Optimal. Leaf size=196 \[ \frac {3 a^2 c^5 \tan ^5(e+f x)}{5 f}+\frac {a^2 c^5 \tan ^3(e+f x)}{3 f}-\frac {a^2 c^5 \tan (e+f x)}{f}-\frac {19 a^2 c^5 \tanh ^{-1}(\sin (e+f x))}{16 f}-\frac {a^2 c^5 \tan ^3(e+f x) \sec ^3(e+f x)}{6 f}+\frac {a^2 c^5 \tan (e+f x) \sec ^3(e+f x)}{8 f}-\frac {3 a^2 c^5 \tan ^3(e+f x) \sec (e+f x)}{4 f}+\frac {17 a^2 c^5 \tan (e+f x) \sec (e+f x)}{16 f}+a^2 c^5 x \]
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Rubi [A] time = 0.30, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {3904, 3886, 3473, 8, 2611, 3770, 2607, 30, 3768} \[ \frac {3 a^2 c^5 \tan ^5(e+f x)}{5 f}+\frac {a^2 c^5 \tan ^3(e+f x)}{3 f}-\frac {a^2 c^5 \tan (e+f x)}{f}-\frac {19 a^2 c^5 \tanh ^{-1}(\sin (e+f x))}{16 f}-\frac {a^2 c^5 \tan ^3(e+f x) \sec ^3(e+f x)}{6 f}+\frac {a^2 c^5 \tan (e+f x) \sec ^3(e+f x)}{8 f}-\frac {3 a^2 c^5 \tan ^3(e+f x) \sec (e+f x)}{4 f}+\frac {17 a^2 c^5 \tan (e+f x) \sec (e+f x)}{16 f}+a^2 c^5 x \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 2607
Rule 2611
Rule 3473
Rule 3768
Rule 3770
Rule 3886
Rule 3904
Rubi steps
\begin {align*} \int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^5 \, dx &=\left (a^2 c^2\right ) \int (c-c \sec (e+f x))^3 \tan ^4(e+f x) \, dx\\ &=\left (a^2 c^2\right ) \int \left (c^3 \tan ^4(e+f x)-3 c^3 \sec (e+f x) \tan ^4(e+f x)+3 c^3 \sec ^2(e+f x) \tan ^4(e+f x)-c^3 \sec ^3(e+f x) \tan ^4(e+f x)\right ) \, dx\\ &=\left (a^2 c^5\right ) \int \tan ^4(e+f x) \, dx-\left (a^2 c^5\right ) \int \sec ^3(e+f x) \tan ^4(e+f x) \, dx-\left (3 a^2 c^5\right ) \int \sec (e+f x) \tan ^4(e+f x) \, dx+\left (3 a^2 c^5\right ) \int \sec ^2(e+f x) \tan ^4(e+f x) \, dx\\ &=\frac {a^2 c^5 \tan ^3(e+f x)}{3 f}-\frac {3 a^2 c^5 \sec (e+f x) \tan ^3(e+f x)}{4 f}-\frac {a^2 c^5 \sec ^3(e+f x) \tan ^3(e+f x)}{6 f}+\frac {1}{2} \left (a^2 c^5\right ) \int \sec ^3(e+f x) \tan ^2(e+f x) \, dx-\left (a^2 c^5\right ) \int \tan ^2(e+f x) \, dx+\frac {1}{4} \left (9 a^2 c^5\right ) \int \sec (e+f x) \tan ^2(e+f x) \, dx+\frac {\left (3 a^2 c^5\right ) \operatorname {Subst}\left (\int x^4 \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {a^2 c^5 \tan (e+f x)}{f}+\frac {9 a^2 c^5 \sec (e+f x) \tan (e+f x)}{8 f}+\frac {a^2 c^5 \sec ^3(e+f x) \tan (e+f x)}{8 f}+\frac {a^2 c^5 \tan ^3(e+f x)}{3 f}-\frac {3 a^2 c^5 \sec (e+f x) \tan ^3(e+f x)}{4 f}-\frac {a^2 c^5 \sec ^3(e+f x) \tan ^3(e+f x)}{6 f}+\frac {3 a^2 c^5 \tan ^5(e+f x)}{5 f}-\frac {1}{8} \left (a^2 c^5\right ) \int \sec ^3(e+f x) \, dx+\left (a^2 c^5\right ) \int 1 \, dx-\frac {1}{8} \left (9 a^2 c^5\right ) \int \sec (e+f x) \, dx\\ &=a^2 c^5 x-\frac {9 a^2 c^5 \tanh ^{-1}(\sin (e+f x))}{8 f}-\frac {a^2 c^5 \tan (e+f x)}{f}+\frac {17 a^2 c^5 \sec (e+f x) \tan (e+f x)}{16 f}+\frac {a^2 c^5 \sec ^3(e+f x) \tan (e+f x)}{8 f}+\frac {a^2 c^5 \tan ^3(e+f x)}{3 f}-\frac {3 a^2 c^5 \sec (e+f x) \tan ^3(e+f x)}{4 f}-\frac {a^2 c^5 \sec ^3(e+f x) \tan ^3(e+f x)}{6 f}+\frac {3 a^2 c^5 \tan ^5(e+f x)}{5 f}-\frac {1}{16} \left (a^2 c^5\right ) \int \sec (e+f x) \, dx\\ &=a^2 c^5 x-\frac {19 a^2 c^5 \tanh ^{-1}(\sin (e+f x))}{16 f}-\frac {a^2 c^5 \tan (e+f x)}{f}+\frac {17 a^2 c^5 \sec (e+f x) \tan (e+f x)}{16 f}+\frac {a^2 c^5 \sec ^3(e+f x) \tan (e+f x)}{8 f}+\frac {a^2 c^5 \tan ^3(e+f x)}{3 f}-\frac {3 a^2 c^5 \sec (e+f x) \tan ^3(e+f x)}{4 f}-\frac {a^2 c^5 \sec ^3(e+f x) \tan ^3(e+f x)}{6 f}+\frac {3 a^2 c^5 \tan ^5(e+f x)}{5 f}\\ \end {align*}
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Mathematica [A] time = 2.13, size = 165, normalized size = 0.84 \[ \frac {a^2 c^5 \sec ^6(e+f x) \left (-210 \sin (e+f x)-120 \sin (2 (e+f x))+865 \sin (3 (e+f x))-768 \sin (4 (e+f x))+435 \sin (5 (e+f x))-88 \sin (6 (e+f x))+1800 (e+f x) \cos (2 (e+f x))+720 e \cos (4 (e+f x))+720 f x \cos (4 (e+f x))+120 e \cos (6 (e+f x))+120 f x \cos (6 (e+f x))-4560 \cos ^6(e+f x) \tanh ^{-1}(\sin (e+f x))+1200 e+1200 f x\right )}{3840 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 179, normalized size = 0.91 \[ \frac {480 \, a^{2} c^{5} f x \cos \left (f x + e\right )^{6} - 285 \, a^{2} c^{5} \cos \left (f x + e\right )^{6} \log \left (\sin \left (f x + e\right ) + 1\right ) + 285 \, a^{2} c^{5} \cos \left (f x + e\right )^{6} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (176 \, a^{2} c^{5} \cos \left (f x + e\right )^{5} - 435 \, a^{2} c^{5} \cos \left (f x + e\right )^{4} + 208 \, a^{2} c^{5} \cos \left (f x + e\right )^{3} + 110 \, a^{2} c^{5} \cos \left (f x + e\right )^{2} - 144 \, a^{2} c^{5} \cos \left (f x + e\right ) + 40 \, a^{2} c^{5}\right )} \sin \left (f x + e\right )}{480 \, f \cos \left (f x + e\right )^{6}} \]
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 = 1.58, size = 186, normalized size = 0.95 \[ -\frac {11 a^{2} c^{5} \left (\sec ^{3}\left (f x +e \right )\right ) \tan \left (f x +e \right )}{24 f}+\frac {29 a^{2} c^{5} \sec \left (f x +e \right ) \tan \left (f x +e \right )}{16 f}-\frac {19 c^{5} a^{2} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16 f}-\frac {11 a^{2} c^{5} \tan \left (f x +e \right )}{15 f}-\frac {13 c^{5} a^{2} \tan \left (f x +e \right ) \left (\sec ^{2}\left (f x +e \right )\right )}{15 f}+a^{2} c^{5} x +\frac {a^{2} c^{5} e}{f}+\frac {3 c^{5} a^{2} \tan \left (f x +e \right ) \left (\sec ^{4}\left (f x +e \right )\right )}{5 f}-\frac {c^{5} a^{2} \tan \left (f x +e \right ) \left (\sec ^{5}\left (f x +e \right )\right )}{6 f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 334, normalized size = 1.70 \[ \frac {96 \, {\left (3 \, \tan \left (f x + e\right )^{5} + 10 \, \tan \left (f x + e\right )^{3} + 15 \, \tan \left (f x + e\right )\right )} a^{2} c^{5} - 800 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{2} c^{5} + 480 \, {\left (f x + e\right )} a^{2} c^{5} + 5 \, a^{2} c^{5} {\left (\frac {2 \, {\left (15 \, \sin \left (f x + e\right )^{5} - 40 \, \sin \left (f x + e\right )^{3} + 33 \, \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{6} - 3 \, \sin \left (f x + e\right )^{4} + 3 \, \sin \left (f x + e\right )^{2} - 1} - 15 \, \log \left (\sin \left (f x + e\right ) + 1\right ) + 15 \, \log \left (\sin \left (f x + e\right ) - 1\right )\right )} + 30 \, a^{2} c^{5} {\left (\frac {2 \, {\left (3 \, \sin \left (f x + e\right )^{3} - 5 \, \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{4} - 2 \, \sin \left (f x + e\right )^{2} + 1} - 3 \, \log \left (\sin \left (f x + e\right ) + 1\right ) + 3 \, \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 600 \, a^{2} c^{5} {\left (\frac {2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 1440 \, a^{2} c^{5} \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) + 480 \, a^{2} c^{5} \tan \left (f x + e\right )}{480 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.53, size = 228, normalized size = 1.16 \[ a^2\,c^5\,x-\frac {-\frac {35\,a^2\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}}{8}+\frac {209\,a^2\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9}{8}-\frac {291\,a^2\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7}{20}+\frac {61\,a^2\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{20}+\frac {19\,a^2\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{24}-\frac {3\,a^2\,c^5\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{8}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}-6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8-20\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}-\frac {19\,a^2\,c^5\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{8\,f} \]
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 \[ - a^{2} c^{5} \left (\int \left (-1\right )\, dx + \int 3 \sec {\left (e + f x \right )}\, dx + \int \left (- \sec ^{2}{\left (e + f x \right )}\right )\, dx + \int \left (- 5 \sec ^{3}{\left (e + f x \right )}\right )\, dx + \int 5 \sec ^{4}{\left (e + f x \right )}\, dx + \int \sec ^{5}{\left (e + f x \right )}\, dx + \int \left (- 3 \sec ^{6}{\left (e + f x \right )}\right )\, dx + \int \sec ^{7}{\left (e + f x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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