Optimal. Leaf size=164 \[ -\frac {7 c^5 \tan ^3(e+f x)}{a^2 f}-\frac {84 c^5 \tan (e+f x)}{a^2 f}+\frac {105 c^5 \tanh ^{-1}(\sin (e+f x))}{2 a^2 f}+\frac {63 c^5 \tan (e+f x) \sec (e+f x)}{2 a^2 f}-\frac {6 c^2 \tan (e+f x) (c-c \sec (e+f x))^3}{f \left (a^2 \sec (e+f x)+a^2\right )}+\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^4}{3 f (a \sec (e+f x)+a)^2} \]
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Rubi [A] time = 0.25, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3957, 3791, 3770, 3767, 8, 3768} \[ -\frac {7 c^5 \tan ^3(e+f x)}{a^2 f}-\frac {84 c^5 \tan (e+f x)}{a^2 f}+\frac {105 c^5 \tanh ^{-1}(\sin (e+f x))}{2 a^2 f}+\frac {63 c^5 \tan (e+f x) \sec (e+f x)}{2 a^2 f}-\frac {6 c^2 \tan (e+f x) (c-c \sec (e+f x))^3}{f \left (a^2 \sec (e+f x)+a^2\right )}+\frac {2 c \tan (e+f x) (c-c \sec (e+f x))^4}{3 f (a \sec (e+f x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3768
Rule 3770
Rule 3791
Rule 3957
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^2} \, dx &=\frac {2 c (c-c \sec (e+f x))^4 \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}-\frac {(3 c) \int \frac {\sec (e+f x) (c-c \sec (e+f x))^4}{a+a \sec (e+f x)} \, dx}{a}\\ &=-\frac {6 c^2 (c-c \sec (e+f x))^3 \tan (e+f x)}{f \left (a^2+a^2 \sec (e+f x)\right )}+\frac {2 c (c-c \sec (e+f x))^4 \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}+\frac {\left (21 c^2\right ) \int \sec (e+f x) (c-c \sec (e+f x))^3 \, dx}{a^2}\\ &=-\frac {6 c^2 (c-c \sec (e+f x))^3 \tan (e+f x)}{f \left (a^2+a^2 \sec (e+f x)\right )}+\frac {2 c (c-c \sec (e+f x))^4 \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}+\frac {\left (21 c^2\right ) \int \left (c^3 \sec (e+f x)-3 c^3 \sec ^2(e+f x)+3 c^3 \sec ^3(e+f x)-c^3 \sec ^4(e+f x)\right ) \, dx}{a^2}\\ &=-\frac {6 c^2 (c-c \sec (e+f x))^3 \tan (e+f x)}{f \left (a^2+a^2 \sec (e+f x)\right )}+\frac {2 c (c-c \sec (e+f x))^4 \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}+\frac {\left (21 c^5\right ) \int \sec (e+f x) \, dx}{a^2}-\frac {\left (21 c^5\right ) \int \sec ^4(e+f x) \, dx}{a^2}-\frac {\left (63 c^5\right ) \int \sec ^2(e+f x) \, dx}{a^2}+\frac {\left (63 c^5\right ) \int \sec ^3(e+f x) \, dx}{a^2}\\ &=\frac {21 c^5 \tanh ^{-1}(\sin (e+f x))}{a^2 f}+\frac {63 c^5 \sec (e+f x) \tan (e+f x)}{2 a^2 f}-\frac {6 c^2 (c-c \sec (e+f x))^3 \tan (e+f x)}{f \left (a^2+a^2 \sec (e+f x)\right )}+\frac {2 c (c-c \sec (e+f x))^4 \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}+\frac {\left (63 c^5\right ) \int \sec (e+f x) \, dx}{2 a^2}+\frac {\left (21 c^5\right ) \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (e+f x)\right )}{a^2 f}+\frac {\left (63 c^5\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (e+f x))}{a^2 f}\\ &=\frac {105 c^5 \tanh ^{-1}(\sin (e+f x))}{2 a^2 f}-\frac {84 c^5 \tan (e+f x)}{a^2 f}+\frac {63 c^5 \sec (e+f x) \tan (e+f x)}{2 a^2 f}-\frac {6 c^2 (c-c \sec (e+f x))^3 \tan (e+f x)}{f \left (a^2+a^2 \sec (e+f x)\right )}+\frac {2 c (c-c \sec (e+f x))^4 \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}-\frac {7 c^5 \tan ^3(e+f x)}{a^2 f}\\ \end {align*}
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Mathematica [B] time = 1.20, size = 380, normalized size = 2.32 \[ \frac {\cot \left (\frac {1}{2} (e+f x)\right ) \csc ^6\left (\frac {1}{2} (e+f x)\right ) (c-c \sec (e+f x))^5 \left (\sec \left (\frac {e}{2}\right ) \sec (e) \left (-2901 \sin \left (e-\frac {f x}{2}\right )+1197 \sin \left (e+\frac {f x}{2}\right )-3027 \sin \left (2 e+\frac {f x}{2}\right )-273 \sin \left (e+\frac {3 f x}{2}\right )+1827 \sin \left (2 e+\frac {3 f x}{2}\right )-1693 \sin \left (3 e+\frac {3 f x}{2}\right )+1995 \sin \left (e+\frac {5 f x}{2}\right )-117 \sin \left (2 e+\frac {5 f x}{2}\right )+1143 \sin \left (3 e+\frac {5 f x}{2}\right )-969 \sin \left (4 e+\frac {5 f x}{2}\right )+1173 \sin \left (2 e+\frac {7 f x}{2}\right )+117 \sin \left (3 e+\frac {7 f x}{2}\right )+747 \sin \left (4 e+\frac {7 f x}{2}\right )-309 \sin \left (5 e+\frac {7 f x}{2}\right )+494 \sin \left (3 e+\frac {9 f x}{2}\right )+142 \sin \left (4 e+\frac {9 f x}{2}\right )+352 \sin \left (5 e+\frac {9 f x}{2}\right )-1323 \sin \left (\frac {f x}{2}\right )+3247 \sin \left (\frac {3 f x}{2}\right )\right ) \csc ^3\left (\frac {1}{2} (e+f x)\right )+20160 \cos ^3(e+f x) \cot ^3\left (\frac {1}{2} (e+f x)\right ) \left (\log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )\right )\right )}{3072 a^2 f (\sec (e+f x)+1)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 210, normalized size = 1.28 \[ \frac {315 \, {\left (c^{5} \cos \left (f x + e\right )^{5} + 2 \, c^{5} \cos \left (f x + e\right )^{4} + c^{5} \cos \left (f x + e\right )^{3}\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) - 315 \, {\left (c^{5} \cos \left (f x + e\right )^{5} + 2 \, c^{5} \cos \left (f x + e\right )^{4} + c^{5} \cos \left (f x + e\right )^{3}\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (494 \, c^{5} \cos \left (f x + e\right )^{4} + 679 \, c^{5} \cos \left (f x + e\right )^{3} + 102 \, c^{5} \cos \left (f x + e\right )^{2} - 17 \, c^{5} \cos \left (f x + e\right ) + 2 \, c^{5}\right )} \sin \left (f x + e\right )}{12 \, {\left (a^{2} f \cos \left (f x + e\right )^{5} + 2 \, a^{2} f \cos \left (f x + e\right )^{4} + a^{2} f \cos \left (f x + e\right )^{3}\right )}} \]
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 = 0.80, size = 234, normalized size = 1.43 \[ -\frac {16 c^{5} \left (\tan ^{3}\left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{3 f \,a^{2}}-\frac {64 c^{5} \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{f \,a^{2}}+\frac {c^{5}}{3 f \,a^{2} \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-1\right )^{3}}+\frac {4 c^{5}}{f \,a^{2} \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-1\right )^{2}}+\frac {55 c^{5}}{2 f \,a^{2} \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-1\right )}-\frac {105 c^{5} \ln \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-1\right )}{2 f \,a^{2}}+\frac {c^{5}}{3 f \,a^{2} \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )+1\right )^{3}}-\frac {4 c^{5}}{f \,a^{2} \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )+1\right )^{2}}+\frac {55 c^{5}}{2 f \,a^{2} \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )+1\right )}+\frac {105 c^{5} \ln \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )+1\right )}{2 f \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 765, normalized size = 4.66 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.73, size = 170, normalized size = 1.04 \[ \frac {55\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5-\frac {280\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{3}+41\,c^5\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,\left (a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6-3\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+3\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-a^2\right )}-\frac {64\,c^5\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{a^2\,f}-\frac {16\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{3\,a^2\,f}+\frac {105\,c^5\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{a^2\,f} \]
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 \[ - \frac {c^{5} \left (\int \left (- \frac {\sec {\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\right )\, dx + \int \frac {5 \sec ^{2}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \left (- \frac {10 \sec ^{3}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\right )\, dx + \int \frac {10 \sec ^{4}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \left (- \frac {5 \sec ^{5}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\right )\, dx + \int \frac {\sec ^{6}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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