Integrand size = 28, antiderivative size = 67 \[ \int \frac {\sec (e+f x) \tan ^4(e+f x)}{(c-c \sec (e+f x))^7} \, dx=\frac {\cot ^5\left (\frac {1}{2} (e+f x)\right )}{20 c^7 f}-\frac {\cot ^7\left (\frac {1}{2} (e+f x)\right )}{14 c^7 f}+\frac {\cot ^9\left (\frac {1}{2} (e+f x)\right )}{36 c^7 f} \]
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Time = 0.36 (sec) , antiderivative size = 67, 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} \[ \int \frac {\sec (e+f x) \tan ^4(e+f x)}{(c-c \sec (e+f x))^7} \, dx=\frac {\cot ^9\left (\frac {1}{2} (e+f x)\right )}{36 c^7 f}-\frac {\cot ^7\left (\frac {1}{2} (e+f x)\right )}{14 c^7 f}+\frac {\cot ^5\left (\frac {1}{2} (e+f x)\right )}{20 c^7 f} \]
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Rule 12
Rule 276
Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {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 {\text {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 {\text {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*}
Leaf count is larger than twice the leaf count of optimal. \(151\) vs. \(2(67)=134\).
Time = 4.03 (sec) , antiderivative size = 151, normalized size of antiderivative = 2.25 \[ \int \frac {\sec (e+f x) \tan ^4(e+f x)}{(c-c \sec (e+f x))^7} \, dx=\frac {\csc \left (\frac {e}{2}\right ) \csc ^9\left (\frac {1}{2} (e+f x)\right ) \left (-971082 \sin \left (\frac {f x}{2}\right )-718830 \sin \left (e+\frac {f x}{2}\right )+467208 \sin \left (e+\frac {3 f x}{2}\right )+659400 \sin \left (2 e+\frac {3 f x}{2}\right )-303192 \sin \left (2 e+\frac {5 f x}{2}\right )-179640 \sin \left (3 e+\frac {5 f x}{2}\right )+30753 \sin \left (3 e+\frac {7 f x}{2}\right )+89955 \sin \left (4 e+\frac {7 f x}{2}\right )-13427 \sin \left (4 e+\frac {9 f x}{2}\right )+15 \sin \left (5 e+\frac {9 f x}{2}\right )\right )}{23063040 c^7 f} \]
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Time = 11.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.73
method | result | size |
derivativedivides | \(\frac {\frac {1}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}-\frac {2}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}+\frac {1}{9 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}}{4 f \,c^{7}}\) | \(49\) |
default | \(\frac {\frac {1}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}-\frac {2}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}+\frac {1}{9 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}}{4 f \,c^{7}}\) | \(49\) |
risch | \(\frac {2 i \left (315 \,{\mathrm e}^{8 i \left (f x +e \right )}-630 \,{\mathrm e}^{7 i \left (f x +e \right )}+2310 \,{\mathrm e}^{6 i \left (f x +e \right )}-2520 \,{\mathrm e}^{5 i \left (f x +e \right )}+3402 \,{\mathrm e}^{4 i \left (f x +e \right )}-1638 \,{\mathrm e}^{3 i \left (f x +e \right )}+1062 \,{\mathrm e}^{2 i \left (f x +e \right )}-108 \,{\mathrm e}^{i \left (f x +e \right )}+47\right )}{315 f \,c^{7} \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{9}}\) | \(113\) |
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Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (58) = 116\).
Time = 0.31 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.81 \[ \int \frac {\sec (e+f x) \tan ^4(e+f x)}{(c-c \sec (e+f x))^7} \, dx=\frac {47 \, \cos \left (f x + e\right )^{5} + 127 \, \cos \left (f x + e\right )^{4} + 101 \, \cos \left (f x + e\right )^{3} + 11 \, \cos \left (f x + e\right )^{2} - 8 \, \cos \left (f x + e\right ) + 2}{315 \, {\left (c^{7} f \cos \left (f x + e\right )^{4} - 4 \, c^{7} f \cos \left (f x + e\right )^{3} + 6 \, c^{7} f \cos \left (f x + e\right )^{2} - 4 \, c^{7} f \cos \left (f x + e\right ) + c^{7} f\right )} \sin \left (f x + e\right )} \]
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\[ \int \frac {\sec (e+f x) \tan ^4(e+f x)}{(c-c \sec (e+f x))^7} \, dx=- \frac {\int \frac {\tan ^{4}{\left (e + f x \right )} \sec {\left (e + f x \right )}}{\sec ^{7}{\left (e + f x \right )} - 7 \sec ^{6}{\left (e + f x \right )} + 21 \sec ^{5}{\left (e + f x \right )} - 35 \sec ^{4}{\left (e + f x \right )} + 35 \sec ^{3}{\left (e + f x \right )} - 21 \sec ^{2}{\left (e + f x \right )} + 7 \sec {\left (e + f x \right )} - 1}\, dx}{c^{7}} \]
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Time = 0.22 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.01 \[ \int \frac {\sec (e+f x) \tan ^4(e+f x)}{(c-c \sec (e+f x))^7} \, dx=-\frac {{\left (\frac {90 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {63 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - 35\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{9}}{1260 \, c^{7} f \sin \left (f x + e\right )^{9}} \]
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Time = 1.10 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.70 \[ \int \frac {\sec (e+f x) \tan ^4(e+f x)}{(c-c \sec (e+f x))^7} \, dx=\frac {63 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 90 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 35}{1260 \, c^{7} f \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9}} \]
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Time = 13.52 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.70 \[ \int \frac {\sec (e+f x) \tan ^4(e+f x)}{(c-c \sec (e+f x))^7} \, dx=\frac {63\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-90\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+35}{1260\,c^7\,f\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9} \]
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