Optimal. Leaf size=55 \[ \frac {a^2 c \tanh ^{-1}(\sin (e+f x))}{2 f}-\frac {c \tan (e+f x) \left (a^2 \sec (e+f x)+2 a^2\right )}{2 f}+a^2 c x \]
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Rubi [A] time = 0.06, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3904, 3881, 3770} \[ \frac {a^2 c \tanh ^{-1}(\sin (e+f x))}{2 f}-\frac {c \tan (e+f x) \left (a^2 \sec (e+f x)+2 a^2\right )}{2 f}+a^2 c x \]
Antiderivative was successfully verified.
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Rule 3770
Rule 3881
Rule 3904
Rubi steps
\begin {align*} \int (a+a \sec (e+f x))^2 (c-c \sec (e+f x)) \, dx &=-\left ((a c) \int (a+a \sec (e+f x)) \tan ^2(e+f x) \, dx\right )\\ &=-\frac {c \left (2 a^2+a^2 \sec (e+f x)\right ) \tan (e+f x)}{2 f}+\frac {1}{2} (a c) \int (2 a+a \sec (e+f x)) \, dx\\ &=a^2 c x-\frac {c \left (2 a^2+a^2 \sec (e+f x)\right ) \tan (e+f x)}{2 f}+\frac {1}{2} \left (a^2 c\right ) \int \sec (e+f x) \, dx\\ &=a^2 c x+\frac {a^2 c \tanh ^{-1}(\sin (e+f x))}{2 f}-\frac {c \left (2 a^2+a^2 \sec (e+f x)\right ) \tan (e+f x)}{2 f}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 72, normalized size = 1.31 \[ \frac {a^2 c \sec ^2(e+f x) \left (-\sin (e+f x)-\sin (2 (e+f x))+(e+f x) \cos (2 (e+f x))+\cos ^2(e+f x) \tanh ^{-1}(\sin (e+f x))+e+f x\right )}{2 f} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 103, normalized size = 1.87 \[ \frac {4 \, a^{2} c f x \cos \left (f x + e\right )^{2} + a^{2} c \cos \left (f x + e\right )^{2} \log \left (\sin \left (f x + e\right ) + 1\right ) - a^{2} c \cos \left (f x + e\right )^{2} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (2 \, a^{2} c \cos \left (f x + e\right ) + a^{2} c\right )} \sin \left (f x + e\right )}{4 \, f \cos \left (f x + e\right )^{2}} \]
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 = 0.86, size = 76, normalized size = 1.38 \[ \frac {a^{2} c \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2 f}+a^{2} c x +\frac {a^{2} c e}{f}-\frac {a^{2} c \tan \left (f x +e \right )}{f}-\frac {a^{2} c \sec \left (f x +e \right ) \tan \left (f x +e \right )}{2 f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 95, normalized size = 1.73 \[ \frac {4 \, {\left (f x + e\right )} a^{2} c + a^{2} c {\left (\frac {2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} + 4 \, a^{2} c \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) - 4 \, a^{2} c \tan \left (f x + e\right )}{4 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.51, size = 91, normalized size = 1.65 \[ a^2\,c\,x-\frac {3\,a^2\,c\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )-a^2\,c\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}+\frac {a^2\,c\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{f} \]
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 \[ - a^{2} c \left (\int \left (-1\right )\, dx + \int \left (- \sec {\left (e + f x \right )}\right )\, dx + \int \sec ^{2}{\left (e + f x \right )}\, dx + \int \sec ^{3}{\left (e + f x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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