Integrand size = 21, antiderivative size = 31 \[ \int \cos ^2(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx=\frac {1}{2} (a+2 b) x+\frac {a \cos (e+f x) \sin (e+f x)}{2 f} \]
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Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {4130, 8} \[ \int \cos ^2(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx=\frac {1}{2} x (a+2 b)+\frac {a \sin (e+f x) \cos (e+f x)}{2 f} \]
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Rule 8
Rule 4130
Rubi steps \begin{align*} \text {integral}& = \frac {a \cos (e+f x) \sin (e+f x)}{2 f}+\frac {1}{2} (a+2 b) \int 1 \, dx \\ & = \frac {1}{2} (a+2 b) x+\frac {a \cos (e+f x) \sin (e+f x)}{2 f} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \cos ^2(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx=b x+\frac {a (e+f x)}{2 f}+\frac {a \sin (2 (e+f x))}{4 f} \]
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Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77
method | result | size |
risch | \(\frac {a x}{2}+x b +\frac {a \sin \left (2 f x +2 e \right )}{4 f}\) | \(24\) |
parallelrisch | \(\frac {a \sin \left (2 f x +2 e \right )+2 \left (a +2 b \right ) x f}{4 f}\) | \(27\) |
derivativedivides | \(\frac {a \left (\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+b \left (f x +e \right )}{f}\) | \(37\) |
default | \(\frac {a \left (\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+b \left (f x +e \right )}{f}\) | \(37\) |
norman | \(\frac {\left (-\frac {a}{2}-b \right ) x +\left (-\frac {a}{2}-b \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+\left (\frac {a}{2}+b \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+\left (\frac {a}{2}+b \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}-\frac {a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f}+\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{f}-\frac {a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{f}}{\left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )}\) | \(147\) |
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Time = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90 \[ \int \cos ^2(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx=\frac {{\left (a + 2 \, b\right )} f x + a \cos \left (f x + e\right ) \sin \left (f x + e\right )}{2 \, f} \]
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Time = 2.47 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.65 \[ \int \cos ^2(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx=a \left (\begin {cases} \frac {x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {x \cos ^{2}{\left (e + f x \right )}}{2} + \frac {\sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} & \text {for}\: f \neq 0 \\x \cos ^{2}{\left (e \right )} & \text {otherwise} \end {cases}\right ) + b x \]
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Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.19 \[ \int \cos ^2(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx=\frac {{\left (f x + e\right )} {\left (a + 2 \, b\right )} + \frac {a \tan \left (f x + e\right )}{\tan \left (f x + e\right )^{2} + 1}}{2 \, f} \]
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Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.19 \[ \int \cos ^2(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx=\frac {{\left (f x + e\right )} {\left (a + 2 \, b\right )} + \frac {a \tan \left (f x + e\right )}{\tan \left (f x + e\right )^{2} + 1}}{2 \, f} \]
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Time = 19.58 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81 \[ \int \cos ^2(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx=\frac {\frac {a\,\sin \left (2\,e+2\,f\,x\right )}{4}+f\,x\,\left (\frac {a}{2}+b\right )}{f} \]
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