Optimal. Leaf size=215 \[ \frac{77 c^6 \tan ^3(e+f x)}{5 a^3 f}+\frac{924 c^6 \tan (e+f x)}{5 a^3 f}-\frac{231 c^6 \tanh ^{-1}(\sin (e+f x))}{2 a^3 f}-\frac{693 c^6 \tan (e+f x) \sec (e+f x)}{10 a^3 f}+\frac{66 \tan (e+f x) \left (c^2-c^2 \sec (e+f x)\right )^3}{5 f \left (a^3 \sec (e+f x)+a^3\right )}-\frac{22 c^2 \tan (e+f x) (c-c \sec (e+f x))^4}{15 a f (a \sec (e+f x)+a)^2}+\frac{2 c \tan (e+f x) (c-c \sec (e+f x))^5}{5 f (a \sec (e+f x)+a)^3} \]
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Rubi [A] time = 0.336145, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 12, 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} \[ \frac{77 c^6 \tan ^3(e+f x)}{5 a^3 f}+\frac{924 c^6 \tan (e+f x)}{5 a^3 f}-\frac{231 c^6 \tanh ^{-1}(\sin (e+f x))}{2 a^3 f}-\frac{693 c^6 \tan (e+f x) \sec (e+f x)}{10 a^3 f}+\frac{66 \tan (e+f x) \left (c^2-c^2 \sec (e+f x)\right )^3}{5 f \left (a^3 \sec (e+f x)+a^3\right )}-\frac{22 c^2 \tan (e+f x) (c-c \sec (e+f x))^4}{15 a f (a \sec (e+f x)+a)^2}+\frac{2 c \tan (e+f x) (c-c \sec (e+f x))^5}{5 f (a \sec (e+f x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 3957
Rule 3791
Rule 3770
Rule 3767
Rule 8
Rule 3768
Rubi steps
\begin{align*} \int \frac{\sec (e+f x) (c-c \sec (e+f x))^6}{(a+a \sec (e+f x))^3} \, dx &=\frac{2 c (c-c \sec (e+f x))^5 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}-\frac{(11 c) \int \frac{\sec (e+f x) (c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^2} \, dx}{5 a}\\ &=-\frac{22 c^2 (c-c \sec (e+f x))^4 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac{2 c (c-c \sec (e+f x))^5 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}+\frac{\left (33 c^2\right ) \int \frac{\sec (e+f x) (c-c \sec (e+f x))^4}{a+a \sec (e+f x)} \, dx}{5 a^2}\\ &=-\frac{22 c^2 (c-c \sec (e+f x))^4 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac{2 c (c-c \sec (e+f x))^5 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}+\frac{66 \left (c^2-c^2 \sec (e+f x)\right )^3 \tan (e+f x)}{5 f \left (a^3+a^3 \sec (e+f x)\right )}-\frac{\left (231 c^3\right ) \int \sec (e+f x) (c-c \sec (e+f x))^3 \, dx}{5 a^3}\\ &=-\frac{22 c^2 (c-c \sec (e+f x))^4 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac{2 c (c-c \sec (e+f x))^5 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}+\frac{66 \left (c^2-c^2 \sec (e+f x)\right )^3 \tan (e+f x)}{5 f \left (a^3+a^3 \sec (e+f x)\right )}-\frac{\left (231 c^3\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}{5 a^3}\\ &=-\frac{22 c^2 (c-c \sec (e+f x))^4 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac{2 c (c-c \sec (e+f x))^5 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}+\frac{66 \left (c^2-c^2 \sec (e+f x)\right )^3 \tan (e+f x)}{5 f \left (a^3+a^3 \sec (e+f x)\right )}-\frac{\left (231 c^6\right ) \int \sec (e+f x) \, dx}{5 a^3}+\frac{\left (231 c^6\right ) \int \sec ^4(e+f x) \, dx}{5 a^3}+\frac{\left (693 c^6\right ) \int \sec ^2(e+f x) \, dx}{5 a^3}-\frac{\left (693 c^6\right ) \int \sec ^3(e+f x) \, dx}{5 a^3}\\ &=-\frac{231 c^6 \tanh ^{-1}(\sin (e+f x))}{5 a^3 f}-\frac{693 c^6 \sec (e+f x) \tan (e+f x)}{10 a^3 f}-\frac{22 c^2 (c-c \sec (e+f x))^4 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac{2 c (c-c \sec (e+f x))^5 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}+\frac{66 \left (c^2-c^2 \sec (e+f x)\right )^3 \tan (e+f x)}{5 f \left (a^3+a^3 \sec (e+f x)\right )}-\frac{\left (693 c^6\right ) \int \sec (e+f x) \, dx}{10 a^3}-\frac{\left (231 c^6\right ) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (e+f x)\right )}{5 a^3 f}-\frac{\left (693 c^6\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (e+f x))}{5 a^3 f}\\ &=-\frac{231 c^6 \tanh ^{-1}(\sin (e+f x))}{2 a^3 f}+\frac{924 c^6 \tan (e+f x)}{5 a^3 f}-\frac{693 c^6 \sec (e+f x) \tan (e+f x)}{10 a^3 f}-\frac{22 c^2 (c-c \sec (e+f x))^4 \tan (e+f x)}{15 a f (a+a \sec (e+f x))^2}+\frac{2 c (c-c \sec (e+f x))^5 \tan (e+f x)}{5 f (a+a \sec (e+f x))^3}+\frac{66 \left (c^2-c^2 \sec (e+f x)\right )^3 \tan (e+f x)}{5 f \left (a^3+a^3 \sec (e+f x)\right )}+\frac{77 c^6 \tan ^3(e+f x)}{5 a^3 f}\\ \end{align*}
Mathematica [A] time = 2.19962, size = 406, normalized size = 1.89 \[ \frac{c^6 \cos \left (\frac{1}{2} (e+f x)\right ) \sec ^3(e+f x) \left (\sec \left (\frac{e}{2}\right ) \sec (e) \left (-130340 \sin \left (e-\frac{f x}{2}\right )+75600 \sin \left (e+\frac{f x}{2}\right )-120176 \sin \left (2 e+\frac{f x}{2}\right )-34230 \sin \left (e+\frac{3 f x}{2}\right )+82278 \sin \left (2 e+\frac{3 f x}{2}\right )-79450 \sin \left (3 e+\frac{3 f x}{2}\right )+91670 \sin \left (e+\frac{5 f x}{2}\right )-14730 \sin \left (2 e+\frac{5 f x}{2}\right )+61920 \sin \left (3 e+\frac{5 f x}{2}\right )-44480 \sin \left (4 e+\frac{5 f x}{2}\right )+53593 \sin \left (2 e+\frac{7 f x}{2}\right )-1735 \sin \left (3 e+\frac{7 f x}{2}\right )+38123 \sin \left (4 e+\frac{7 f x}{2}\right )-17205 \sin \left (5 e+\frac{7 f x}{2}\right )+23735 \sin \left (3 e+\frac{9 f x}{2}\right )+2455 \sin \left (4 e+\frac{9 f x}{2}\right )+17785 \sin \left (5 e+\frac{9 f x}{2}\right )-3495 \sin \left (6 e+\frac{9 f x}{2}\right )+5446 \sin \left (4 e+\frac{11 f x}{2}\right )+1190 \sin \left (5 e+\frac{11 f x}{2}\right )+4256 \sin \left (6 e+\frac{11 f x}{2}\right )-65436 \sin \left (\frac{f x}{2}\right )+127498 \sin \left (\frac{3 f x}{2}\right )\right ) \sec ^3(e+f x)+887040 \cos ^5\left (\frac{1}{2} (e+f x)\right ) \left (\log \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )\right )\right )}{960 a^3 f (\sec (e+f x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.123, size = 256, normalized size = 1.2 \begin{align*}{\frac{16\,{c}^{6}}{5\,f{a}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{5}}+{\frac{64\,{c}^{6}}{3\,f{a}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{3}}+160\,{\frac{{c}^{6}\tan \left ( 1/2\,fx+e/2 \right ) }{f{a}^{3}}}-{\frac{{c}^{6}}{3\,f{a}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) +1 \right ) ^{-3}}+5\,{\frac{{c}^{6}}{f{a}^{3} \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{2}}}-{\frac{89\,{c}^{6}}{2\,f{a}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) +1 \right ) ^{-1}}-{\frac{231\,{c}^{6}}{2\,f{a}^{3}}\ln \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) +1 \right ) }-{\frac{{c}^{6}}{3\,f{a}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) -1 \right ) ^{-3}}-5\,{\frac{{c}^{6}}{f{a}^{3} \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{2}}}-{\frac{89\,{c}^{6}}{2\,f{a}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) -1 \right ) ^{-1}}+{\frac{231\,{c}^{6}}{2\,f{a}^{3}}\ln \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) -1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.1138, size = 1262, normalized size = 5.87 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.516365, size = 663, normalized size = 3.08 \begin{align*} -\frac{3465 \,{\left (c^{6} \cos \left (f x + e\right )^{6} + 3 \, c^{6} \cos \left (f x + e\right )^{5} + 3 \, c^{6} \cos \left (f x + e\right )^{4} + c^{6} \cos \left (f x + e\right )^{3}\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) - 3465 \,{\left (c^{6} \cos \left (f x + e\right )^{6} + 3 \, c^{6} \cos \left (f x + e\right )^{5} + 3 \, c^{6} \cos \left (f x + e\right )^{4} + c^{6} \cos \left (f x + e\right )^{3}\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \,{\left (5446 \, c^{6} \cos \left (f x + e\right )^{5} + 12843 \, c^{6} \cos \left (f x + e\right )^{4} + 8711 \, c^{6} \cos \left (f x + e\right )^{3} + 815 \, c^{6} \cos \left (f x + e\right )^{2} - 105 \, c^{6} \cos \left (f x + e\right ) + 10 \, c^{6}\right )} \sin \left (f x + e\right )}{60 \,{\left (a^{3} f \cos \left (f x + e\right )^{6} + 3 \, a^{3} f \cos \left (f x + e\right )^{5} + 3 \, a^{3} f \cos \left (f x + e\right )^{4} + a^{3} f \cos \left (f x + e\right )^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.33142, size = 250, normalized size = 1.16 \begin{align*} -\frac{\frac{3465 \, c^{6} \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1 \right |}\right )}{a^{3}} - \frac{3465 \, c^{6} \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1 \right |}\right )}{a^{3}} + \frac{10 \,{\left (267 \, c^{6} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} - 472 \, c^{6} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 213 \, c^{6} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 1\right )}^{3} a^{3}} - \frac{32 \,{\left (3 \, a^{12} c^{6} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 20 \, a^{12} c^{6} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 150 \, a^{12} c^{6} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )}}{a^{15}}}{30 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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