Optimal. Leaf size=67 \[ \frac {\cot ^9\left (\frac {1}{2} (e+f x)\right )}{36 c^7 f}-\frac {\cot ^7\left (\frac {1}{2} (e+f x)\right )}{14 c^7 f}+\frac {\cot ^5\left (\frac {1}{2} (e+f x)\right )}{20 c^7 f} \]
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Rubi [A] time = 0.30, antiderivative size = 67, 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} \[ \frac {\cot ^9\left (\frac {1}{2} (e+f x)\right )}{36 c^7 f}-\frac {\cot ^7\left (\frac {1}{2} (e+f x)\right )}{14 c^7 f}+\frac {\cot ^5\left (\frac {1}{2} (e+f x)\right )}{20 c^7 f} \]
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))^7} \, dx &=\frac {2 \operatorname {Subst}\left (\int -\frac {\left (1-x^2\right )^2}{8 c^7 x^{10}} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{f}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^2}{x^{10}} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{4 c^7 f}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {1}{x^{10}}-\frac {2}{x^8}+\frac {1}{x^6}\right ) \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{4 c^7 f}\\ &=\frac {\cot ^5\left (\frac {1}{2} (e+f x)\right )}{20 c^7 f}-\frac {\cot ^7\left (\frac {1}{2} (e+f x)\right )}{14 c^7 f}+\frac {\cot ^9\left (\frac {1}{2} (e+f x)\right )}{36 c^7 f}\\ \end {align*}
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Mathematica [B] time = 0.96, size = 151, normalized size = 2.25 \[ \frac {\csc \left (\frac {e}{2}\right ) \left (-718830 \sin \left (e+\frac {f x}{2}\right )+467208 \sin \left (e+\frac {3 f x}{2}\right )+659400 \sin \left (2 e+\frac {3 f x}{2}\right )-303192 \sin \left (2 e+\frac {5 f x}{2}\right )-179640 \sin \left (3 e+\frac {5 f x}{2}\right )+30753 \sin \left (3 e+\frac {7 f x}{2}\right )+89955 \sin \left (4 e+\frac {7 f x}{2}\right )-13427 \sin \left (4 e+\frac {9 f x}{2}\right )+15 \sin \left (5 e+\frac {9 f x}{2}\right )-971082 \sin \left (\frac {f x}{2}\right )\right ) \csc ^9\left (\frac {1}{2} (e+f x)\right )}{23063040 c^7 f} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.18, size = 121, normalized size = 1.81 \[ \frac {47 \, \cos \left (f x + e\right )^{5} + 127 \, \cos \left (f x + e\right )^{4} + 101 \, \cos \left (f x + e\right )^{3} + 11 \, \cos \left (f x + e\right )^{2} - 8 \, \cos \left (f x + e\right ) + 2}{315 \, {\left (c^{7} f \cos \left (f x + e\right )^{4} - 4 \, c^{7} f \cos \left (f x + e\right )^{3} + 6 \, c^{7} f \cos \left (f x + e\right )^{2} - 4 \, c^{7} f \cos \left (f x + e\right ) + c^{7} f\right )} \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 3.49, size = 50, normalized size = 0.75 \[ \frac {63 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 90 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 35}{1260 \, c^{7} f \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.04, size = 49, normalized size = 0.73 \[ \frac {-\frac {2}{7 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{7}}+\frac {1}{9 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{9}}+\frac {1}{5 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{5}}}{4 f \,c^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 68, normalized size = 1.01 \[ -\frac {{\left (\frac {90 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {63 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - 35\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{9}}{1260 \, c^{7} f \sin \left (f x + e\right )^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.93, size = 47, normalized size = 0.70 \[ \frac {63\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-90\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+35}{1260\,c^7\,f\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9} \]
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 \[ - \frac {\int \frac {\tan ^{4}{\left (e + f x \right )} \sec {\left (e + f x \right )}}{\sec ^{7}{\left (e + f x \right )} - 7 \sec ^{6}{\left (e + f x \right )} + 21 \sec ^{5}{\left (e + f x \right )} - 35 \sec ^{4}{\left (e + f x \right )} + 35 \sec ^{3}{\left (e + f x \right )} - 21 \sec ^{2}{\left (e + f x \right )} + 7 \sec {\left (e + f x \right )} - 1}\, dx}{c^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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