Optimal. Leaf size=132 \[ -\frac {5 a^3 c^4 \tanh ^{-1}(\sin (e+f x))}{16 f}-\frac {a^3 \tan ^5(e+f x) \left (6 c^4-5 c^4 \sec (e+f x)\right )}{30 f}+\frac {a^3 \tan ^3(e+f x) \left (8 c^4-5 c^4 \sec (e+f x)\right )}{24 f}-\frac {a^3 \tan (e+f x) \left (16 c^4-5 c^4 \sec (e+f x)\right )}{16 f}+a^3 c^4 x \]
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Rubi [A] time = 0.15, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {3904, 3881, 3770} \[ -\frac {5 a^3 c^4 \tanh ^{-1}(\sin (e+f x))}{16 f}-\frac {a^3 \tan ^5(e+f x) \left (6 c^4-5 c^4 \sec (e+f x)\right )}{30 f}+\frac {a^3 \tan ^3(e+f x) \left (8 c^4-5 c^4 \sec (e+f x)\right )}{24 f}-\frac {a^3 \tan (e+f x) \left (16 c^4-5 c^4 \sec (e+f x)\right )}{16 f}+a^3 c^4 x \]
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))^3 (c-c \sec (e+f x))^4 \, dx &=-\left (\left (a^3 c^3\right ) \int (c-c \sec (e+f x)) \tan ^6(e+f x) \, dx\right )\\ &=-\frac {a^3 \left (6 c^4-5 c^4 \sec (e+f x)\right ) \tan ^5(e+f x)}{30 f}+\frac {1}{6} \left (a^3 c^3\right ) \int (6 c-5 c \sec (e+f x)) \tan ^4(e+f x) \, dx\\ &=\frac {a^3 \left (8 c^4-5 c^4 \sec (e+f x)\right ) \tan ^3(e+f x)}{24 f}-\frac {a^3 \left (6 c^4-5 c^4 \sec (e+f x)\right ) \tan ^5(e+f x)}{30 f}-\frac {1}{24} \left (a^3 c^3\right ) \int (24 c-15 c \sec (e+f x)) \tan ^2(e+f x) \, dx\\ &=-\frac {a^3 \left (16 c^4-5 c^4 \sec (e+f x)\right ) \tan (e+f x)}{16 f}+\frac {a^3 \left (8 c^4-5 c^4 \sec (e+f x)\right ) \tan ^3(e+f x)}{24 f}-\frac {a^3 \left (6 c^4-5 c^4 \sec (e+f x)\right ) \tan ^5(e+f x)}{30 f}+\frac {1}{48} \left (a^3 c^3\right ) \int (48 c-15 c \sec (e+f x)) \, dx\\ &=a^3 c^4 x-\frac {a^3 \left (16 c^4-5 c^4 \sec (e+f x)\right ) \tan (e+f x)}{16 f}+\frac {a^3 \left (8 c^4-5 c^4 \sec (e+f x)\right ) \tan ^3(e+f x)}{24 f}-\frac {a^3 \left (6 c^4-5 c^4 \sec (e+f x)\right ) \tan ^5(e+f x)}{30 f}-\frac {1}{16} \left (5 a^3 c^4\right ) \int \sec (e+f x) \, dx\\ &=a^3 c^4 x-\frac {5 a^3 c^4 \tanh ^{-1}(\sin (e+f x))}{16 f}-\frac {a^3 \left (16 c^4-5 c^4 \sec (e+f x)\right ) \tan (e+f x)}{16 f}+\frac {a^3 \left (8 c^4-5 c^4 \sec (e+f x)\right ) \tan ^3(e+f x)}{24 f}-\frac {a^3 \left (6 c^4-5 c^4 \sec (e+f x)\right ) \tan ^5(e+f x)}{30 f}\\ \end {align*}
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Mathematica [A] time = 1.87, size = 165, normalized size = 1.25 \[ \frac {a^3 c^4 \sec ^6(e+f x) \left (450 \sin (e+f x)-600 \sin (2 (e+f x))-25 \sin (3 (e+f x))-384 \sin (4 (e+f x))+165 \sin (5 (e+f x))-184 \sin (6 (e+f x))+1800 (e+f x) \cos (2 (e+f x))+720 e \cos (4 (e+f x))+720 f x \cos (4 (e+f x))+120 e \cos (6 (e+f x))+120 f x \cos (6 (e+f x))-1200 \cos ^6(e+f x) \tanh ^{-1}(\sin (e+f x))+1200 e+1200 f x\right )}{3840 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 179, normalized size = 1.36 \[ \frac {480 \, a^{3} c^{4} f x \cos \left (f x + e\right )^{6} - 75 \, a^{3} c^{4} \cos \left (f x + e\right )^{6} \log \left (\sin \left (f x + e\right ) + 1\right ) + 75 \, a^{3} c^{4} \cos \left (f x + e\right )^{6} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (368 \, a^{3} c^{4} \cos \left (f x + e\right )^{5} - 165 \, a^{3} c^{4} \cos \left (f x + e\right )^{4} - 176 \, a^{3} c^{4} \cos \left (f x + e\right )^{3} + 130 \, a^{3} c^{4} \cos \left (f x + e\right )^{2} + 48 \, a^{3} c^{4} \cos \left (f x + e\right ) - 40 \, a^{3} c^{4}\right )} \sin \left (f x + e\right )}{480 \, f \cos \left (f x + e\right )^{6}} \]
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 = 1.65, size = 186, normalized size = 1.41 \[ -\frac {23 a^{3} c^{4} \tan \left (f x +e \right )}{15 f}+\frac {11 c^{4} a^{3} \tan \left (f x +e \right ) \left (\sec ^{2}\left (f x +e \right )\right )}{15 f}+\frac {11 c^{4} a^{3} \sec \left (f x +e \right ) \tan \left (f x +e \right )}{16 f}-\frac {5 c^{4} a^{3} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16 f}+a^{3} c^{4} x +\frac {a^{3} c^{4} e}{f}-\frac {13 c^{4} a^{3} \tan \left (f x +e \right ) \left (\sec ^{3}\left (f x +e \right )\right )}{24 f}-\frac {c^{4} a^{3} \tan \left (f x +e \right ) \left (\sec ^{4}\left (f x +e \right )\right )}{5 f}+\frac {c^{4} a^{3} \tan \left (f x +e \right ) \left (\sec ^{5}\left (f x +e \right )\right )}{6 f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 334, normalized size = 2.53 \[ -\frac {32 \, {\left (3 \, \tan \left (f x + e\right )^{5} + 10 \, \tan \left (f x + e\right )^{3} + 15 \, \tan \left (f x + e\right )\right )} a^{3} c^{4} - 480 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{3} c^{4} - 480 \, {\left (f x + e\right )} a^{3} c^{4} + 5 \, a^{3} c^{4} {\left (\frac {2 \, {\left (15 \, \sin \left (f x + e\right )^{5} - 40 \, \sin \left (f x + e\right )^{3} + 33 \, \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{6} - 3 \, \sin \left (f x + e\right )^{4} + 3 \, \sin \left (f x + e\right )^{2} - 1} - 15 \, \log \left (\sin \left (f x + e\right ) + 1\right ) + 15 \, \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 90 \, a^{3} c^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (f x + e\right )^{3} - 5 \, \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{4} - 2 \, \sin \left (f x + e\right )^{2} + 1} - 3 \, \log \left (\sin \left (f x + e\right ) + 1\right ) + 3 \, \log \left (\sin \left (f x + e\right ) - 1\right )\right )} + 360 \, a^{3} c^{4} {\left (\frac {2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} + 480 \, a^{3} c^{4} \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) + 1440 \, a^{3} c^{4} \tan \left (f x + e\right )}{480 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.59, size = 227, normalized size = 1.72 \[ a^3\,c^4\,x+\frac {\frac {21\,a^3\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}}{8}-\frac {389\,a^3\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9}{24}+\frac {853\,a^3\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7}{20}-\frac {523\,a^3\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{20}+\frac {73\,a^3\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{8}-\frac {11\,a^3\,c^4\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{8}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}-6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8-20\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}-\frac {5\,a^3\,c^4\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{8\,f} \]
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 \[ a^{3} c^{4} \left (\int 1\, dx + \int \left (- \sec {\left (e + f x \right )}\right )\, dx + \int \left (- 3 \sec ^{2}{\left (e + f x \right )}\right )\, dx + \int 3 \sec ^{3}{\left (e + f x \right )}\, dx + \int 3 \sec ^{4}{\left (e + f x \right )}\, dx + \int \left (- 3 \sec ^{5}{\left (e + f x \right )}\right )\, dx + \int \left (- \sec ^{6}{\left (e + f x \right )}\right )\, dx + \int \sec ^{7}{\left (e + f x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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