Optimal. Leaf size=42 \[ -\frac {a \tanh ^{-1}(\sin (e+f x))}{c f}-\frac {2 a \tan (e+f x)}{f (c-c \sec (e+f x))} \]
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Rubi [A] time = 0.05, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {3957, 3770} \[ -\frac {a \tanh ^{-1}(\sin (e+f x))}{c f}-\frac {2 a \tan (e+f x)}{f (c-c \sec (e+f x))} \]
Antiderivative was successfully verified.
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Rule 3770
Rule 3957
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (a+a \sec (e+f x))}{c-c \sec (e+f x)} \, dx &=-\frac {2 a \tan (e+f x)}{f (c-c \sec (e+f x))}-\frac {a \int \sec (e+f x) \, dx}{c}\\ &=-\frac {a \tanh ^{-1}(\sin (e+f x))}{c f}-\frac {2 a \tan (e+f x)}{f (c-c \sec (e+f x))}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 77, normalized size = 1.83 \[ -\frac {a \left (-\frac {2 \cot \left (\frac {1}{2} (e+f x)\right )}{f}-\frac {\log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )}{f}+\frac {\log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )}{f}\right )}{c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 66, normalized size = 1.57 \[ -\frac {a \log \left (\sin \left (f x + e\right ) + 1\right ) \sin \left (f x + e\right ) - a \log \left (-\sin \left (f x + e\right ) + 1\right ) \sin \left (f x + e\right ) - 4 \, a \cos \left (f x + e\right ) - 4 \, a}{2 \, c f \sin \left (f x + e\right )} \]
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 = 0.87, size = 63, normalized size = 1.50 \[ \frac {a \ln \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-1\right )}{f c}-\frac {a \ln \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )+1\right )}{f c}+\frac {2 a}{f c \tan \left (\frac {e}{2}+\frac {f x}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.17, size = 101, normalized size = 2.40 \[ -\frac {a {\left (\frac {\log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{c} - \frac {\log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{c} - \frac {\cos \left (f x + e\right ) + 1}{c \sin \left (f x + e\right )}\right )} - \frac {a {\left (\cos \left (f x + e\right ) + 1\right )}}{c \sin \left (f x + e\right )}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.85, size = 31, normalized size = 0.74 \[ -\frac {2\,a\,\left (\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )-\mathrm {cot}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{c\,f} \]
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 \[ - \frac {a \left (\int \frac {\sec {\left (e + f x \right )}}{\sec {\left (e + f x \right )} - 1}\, dx + \int \frac {\sec ^{2}{\left (e + f x \right )}}{\sec {\left (e + f x \right )} - 1}\, dx\right )}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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