Optimal. Leaf size=41 \[ \frac {2 c \tan (e+f x)}{f (a \sec (e+f x)+a)}-\frac {c \tanh ^{-1}(\sin (e+f x))}{a f} \]
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Rubi [A] time = 0.05, antiderivative size = 41, 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} \[ \frac {2 c \tan (e+f x)}{f (a \sec (e+f x)+a)}-\frac {c \tanh ^{-1}(\sin (e+f x))}{a f} \]
Antiderivative was successfully verified.
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Rule 3770
Rule 3957
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (c-c \sec (e+f x))}{a+a \sec (e+f x)} \, dx &=\frac {2 c \tan (e+f x)}{f (a+a \sec (e+f x))}-\frac {c \int \sec (e+f x) \, dx}{a}\\ &=-\frac {c \tanh ^{-1}(\sin (e+f x))}{a f}+\frac {2 c \tan (e+f x)}{f (a+a \sec (e+f x))}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 77, normalized size = 1.88 \[ -\frac {c \left (-\frac {2 \tan \left (\frac {1}{2} (e+f x)\right )}{f}-\frac {\log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )}{f}+\frac {\log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )}{f}\right )}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 70, normalized size = 1.71 \[ -\frac {{\left (c \cos \left (f x + e\right ) + c\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) - {\left (c \cos \left (f x + e\right ) + c\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 4 \, c \sin \left (f x + e\right )}{2 \, {\left (a f \cos \left (f x + e\right ) + a f\right )}} \]
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 = 0.78, size = 61, normalized size = 1.49 \[ \frac {2 c \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{f a}+\frac {c \ln \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-1\right )}{f a}-\frac {c \ln \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )+1\right )}{f a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 101, normalized size = 2.46 \[ -\frac {c {\left (\frac {\log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a} - \frac {\log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a} - \frac {\sin \left (f x + e\right )}{a {\left (\cos \left (f x + e\right ) + 1\right )}}\right )} - \frac {c \sin \left (f x + e\right )}{a {\left (\cos \left (f x + e\right ) + 1\right )}}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.58, size = 31, normalized size = 0.76 \[ -\frac {2\,c\,\left (\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )-\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{a\,f} \]
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 \[ - \frac {c \left (\int \left (- \frac {\sec {\left (e + f x \right )}}{\sec {\left (e + f x \right )} + 1}\right )\, dx + \int \frac {\sec ^{2}{\left (e + f x \right )}}{\sec {\left (e + f x \right )} + 1}\, dx\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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