Integrand size = 28, antiderivative size = 89 \[ \int \frac {\sec (e+f x) \tan ^4(e+f x)}{(c-c \sec (e+f x))^8} \, dx=\frac {\cot ^5\left (\frac {1}{2} (e+f x)\right )}{40 c^8 f}-\frac {3 \cot ^7\left (\frac {1}{2} (e+f x)\right )}{56 c^8 f}+\frac {\cot ^9\left (\frac {1}{2} (e+f x)\right )}{24 c^8 f}-\frac {\cot ^{11}\left (\frac {1}{2} (e+f x)\right )}{88 c^8 f} \]
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Time = 0.41 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {12, 276} \[ \int \frac {\sec (e+f x) \tan ^4(e+f x)}{(c-c \sec (e+f x))^8} \, dx=-\frac {\cot ^{11}\left (\frac {1}{2} (e+f x)\right )}{88 c^8 f}+\frac {\cot ^9\left (\frac {1}{2} (e+f x)\right )}{24 c^8 f}-\frac {3 \cot ^7\left (\frac {1}{2} (e+f x)\right )}{56 c^8 f}+\frac {\cot ^5\left (\frac {1}{2} (e+f x)\right )}{40 c^8 f} \]
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Rule 12
Rule 276
Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {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 {\text {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 {\text {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*}
Time = 5.02 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.97 \[ \int \frac {\sec (e+f x) \tan ^4(e+f x)}{(c-c \sec (e+f x))^8} \, dx=-\frac {\csc \left (\frac {e}{2}\right ) \csc ^{11}\left (\frac {1}{2} (e+f x)\right ) \left (425964 \sin \left (\frac {f x}{2}\right )+486024 \sin \left (e+\frac {f x}{2}\right )-351450 \sin \left (e+\frac {3 f x}{2}\right )-299970 \sin \left (2 e+\frac {3 f x}{2}\right )+145695 \sin \left (2 e+\frac {5 f x}{2}\right )+180015 \sin \left (3 e+\frac {5 f x}{2}\right )-63580 \sin \left (3 e+\frac {7 f x}{2}\right )-44990 \sin \left (4 e+\frac {7 f x}{2}\right )+6710 \sin \left (4 e+\frac {9 f x}{2}\right )+15004 \sin \left (5 e+\frac {9 f x}{2}\right )-1975 \sin \left (5 e+\frac {11 f x}{2}\right )+\sin \left (6 e+\frac {11 f x}{2}\right )\right )}{15375360 c^8 f} \]
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Time = 17.28 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.70
method | result | size |
derivativedivides | \(\frac {-\frac {3}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}-\frac {1}{11 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}+\frac {1}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}+\frac {1}{3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}}{8 f \,c^{8}}\) | \(62\) |
default | \(\frac {-\frac {3}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}-\frac {1}{11 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}+\frac {1}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}+\frac {1}{3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}}{8 f \,c^{8}}\) | \(62\) |
risch | \(\frac {2 i \left (1155 \,{\mathrm e}^{10 i \left (f x +e \right )}-3465 \,{\mathrm e}^{9 i \left (f x +e \right )}+13860 \,{\mathrm e}^{8 i \left (f x +e \right )}-23100 \,{\mathrm e}^{7 i \left (f x +e \right )}+37422 \,{\mathrm e}^{6 i \left (f x +e \right )}-32802 \,{\mathrm e}^{5 i \left (f x +e \right )}+27060 \,{\mathrm e}^{4 i \left (f x +e \right )}-11220 \,{\mathrm e}^{3 i \left (f x +e \right )}+4895 \,{\mathrm e}^{2 i \left (f x +e \right )}-517 \,{\mathrm e}^{i \left (f x +e \right )}+152\right )}{1155 f \,c^{8} \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{11}}\) | \(135\) |
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Time = 0.29 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.64 \[ \int \frac {\sec (e+f x) \tan ^4(e+f x)}{(c-c \sec (e+f x))^8} \, dx=\frac {152 \, \cos \left (f x + e\right )^{6} + 395 \, \cos \left (f x + e\right )^{5} + 289 \, \cos \left (f x + e\right )^{4} + 15 \, \cos \left (f x + e\right )^{3} - 19 \, \cos \left (f x + e\right )^{2} + 10 \, \cos \left (f x + e\right ) - 2}{1155 \, {\left (c^{8} f \cos \left (f x + e\right )^{5} - 5 \, c^{8} f \cos \left (f x + e\right )^{4} + 10 \, c^{8} f \cos \left (f x + e\right )^{3} - 10 \, c^{8} f \cos \left (f x + e\right )^{2} + 5 \, c^{8} f \cos \left (f x + e\right ) - c^{8} f\right )} \sin \left (f x + e\right )} \]
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\[ \int \frac {\sec (e+f x) \tan ^4(e+f x)}{(c-c \sec (e+f x))^8} \, dx=\frac {\int \frac {\tan ^{4}{\left (e + f x \right )} \sec {\left (e + f x \right )}}{\sec ^{8}{\left (e + f x \right )} - 8 \sec ^{7}{\left (e + f x \right )} + 28 \sec ^{6}{\left (e + f x \right )} - 56 \sec ^{5}{\left (e + f x \right )} + 70 \sec ^{4}{\left (e + f x \right )} - 56 \sec ^{3}{\left (e + f x \right )} + 28 \sec ^{2}{\left (e + f x \right )} - 8 \sec {\left (e + f x \right )} + 1}\, dx}{c^{8}} \]
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Time = 0.22 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.99 \[ \int \frac {\sec (e+f x) \tan ^4(e+f x)}{(c-c \sec (e+f x))^8} \, dx=\frac {{\left (\frac {385 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {495 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {231 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - 105\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{11}}{9240 \, c^{8} f \sin \left (f x + e\right )^{11}} \]
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Time = 1.17 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.67 \[ \int \frac {\sec (e+f x) \tan ^4(e+f x)}{(c-c \sec (e+f x))^8} \, dx=\frac {231 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 495 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 385 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 105}{9240 \, c^{8} f \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11}} \]
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Time = 14.18 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.67 \[ \int \frac {\sec (e+f x) \tan ^4(e+f x)}{(c-c \sec (e+f x))^8} \, dx=\frac {\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6}{5}-\frac {3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4}{7}+\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2}{3}-\frac {1}{11}}{8\,c^8\,f\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}} \]
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