Optimal. Leaf size=77 \[ -\frac {a^3 c \tan ^3(e+f x)}{3 f}-\frac {a^3 c \tan (e+f x)}{f}+\frac {a^3 c \tanh ^{-1}(\sin (e+f x))}{f}-\frac {a^3 c \tan (e+f x) \sec (e+f x)}{f}+a^3 c x \]
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Rubi [A] time = 0.15, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3904, 3886, 3473, 8, 2611, 3770, 2607, 30} \[ -\frac {a^3 c \tan ^3(e+f x)}{3 f}-\frac {a^3 c \tan (e+f x)}{f}+\frac {a^3 c \tanh ^{-1}(\sin (e+f x))}{f}-\frac {a^3 c \tan (e+f x) \sec (e+f x)}{f}+a^3 c x \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 2607
Rule 2611
Rule 3473
Rule 3770
Rule 3886
Rule 3904
Rubi steps
\begin {align*} \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x)) \, dx &=-\left ((a c) \int (a+a \sec (e+f x))^2 \tan ^2(e+f x) \, dx\right )\\ &=-\left ((a c) \int \left (a^2 \tan ^2(e+f x)+2 a^2 \sec (e+f x) \tan ^2(e+f x)+a^2 \sec ^2(e+f x) \tan ^2(e+f x)\right ) \, dx\right )\\ &=-\left (\left (a^3 c\right ) \int \tan ^2(e+f x) \, dx\right )-\left (a^3 c\right ) \int \sec ^2(e+f x) \tan ^2(e+f x) \, dx-\left (2 a^3 c\right ) \int \sec (e+f x) \tan ^2(e+f x) \, dx\\ &=-\frac {a^3 c \tan (e+f x)}{f}-\frac {a^3 c \sec (e+f x) \tan (e+f x)}{f}+\left (a^3 c\right ) \int 1 \, dx+\left (a^3 c\right ) \int \sec (e+f x) \, dx-\frac {\left (a^3 c\right ) \operatorname {Subst}\left (\int x^2 \, dx,x,\tan (e+f x)\right )}{f}\\ &=a^3 c x+\frac {a^3 c \tanh ^{-1}(\sin (e+f x))}{f}-\frac {a^3 c \tan (e+f x)}{f}-\frac {a^3 c \sec (e+f x) \tan (e+f x)}{f}-\frac {a^3 c \tan ^3(e+f x)}{3 f}\\ \end {align*}
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Mathematica [A] time = 0.45, size = 101, normalized size = 1.31 \[ \frac {a^3 c \sec ^3(e+f x) \left (-6 \sin (e+f x)-6 \sin (2 (e+f x))-2 \sin (3 (e+f x))+9 (e+f x) \cos (e+f x)+3 e \cos (3 (e+f x))+3 f x \cos (3 (e+f x))+12 \cos ^3(e+f x) \tanh ^{-1}(\sin (e+f x))\right )}{12 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 118, normalized size = 1.53 \[ \frac {6 \, a^{3} c f x \cos \left (f x + e\right )^{3} + 3 \, a^{3} c \cos \left (f x + e\right )^{3} \log \left (\sin \left (f x + e\right ) + 1\right ) - 3 \, a^{3} c \cos \left (f x + e\right )^{3} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (2 \, a^{3} c \cos \left (f x + e\right )^{2} + 3 \, a^{3} c \cos \left (f x + e\right ) + a^{3} c\right )} \sin \left (f x + e\right )}{6 \, f \cos \left (f x + e\right )^{3}} \]
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 = 0.86, size = 98, normalized size = 1.27 \[ \frac {a^{3} c \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{f}+a^{3} c x +\frac {a^{3} c e}{f}-\frac {a^{3} c \sec \left (f x +e \right ) \tan \left (f x +e \right )}{f}-\frac {2 a^{3} c \tan \left (f x +e \right )}{3 f}-\frac {a^{3} c \tan \left (f x +e \right ) \left (\sec ^{2}\left (f x +e \right )\right )}{3 f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 107, normalized size = 1.39 \[ -\frac {2 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{3} c - 6 \, {\left (f x + e\right )} a^{3} c - 3 \, a^{3} c {\left (\frac {2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 12 \, a^{3} c \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right )}{6 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.55, size = 104, normalized size = 1.35 \[ \frac {4\,a^3\,c\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )-\frac {4\,a^3\,c\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{3}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6-3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1\right )}+a^3\,c\,x+\frac {2\,a^3\,c\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{f} \]
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 \[ - a^{3} c \left (\int \left (-1\right )\, dx + \int \left (- 2 \sec {\left (e + f x \right )}\right )\, dx + \int 2 \sec ^{3}{\left (e + f x \right )}\, dx + \int \sec ^{4}{\left (e + f x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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