Optimal. Leaf size=89 \[ -\frac {\cot ^{11}\left (\frac {1}{2} (e+f x)\right )}{88 c^8 f}+\frac {\cot ^9\left (\frac {1}{2} (e+f x)\right )}{24 c^8 f}-\frac {3 \cot ^7\left (\frac {1}{2} (e+f x)\right )}{56 c^8 f}+\frac {\cot ^5\left (\frac {1}{2} (e+f x)\right )}{40 c^8 f} \]
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Rubi [A] time = 0.34, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {12, 270} \[ -\frac {\cot ^{11}\left (\frac {1}{2} (e+f x)\right )}{88 c^8 f}+\frac {\cot ^9\left (\frac {1}{2} (e+f x)\right )}{24 c^8 f}-\frac {3 \cot ^7\left (\frac {1}{2} (e+f x)\right )}{56 c^8 f}+\frac {\cot ^5\left (\frac {1}{2} (e+f x)\right )}{40 c^8 f} \]
Antiderivative was successfully verified.
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Rule 12
Rule 270
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) \tan ^4(e+f x)}{(c-c \sec (e+f x))^8} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^3}{16 c^8 x^{12}} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^3}{x^{12}} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{8 c^8 f}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{x^{12}}-\frac {3}{x^{10}}+\frac {3}{x^8}-\frac {1}{x^6}\right ) \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{8 c^8 f}\\ &=\frac {\cot ^5\left (\frac {1}{2} (e+f x)\right )}{40 c^8 f}-\frac {3 \cot ^7\left (\frac {1}{2} (e+f x)\right )}{56 c^8 f}+\frac {\cot ^9\left (\frac {1}{2} (e+f x)\right )}{24 c^8 f}-\frac {\cot ^{11}\left (\frac {1}{2} (e+f x)\right )}{88 c^8 f}\\ \end {align*}
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Mathematica [A] time = 1.12, size = 175, normalized size = 1.97 \[ -\frac {\csc \left (\frac {e}{2}\right ) \left (486024 \sin \left (e+\frac {f x}{2}\right )-351450 \sin \left (e+\frac {3 f x}{2}\right )-299970 \sin \left (2 e+\frac {3 f x}{2}\right )+145695 \sin \left (2 e+\frac {5 f x}{2}\right )+180015 \sin \left (3 e+\frac {5 f x}{2}\right )-63580 \sin \left (3 e+\frac {7 f x}{2}\right )-44990 \sin \left (4 e+\frac {7 f x}{2}\right )+6710 \sin \left (4 e+\frac {9 f x}{2}\right )+15004 \sin \left (5 e+\frac {9 f x}{2}\right )-1975 \sin \left (5 e+\frac {11 f x}{2}\right )+\sin \left (6 e+\frac {11 f x}{2}\right )+425964 \sin \left (\frac {f x}{2}\right )\right ) \csc ^{11}\left (\frac {1}{2} (e+f x)\right )}{15375360 c^8 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 146, normalized size = 1.64 \[ \frac {152 \, \cos \left (f x + e\right )^{6} + 395 \, \cos \left (f x + e\right )^{5} + 289 \, \cos \left (f x + e\right )^{4} + 15 \, \cos \left (f x + e\right )^{3} - 19 \, \cos \left (f x + e\right )^{2} + 10 \, \cos \left (f x + e\right ) - 2}{1155 \, {\left (c^{8} f \cos \left (f x + e\right )^{5} - 5 \, c^{8} f \cos \left (f x + e\right )^{4} + 10 \, c^{8} f \cos \left (f x + e\right )^{3} - 10 \, c^{8} f \cos \left (f x + e\right )^{2} + 5 \, c^{8} f \cos \left (f x + e\right ) - c^{8} f\right )} \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 10.61, size = 64, normalized size = 0.72 \[ \frac {231 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 495 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 385 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 105}{9240 \, c^{8} f \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.20, size = 62, normalized size = 0.70 \[ \frac {-\frac {1}{11 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{11}}-\frac {3}{7 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{7}}+\frac {1}{3 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{9}}+\frac {1}{5 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{5}}}{8 f \,c^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 88, normalized size = 0.99 \[ \frac {{\left (\frac {385 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {495 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {231 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - 105\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{11}}{9240 \, c^{8} f \sin \left (f x + e\right )^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.43, size = 60, normalized size = 0.67 \[ \frac {\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6}{5}-\frac {3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4}{7}+\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2}{3}-\frac {1}{11}}{8\,c^8\,f\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}} \]
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 \[ \frac {\int \frac {\tan ^{4}{\left (e + f x \right )} \sec {\left (e + f x \right )}}{\sec ^{8}{\left (e + f x \right )} - 8 \sec ^{7}{\left (e + f x \right )} + 28 \sec ^{6}{\left (e + f x \right )} - 56 \sec ^{5}{\left (e + f x \right )} + 70 \sec ^{4}{\left (e + f x \right )} - 56 \sec ^{3}{\left (e + f x \right )} + 28 \sec ^{2}{\left (e + f x \right )} - 8 \sec {\left (e + f x \right )} + 1}\, dx}{c^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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