Optimal. Leaf size=61 \[ -\frac {a^2 c \tan ^3(e+f x)}{3 f}+\frac {a^2 c \tanh ^{-1}(\sin (e+f x))}{2 f}-\frac {a^2 c \tan (e+f x) \sec (e+f x)}{2 f} \]
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Rubi [A] time = 0.10, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3958, 2611, 3770, 2607, 30} \[ -\frac {a^2 c \tan ^3(e+f x)}{3 f}+\frac {a^2 c \tanh ^{-1}(\sin (e+f x))}{2 f}-\frac {a^2 c \tan (e+f x) \sec (e+f x)}{2 f} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2607
Rule 2611
Rule 3770
Rule 3958
Rubi steps
\begin {align*} \int \sec (e+f x) (a+a \sec (e+f x))^2 (c-c \sec (e+f x)) \, dx &=-\left ((a c) \int \left (a \sec (e+f x) \tan ^2(e+f x)+a \sec ^2(e+f x) \tan ^2(e+f x)\right ) \, dx\right )\\ &=-\left (\left (a^2 c\right ) \int \sec (e+f x) \tan ^2(e+f x) \, dx\right )-\left (a^2 c\right ) \int \sec ^2(e+f x) \tan ^2(e+f x) \, dx\\ &=-\frac {a^2 c \sec (e+f x) \tan (e+f x)}{2 f}+\frac {1}{2} \left (a^2 c\right ) \int \sec (e+f x) \, dx-\frac {\left (a^2 c\right ) \operatorname {Subst}\left (\int x^2 \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {a^2 c \tanh ^{-1}(\sin (e+f x))}{2 f}-\frac {a^2 c \sec (e+f x) \tan (e+f x)}{2 f}-\frac {a^2 c \tan ^3(e+f x)}{3 f}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 45, normalized size = 0.74 \[ \frac {a^2 c \left (-2 \tan ^3(e+f x)+3 \tanh ^{-1}(\sin (e+f x))-3 \tan (e+f x) \sec (e+f x)\right )}{6 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 103, normalized size = 1.69 \[ \frac {3 \, a^{2} c \cos \left (f x + e\right )^{3} \log \left (\sin \left (f x + e\right ) + 1\right ) - 3 \, a^{2} c \cos \left (f x + e\right )^{3} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left (2 \, a^{2} c \cos \left (f x + e\right )^{2} - 3 \, a^{2} c \cos \left (f x + e\right ) - 2 \, a^{2} c\right )} \sin \left (f x + e\right )}{12 \, f \cos \left (f x + e\right )^{3}} \]
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 = 1.10, size = 84, normalized size = 1.38 \[ \frac {a^{2} c \tan \left (f x +e \right )}{3 f}+\frac {a^{2} c \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2 f}-\frac {a^{2} c \sec \left (f x +e \right ) \tan \left (f x +e \right )}{2 f}-\frac {a^{2} c \tan \left (f x +e \right ) \left (\sec ^{2}\left (f x +e \right )\right )}{3 f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 108, normalized size = 1.77 \[ -\frac {4 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{2} c - 3 \, a^{2} c {\left (\frac {2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 12 \, a^{2} c \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) - 12 \, a^{2} c \tan \left (f x + e\right )}{12 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.79, size = 113, normalized size = 1.85 \[ \frac {-c\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+\frac {8\,c\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{3}+c\,a^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6-3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1\right )}+\frac {a^2\,c\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{f} \]
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 \[ - a^{2} c \left (\int \left (- \sec {\left (e + f x \right )}\right )\, dx + \int \left (- \sec ^{2}{\left (e + f x \right )}\right )\, dx + \int \sec ^{3}{\left (e + f x \right )}\, dx + \int \sec ^{4}{\left (e + f x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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