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 = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, 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 3770
Rule 4046
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*}
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Mathematica [A] time = 0.01, size = 48, normalized size = 1.20 \[ \frac {a \tanh ^{-1}(\sin (e+f x))}{f}+\frac {b \tanh ^{-1}(\sin (e+f x))}{2 f}+\frac {b \tan (e+f x) \sec (e+f x)}{2 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 72, normalized size = 1.80 \[ \frac {{\left (2 \, a + b\right )} \cos \left (f x + e\right )^{2} \log \left (\sin \left (f x + e\right ) + 1\right ) - {\left (2 \, a + b\right )} \cos \left (f x + e\right )^{2} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, b \sin \left (f x + e\right )}{4 \, f \cos \left (f x + e\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.79, size = 59, normalized size = 1.48 \[ \frac {a \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{f}+\frac {b \sec \left (f x +e \right ) \tan \left (f x +e \right )}{2 f}+\frac {b \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2 f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 58, normalized size = 1.45 \[ \frac {{\left (2 \, a + b\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) - {\left (2 \, a + b\right )} \log \left (\sin \left (f x + e\right ) - 1\right ) - \frac {2 \, b \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1}}{4 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.41, size = 41, normalized size = 1.02 \[ \frac {\mathrm {atanh}\left (\sin \left (e+f\,x\right )\right )\,\left (a+\frac {b}{2}\right )}{f}-\frac {b\,\sin \left (e+f\,x\right )}{2\,f\,\left ({\sin \left (e+f\,x\right )}^2-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \sec ^{2}{\left (e + f x \right )}\right ) \sec {\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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