin \left (f x + e\right )^{2} - 1\right )}^{2} f} \]
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Time = 0.06 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int \sec ^3(e+f x) \left (3-4 \sec ^2(e+f x)\right ) \, dx=-\frac {\sin \left (e+f\,x\right )}{f\,{\left ({\sin \left (e+f\,x\right )}^2-1\right )}^2} \]
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