Integrand size = 26, antiderivative size = 136 \[ \int \frac {(c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^2} \, dx=\frac {c^5 x}{a^2}-\frac {47 c^5 \text {arctanh}(\sin (e+f x))}{2 a^2 f}+\frac {13 c^5 \tan (e+f x)}{2 a^2 f}+\frac {112 c^5 \tan (e+f x)}{3 a^2 f (1+\sec (e+f x))}-\frac {32 c^5 \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}+\frac {\left (c^5-c^5 \sec (e+f x)\right ) \tan (e+f x)}{2 a^2 f} \]
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Time = 0.48 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.12, number of steps used = 26, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {3989, 3971, 3554, 8, 2686, 2687, 30, 3852, 2701, 308, 213, 2700, 276, 294} \[ \int \frac {(c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^2} \, dx=-\frac {47 c^5 \text {arctanh}(\sin (e+f x))}{2 a^2 f}+\frac {7 c^5 \tan (e+f x)}{a^2 f}-\frac {64 c^5 \cot ^3(e+f x)}{3 a^2 f}-\frac {48 c^5 \cot (e+f x)}{a^2 f}+\frac {131 c^5 \csc ^3(e+f x)}{6 a^2 f}+\frac {33 c^5 \csc (e+f x)}{2 a^2 f}-\frac {c^5 \csc ^3(e+f x) \sec ^2(e+f x)}{2 a^2 f}+\frac {c^5 x}{a^2} \]
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Rule 8
Rule 30
Rule 213
Rule 276
Rule 294
Rule 308
Rule 2686
Rule 2687
Rule 2700
Rule 2701
Rule 3554
Rule 3852
Rule 3971
Rule 3989
Rubi steps \begin{align*} \text {integral}& = \frac {\int \cot ^4(e+f x) (c-c \sec (e+f x))^7 \, dx}{a^2 c^2} \\ & = \frac {\int \left (c^7 \cot ^4(e+f x)-7 c^7 \cot ^3(e+f x) \csc (e+f x)+21 c^7 \cot ^2(e+f x) \csc ^2(e+f x)-35 c^7 \cot (e+f x) \csc ^3(e+f x)+35 c^7 \csc ^4(e+f x)-21 c^7 \csc ^4(e+f x) \sec (e+f x)+7 c^7 \csc ^4(e+f x) \sec ^2(e+f x)-c^7 \csc ^4(e+f x) \sec ^3(e+f x)\right ) \, dx}{a^2 c^2} \\ & = \frac {c^5 \int \cot ^4(e+f x) \, dx}{a^2}-\frac {c^5 \int \csc ^4(e+f x) \sec ^3(e+f x) \, dx}{a^2}-\frac {\left (7 c^5\right ) \int \cot ^3(e+f x) \csc (e+f x) \, dx}{a^2}+\frac {\left (7 c^5\right ) \int \csc ^4(e+f x) \sec ^2(e+f x) \, dx}{a^2}+\frac {\left (21 c^5\right ) \int \cot ^2(e+f x) \csc ^2(e+f x) \, dx}{a^2}-\frac {\left (21 c^5\right ) \int \csc ^4(e+f x) \sec (e+f x) \, dx}{a^2}-\frac {\left (35 c^5\right ) \int \cot (e+f x) \csc ^3(e+f x) \, dx}{a^2}+\frac {\left (35 c^5\right ) \int \csc ^4(e+f x) \, dx}{a^2} \\ & = -\frac {c^5 \cot ^3(e+f x)}{3 a^2 f}-\frac {c^5 \int \cot ^2(e+f x) \, dx}{a^2}+\frac {c^5 \text {Subst}\left (\int \frac {x^6}{\left (-1+x^2\right )^2} \, dx,x,\csc (e+f x)\right )}{a^2 f}+\frac {\left (7 c^5\right ) \text {Subst}\left (\int \left (-1+x^2\right ) \, dx,x,\csc (e+f x)\right )}{a^2 f}+\frac {\left (7 c^5\right ) \text {Subst}\left (\int \frac {\left (1+x^2\right )^2}{x^4} \, dx,x,\tan (e+f x)\right )}{a^2 f}+\frac {\left (21 c^5\right ) \text {Subst}\left (\int x^2 \, dx,x,-\cot (e+f x)\right )}{a^2 f}+\frac {\left (21 c^5\right ) \text {Subst}\left (\int \frac {x^4}{-1+x^2} \, dx,x,\csc (e+f x)\right )}{a^2 f}+\frac {\left (35 c^5\right ) \text {Subst}\left (\int x^2 \, dx,x,\csc (e+f x)\right )}{a^2 f}-\frac {\left (35 c^5\right ) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (e+f x)\right )}{a^2 f} \\ & = -\frac {34 c^5 \cot (e+f x)}{a^2 f}-\frac {19 c^5 \cot ^3(e+f x)}{a^2 f}-\frac {7 c^5 \csc (e+f x)}{a^2 f}+\frac {14 c^5 \csc ^3(e+f x)}{a^2 f}-\frac {c^5 \csc ^3(e+f x) \sec ^2(e+f x)}{2 a^2 f}+\frac {c^5 \int 1 \, dx}{a^2}+\frac {\left (5 c^5\right ) \text {Subst}\left (\int \frac {x^4}{-1+x^2} \, dx,x,\csc (e+f x)\right )}{2 a^2 f}+\frac {\left (7 c^5\right ) \text {Subst}\left (\int \left (1+\frac {1}{x^4}+\frac {2}{x^2}\right ) \, dx,x,\tan (e+f x)\right )}{a^2 f}+\frac {\left (21 c^5\right ) \text {Subst}\left (\int \left (1+x^2+\frac {1}{-1+x^2}\right ) \, dx,x,\csc (e+f x)\right )}{a^2 f} \\ & = \frac {c^5 x}{a^2}-\frac {48 c^5 \cot (e+f x)}{a^2 f}-\frac {64 c^5 \cot ^3(e+f x)}{3 a^2 f}+\frac {14 c^5 \csc (e+f x)}{a^2 f}+\frac {21 c^5 \csc ^3(e+f x)}{a^2 f}-\frac {c^5 \csc ^3(e+f x) \sec ^2(e+f x)}{2 a^2 f}+\frac {7 c^5 \tan (e+f x)}{a^2 f}+\frac {\left (5 c^5\right ) \text {Subst}\left (\int \left (1+x^2+\frac {1}{-1+x^2}\right ) \, dx,x,\csc (e+f x)\right )}{2 a^2 f}+\frac {\left (21 c^5\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (e+f x)\right )}{a^2 f} \\ & = \frac {c^5 x}{a^2}-\frac {21 c^5 \text {arctanh}(\sin (e+f x))}{a^2 f}-\frac {48 c^5 \cot (e+f x)}{a^2 f}-\frac {64 c^5 \cot ^3(e+f x)}{3 a^2 f}+\frac {33 c^5 \csc (e+f x)}{2 a^2 f}+\frac {131 c^5 \csc ^3(e+f x)}{6 a^2 f}-\frac {c^5 \csc ^3(e+f x) \sec ^2(e+f x)}{2 a^2 f}+\frac {7 c^5 \tan (e+f x)}{a^2 f}+\frac {\left (5 c^5\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (e+f x)\right )}{2 a^2 f} \\ & = \frac {c^5 x}{a^2}-\frac {47 c^5 \text {arctanh}(\sin (e+f x))}{2 a^2 f}-\frac {48 c^5 \cot (e+f x)}{a^2 f}-\frac {64 c^5 \cot ^3(e+f x)}{3 a^2 f}+\frac {33 c^5 \csc (e+f x)}{2 a^2 f}+\frac {131 c^5 \csc ^3(e+f x)}{6 a^2 f}-\frac {c^5 \csc ^3(e+f x) \sec ^2(e+f x)}{2 a^2 f}+\frac {7 c^5 \tan (e+f x)}{a^2 f} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 3.18 (sec) , antiderivative size = 351, normalized size of antiderivative = 2.58 \[ \int \frac {(c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^2} \, dx=\frac {c^{9/2} \tan (e+f x) \left (8 \sqrt {a} \sqrt {c}+16 \sqrt {2} \sqrt {a} \sqrt {c} \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},-\frac {3}{2},-\frac {1}{2},\frac {1}{2} (1+\sec (e+f x))\right ) \sqrt {1-\sec (e+f x)}+8 \sqrt {2} \sqrt {a} \sqrt {c} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},-\frac {3}{2},-\frac {1}{2},\frac {1}{2} (1+\sec (e+f x))\right ) \sqrt {1-\sec (e+f x)}+4 \sqrt {2} \sqrt {a} \sqrt {c} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{2},-\frac {1}{2},\frac {1}{2} (1+\sec (e+f x))\right ) \sqrt {1-\sec (e+f x)}-4 \sqrt {a} \sqrt {c} \sec (e+f x)-4 \sqrt {a} \sqrt {c} \sec ^2(e+f x)-3 \text {arctanh}\left (\frac {\sqrt {-a c \tan ^2(e+f x)}}{\sqrt {a} \sqrt {c}}\right ) \sqrt {-a c \tan ^2(e+f x)}-3 \text {arctanh}\left (\frac {\sqrt {-a c \tan ^2(e+f x)}}{\sqrt {a} \sqrt {c}}\right ) \sec (e+f x) \sqrt {-a c \tan ^2(e+f x)}\right )}{3 a^{5/2} f (-1+\sec (e+f x)) (1+\sec (e+f x))^2} \]
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Time = 0.76 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.01
method | result | size |
derivativedivides | \(\frac {16 c^{5} \left (\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}+2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+\frac {\arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8}+\frac {1}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {15}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {47 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{32}-\frac {1}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {15}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}+\frac {47 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{32}\right )}{f \,a^{2}}\) | \(137\) |
default | \(\frac {16 c^{5} \left (\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}+2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+\frac {\arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8}+\frac {1}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {15}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {47 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{32}-\frac {1}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {15}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}+\frac {47 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{32}\right )}{f \,a^{2}}\) | \(137\) |
parallelrisch | \(\frac {125 \left (\frac {94 \left (1+\cos \left (2 f x +2 e \right )\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{125}+\frac {94 \left (-1-\cos \left (2 f x +2 e \right )\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{125}+\tan \left (\frac {f x}{2}+\frac {e}{2}\right ) \left (\cos \left (f x +e \right )+\frac {61 \cos \left (2 f x +2 e \right )}{75}+\frac {101 \cos \left (3 f x +3 e \right )}{375}+\frac {299}{375}\right ) \sec \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+\frac {4 f x \left (1+\cos \left (2 f x +2 e \right )\right )}{125}\right ) c^{5}}{4 f \,a^{2} \left (1+\cos \left (2 f x +2 e \right )\right )}\) | \(144\) |
risch | \(\frac {c^{5} x}{a^{2}}+\frac {i c^{5} \left (99 \,{\mathrm e}^{6 i \left (f x +e \right )}+435 \,{\mathrm e}^{5 i \left (f x +e \right )}+484 \,{\mathrm e}^{4 i \left (f x +e \right )}+930 \,{\mathrm e}^{3 i \left (f x +e \right )}+575 \,{\mathrm e}^{2 i \left (f x +e \right )}+507 \,{\mathrm e}^{i \left (f x +e \right )}+202\right )}{3 f \,a^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{3} \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )^{2}}-\frac {47 c^{5} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{2 a^{2} f}+\frac {47 c^{5} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{2 a^{2} f}\) | \(164\) |
norman | \(\frac {\frac {c^{5} x}{a}+\frac {c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{a}-\frac {4 c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{a}+\frac {6 c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{a}-\frac {4 c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{a}+\frac {45 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a f}-\frac {491 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3 a f}+\frac {641 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{3 a f}-\frac {111 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{a f}+\frac {32 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{3 a f}+\frac {16 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{3 a f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{4} a}+\frac {47 c^{5} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{2 a^{2} f}-\frac {47 c^{5} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{2 a^{2} f}\) | \(285\) |
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Time = 0.26 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.78 \[ \int \frac {(c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^2} \, dx=\frac {12 \, c^{5} f x \cos \left (f x + e\right )^{4} + 24 \, c^{5} f x \cos \left (f x + e\right )^{3} + 12 \, c^{5} f x \cos \left (f x + e\right )^{2} - 141 \, {\left (c^{5} \cos \left (f x + e\right )^{4} + 2 \, c^{5} \cos \left (f x + e\right )^{3} + c^{5} \cos \left (f x + e\right )^{2}\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) + 141 \, {\left (c^{5} \cos \left (f x + e\right )^{4} + 2 \, c^{5} \cos \left (f x + e\right )^{3} + c^{5} \cos \left (f x + e\right )^{2}\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left (202 \, c^{5} \cos \left (f x + e\right )^{3} + 305 \, c^{5} \cos \left (f x + e\right )^{2} + 36 \, c^{5} \cos \left (f x + e\right ) - 3 \, c^{5}\right )} \sin \left (f x + e\right )}{12 \, {\left (a^{2} f \cos \left (f x + e\right )^{4} + 2 \, a^{2} f \cos \left (f x + e\right )^{3} + a^{2} f \cos \left (f x + e\right )^{2}\right )}} \]
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\[ \int \frac {(c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^2} \, dx=- \frac {c^{5} \left (\int \frac {5 \sec {\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \left (- \frac {10 \sec ^{2}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\right )\, dx + \int \frac {10 \sec ^{3}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \left (- \frac {5 \sec ^{4}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\right )\, dx + \int \frac {\sec ^{5}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \left (- \frac {1}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\right )\, dx\right )}{a^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 603 vs. \(2 (127) = 254\).
Time = 0.35 (sec) , antiderivative size = 603, normalized size of antiderivative = 4.43 \[ \int \frac {(c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^2} \, dx=\frac {c^{5} {\left (\frac {6 \, {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {5 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{2} - \frac {2 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}} + \frac {\frac {21 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}{a^{2}} - \frac {21 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a^{2}} + \frac {21 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a^{2}}\right )} + 5 \, c^{5} {\left (\frac {\frac {15 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}{a^{2}} - \frac {12 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a^{2}} + \frac {12 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a^{2}} + \frac {12 \, \sin \left (f x + e\right )}{{\left (a^{2} - \frac {a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (f x + e\right ) + 1\right )}}\right )} + 10 \, c^{5} {\left (\frac {\frac {9 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}{a^{2}} - \frac {6 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a^{2}} + \frac {6 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a^{2}}\right )} - c^{5} {\left (\frac {\frac {9 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}{a^{2}} - \frac {12 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{2}}\right )} + \frac {10 \, c^{5} {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{2}} - \frac {5 \, c^{5} {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{2}}}{6 \, f} \]
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Time = 0.38 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.12 \[ \int \frac {(c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^2} \, dx=\frac {\frac {6 \, {\left (f x + e\right )} c^{5}}{a^{2}} - \frac {141 \, c^{5} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{a^{2}} + \frac {141 \, c^{5} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{a^{2}} - \frac {6 \, {\left (15 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 13 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{2} a^{2}} + \frac {32 \, {\left (a^{4} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 6 \, a^{4} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{6}}}{6 \, f} \]
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Time = 13.64 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.07 \[ \int \frac {(c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^2} \, dx=\frac {c^5\,x}{a^2}-\frac {15\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3-13\,c^5\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,\left (a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-2\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+a^2\right )}+\frac {32\,c^5\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{a^2\,f}+\frac {16\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{3\,a^2\,f}-\frac {47\,c^5\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{a^2\,f} \]
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