Integrand size = 19, antiderivative size = 24 \[ \int \left (a+b \sec ^2(e+f x)\right ) \sin (e+f x) \, dx=-\frac {a \cos (e+f x)}{f}+\frac {b \sec (e+f x)}{f} \]
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Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {4218, 14} \[ \int \left (a+b \sec ^2(e+f x)\right ) \sin (e+f x) \, dx=\frac {b \sec (e+f x)}{f}-\frac {a \cos (e+f x)}{f} \]
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Rule 14
Rule 4218
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {b+a x^2}{x^2} \, dx,x,\cos (e+f x)\right )}{f} \\ & = -\frac {\text {Subst}\left (\int \left (a+\frac {b}{x^2}\right ) \, dx,x,\cos (e+f x)\right )}{f} \\ & = -\frac {a \cos (e+f x)}{f}+\frac {b \sec (e+f x)}{f} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46 \[ \int \left (a+b \sec ^2(e+f x)\right ) \sin (e+f x) \, dx=-\frac {a \cos (e) \cos (f x)}{f}+\frac {b \sec (e+f x)}{f}+\frac {a \sin (e) \sin (f x)}{f} \]
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Time = 0.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04
method | result | size |
derivativedivides | \(\frac {\sec \left (f x +e \right ) b -\frac {a}{\sec \left (f x +e \right )}}{f}\) | \(25\) |
default | \(\frac {\sec \left (f x +e \right ) b -\frac {a}{\sec \left (f x +e \right )}}{f}\) | \(25\) |
parts | \(-\frac {a \cos \left (f x +e \right )}{f}+\frac {b \sec \left (f x +e \right )}{f}\) | \(25\) |
parallelrisch | \(\frac {-a \cos \left (2 f x +2 e \right )+\left (-2 a +2 b \right ) \cos \left (f x +e \right )-a +2 b}{2 f \cos \left (f x +e \right )}\) | \(47\) |
risch | \(-\frac {a \,{\mathrm e}^{3 i \left (f x +e \right )}+\left (3 a -4 b \right ) \cos \left (f x +e \right )+i \left (a -4 b \right ) \sin \left (f x +e \right )}{2 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}\) | \(59\) |
norman | \(\frac {\frac {2 a -2 b}{f}-\frac {2 \left (a +b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right ) \left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )}\) | \(63\) |
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none
Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int \left (a+b \sec ^2(e+f x)\right ) \sin (e+f x) \, dx=-\frac {a \cos \left (f x + e\right )^{2} - b}{f \cos \left (f x + e\right )} \]
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\[ \int \left (a+b \sec ^2(e+f x)\right ) \sin (e+f x) \, dx=\int \left (a + b \sec ^{2}{\left (e + f x \right )}\right ) \sin {\left (e + f x \right )}\, dx \]
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none
Time = 0.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04 \[ \int \left (a+b \sec ^2(e+f x)\right ) \sin (e+f x) \, dx=-\frac {a \cos \left (f x + e\right ) - \frac {b}{\cos \left (f x + e\right )}}{f} \]
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Time = 0.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \left (a+b \sec ^2(e+f x)\right ) \sin (e+f x) \, dx=-\frac {a \cos \left (f x + e\right )}{f} + \frac {b}{f \cos \left (f x + e\right )} \]
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Time = 0.05 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04 \[ \int \left (a+b \sec ^2(e+f x)\right ) \sin (e+f x) \, dx=-\frac {a\,\cos \left (e+f\,x\right )-\frac {b}{\cos \left (e+f\,x\right )}}{f} \]
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