Optimal. Leaf size=61 \[ \frac {a c^2 \tan ^3(e+f x)}{3 f}+\frac {a c^2 \tanh ^{-1}(\sin (e+f x))}{2 f}-\frac {a c^2 \tan (e+f x) \sec (e+f x)}{2 f} \]
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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} \[ \frac {a c^2 \tan ^3(e+f x)}{3 f}+\frac {a c^2 \tanh ^{-1}(\sin (e+f x))}{2 f}-\frac {a c^2 \tan (e+f x) \sec (e+f x)}{2 f} \]
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)) (c-c \sec (e+f x))^2 \, dx &=-\left ((a c) \int \left (c \sec (e+f x) \tan ^2(e+f x)-c \sec ^2(e+f x) \tan ^2(e+f x)\right ) \, dx\right )\\ &=-\left (\left (a c^2\right ) \int \sec (e+f x) \tan ^2(e+f x) \, dx\right )+\left (a c^2\right ) \int \sec ^2(e+f x) \tan ^2(e+f x) \, dx\\ &=-\frac {a c^2 \sec (e+f x) \tan (e+f x)}{2 f}+\frac {1}{2} \left (a c^2\right ) \int \sec (e+f x) \, dx+\frac {\left (a c^2\right ) \operatorname {Subst}\left (\int x^2 \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {a c^2 \tanh ^{-1}(\sin (e+f x))}{2 f}-\frac {a c^2 \sec (e+f x) \tan (e+f x)}{2 f}+\frac {a c^2 \tan ^3(e+f x)}{3 f}\\ \end {align*}
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Mathematica [B] time = 0.72, size = 313, normalized size = 5.13 \[ -\frac {a c^2 \sec (e) \sec ^3(e+f x) \left (-12 \sin (2 e+f x)+6 \sin (e+2 f x)+6 \sin (3 e+2 f x)+4 \sin (2 e+3 f x)+3 \cos (2 e+3 f x) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )+3 \cos (4 e+3 f x) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )+9 \cos (f x) \left (\log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )\right )+9 \cos (2 e+f x) \left (\log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )\right )-3 \cos (2 e+3 f x) \log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )-3 \cos (4 e+3 f x) \log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )\right )}{48 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 103, normalized size = 1.69 \[ \frac {3 \, a c^{2} \cos \left (f x + e\right )^{3} \log \left (\sin \left (f x + e\right ) + 1\right ) - 3 \, a c^{2} \cos \left (f x + e\right )^{3} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (2 \, a c^{2} \cos \left (f x + e\right )^{2} + 3 \, a c^{2} \cos \left (f x + e\right ) - 2 \, a c^{2}\right )} \sin \left (f x + e\right )}{12 \, f \cos \left (f x + e\right )^{3}} \]
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 = 1.17, size = 84, normalized size = 1.38 \[ -\frac {a \,c^{2} \sec \left (f x +e \right ) \tan \left (f x +e \right )}{2 f}+\frac {a \,c^{2} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2 f}-\frac {a \,c^{2} \tan \left (f x +e \right )}{3 f}+\frac {a \,c^{2} \tan \left (f x +e \right ) \left (\sec ^{2}\left (f x +e \right )\right )}{3 f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 108, normalized size = 1.77 \[ \frac {4 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a c^{2} + 3 \, a c^{2} {\left (\frac {2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} + 12 \, a c^{2} \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) - 12 \, a c^{2} \tan \left (f x + e\right )}{12 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.81, size = 114, normalized size = 1.87 \[ \frac {a\,c^2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{f}-\frac {a\,c^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+\frac {8\,a\,c^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{3}-a\,c^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6-3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1\right )} \]
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 \[ a c^{2} \left (\int \sec {\left (e + f x \right )}\, dx + \int \left (- \sec ^{2}{\left (e + f x \right )}\right )\, dx + \int \left (- \sec ^{3}{\left (e + f x \right )}\right )\, dx + \int \sec ^{4}{\left (e + f x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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