Integrand size = 12, antiderivative size = 15 \[ \int \left (a+b \sec ^2(e+f x)\right ) \, dx=a x+\frac {b \tan (e+f x)}{f} \]
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Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3852, 8} \[ \int \left (a+b \sec ^2(e+f x)\right ) \, dx=a x+\frac {b \tan (e+f x)}{f} \]
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Rule 8
Rule 3852
Rubi steps \begin{align*} \text {integral}& = a x+b \int \sec ^2(e+f x) \, dx \\ & = a x-\frac {b \text {Subst}(\int 1 \, dx,x,-\tan (e+f x))}{f} \\ & = a x+\frac {b \tan (e+f x)}{f} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \left (a+b \sec ^2(e+f x)\right ) \, dx=a x+\frac {b \tan (e+f x)}{f} \]
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Time = 0.16 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07
method | result | size |
default | \(a x +\frac {b \tan \left (f x +e \right )}{f}\) | \(16\) |
parts | \(a x +\frac {b \tan \left (f x +e \right )}{f}\) | \(16\) |
derivativedivides | \(\frac {\left (f x +e \right ) a +b \tan \left (f x +e \right )}{f}\) | \(21\) |
risch | \(a x +\frac {2 i b}{f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}\) | \(25\) |
parallelrisch | \(-\frac {2 b \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )}+a x\) | \(35\) |
norman | \(\frac {a x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-a x -\frac {2 b \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f}}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1}\) | \(51\) |
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Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (15) = 30\).
Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 2.07 \[ \int \left (a+b \sec ^2(e+f x)\right ) \, dx=\frac {a f x \cos \left (f x + e\right ) + b \sin \left (f x + e\right )}{f \cos \left (f x + e\right )} \]
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\[ \int \left (a+b \sec ^2(e+f x)\right ) \, dx=\int \left (a + b \sec ^{2}{\left (e + f x \right )}\right )\, dx \]
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none
Time = 0.18 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \left (a+b \sec ^2(e+f x)\right ) \, dx=a x + \frac {b \tan \left (f x + e\right )}{f} \]
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none
Time = 0.27 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \left (a+b \sec ^2(e+f x)\right ) \, dx=a x + \frac {b \tan \left (f x + e\right )}{f} \]
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Time = 18.42 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \left (a+b \sec ^2(e+f x)\right ) \, dx=\frac {b\,\mathrm {tan}\left (e+f\,x\right )+a\,f\,x}{f} \]
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