Optimal. Leaf size=40 \[ \frac{(2 a+b) \tanh ^{-1}(\sin (e+f x))}{2 f}+\frac{b \tan (e+f x) \sec (e+f x)}{2 f} \]
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Rubi [A] time = 0.0248988, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {4046, 3770} \[ \frac{(2 a+b) \tanh ^{-1}(\sin (e+f x))}{2 f}+\frac{b \tan (e+f x) \sec (e+f x)}{2 f} \]
Antiderivative was successfully verified.
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Rule 4046
Rule 3770
Rubi steps
\begin{align*} \int \sec (e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx &=\frac{b \sec (e+f x) \tan (e+f x)}{2 f}+\frac{1}{2} (2 a+b) \int \sec (e+f x) \, dx\\ &=\frac{(2 a+b) \tanh ^{-1}(\sin (e+f x))}{2 f}+\frac{b \sec (e+f x) \tan (e+f x)}{2 f}\\ \end{align*}
Mathematica [A] time = 0.0206032, size = 48, normalized size = 1.2 \[ \frac{a \tanh ^{-1}(\sin (e+f x))}{f}+\frac{b \tanh ^{-1}(\sin (e+f x))}{2 f}+\frac{b \tan (e+f x) \sec (e+f x)}{2 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 59, normalized size = 1.5 \begin{align*}{\frac{a\ln \left ( \sec \left ( fx+e \right ) +\tan \left ( fx+e \right ) \right ) }{f}}+{\frac{b\sec \left ( fx+e \right ) \tan \left ( fx+e \right ) }{2\,f}}+{\frac{b\ln \left ( \sec \left ( fx+e \right ) +\tan \left ( fx+e \right ) \right ) }{2\,f}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.986785, size = 78, normalized size = 1.95 \begin{align*} \frac{{\left (2 \, a + b\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) -{\left (2 \, a + b\right )} \log \left (\sin \left (f x + e\right ) - 1\right ) - \frac{2 \, b \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1}}{4 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.494837, size = 192, normalized size = 4.8 \begin{align*} \frac{{\left (2 \, a + b\right )} \cos \left (f x + e\right )^{2} \log \left (\sin \left (f x + e\right ) + 1\right ) -{\left (2 \, a + b\right )} \cos \left (f x + e\right )^{2} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, b \sin \left (f x + e\right )}{4 \, f \cos \left (f x + e\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \sec ^{2}{\left (e + f x \right )}\right ) \sec{\left (e + f x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29418, size = 86, normalized size = 2.15 \begin{align*} \frac{{\left (2 \, a + b\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) -{\left (2 \, a + b\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) - \frac{2 \, b \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1}}{4 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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