Integrand size = 26, antiderivative size = 162 \[ \int \frac {(c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^3} \, dx=\frac {c^5 x}{a^3}+\frac {8 c^5 \text {arctanh}(\sin (e+f x))}{a^3 f}+\frac {32 c^5 \cot (e+f x)}{a^3 f}+\frac {128 c^5 \cot ^3(e+f x)}{3 a^3 f}+\frac {128 c^5 \cot ^5(e+f x)}{5 a^3 f}-\frac {16 c^5 \csc (e+f x)}{a^3 f}+\frac {64 c^5 \csc ^3(e+f x)}{3 a^3 f}-\frac {128 c^5 \csc ^5(e+f x)}{5 a^3 f}-\frac {c^5 \tan (e+f x)}{a^3 f} \]
[Out]
Time = 0.59 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.577, Rules used = {3989, 3971, 3554, 8, 2686, 200, 2687, 30, 14, 3852, 2701, 308, 213, 2700, 276} \[ \int \frac {(c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^3} \, dx=\frac {8 c^5 \text {arctanh}(\sin (e+f x))}{a^3 f}-\frac {c^5 \tan (e+f x)}{a^3 f}+\frac {128 c^5 \cot ^5(e+f x)}{5 a^3 f}+\frac {128 c^5 \cot ^3(e+f x)}{3 a^3 f}+\frac {32 c^5 \cot (e+f x)}{a^3 f}-\frac {128 c^5 \csc ^5(e+f x)}{5 a^3 f}+\frac {64 c^5 \csc ^3(e+f x)}{3 a^3 f}-\frac {16 c^5 \csc (e+f x)}{a^3 f}+\frac {c^5 x}{a^3} \]
[In]
[Out]
Rule 8
Rule 14
Rule 30
Rule 200
Rule 213
Rule 276
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 ^6(e+f x) (c-c \sec (e+f x))^8 \, dx}{a^3 c^3} \\ & = -\frac {\int \left (c^8 \cot ^6(e+f x)-8 c^8 \cot ^5(e+f x) \csc (e+f x)+28 c^8 \cot ^4(e+f x) \csc ^2(e+f x)-56 c^8 \cot ^3(e+f x) \csc ^3(e+f x)+70 c^8 \cot ^2(e+f x) \csc ^4(e+f x)-56 c^8 \cot (e+f x) \csc ^5(e+f x)+28 c^8 \csc ^6(e+f x)-8 c^8 \csc ^6(e+f x) \sec (e+f x)+c^8 \csc ^6(e+f x) \sec ^2(e+f x)\right ) \, dx}{a^3 c^3} \\ & = -\frac {c^5 \int \cot ^6(e+f x) \, dx}{a^3}-\frac {c^5 \int \csc ^6(e+f x) \sec ^2(e+f x) \, dx}{a^3}+\frac {\left (8 c^5\right ) \int \cot ^5(e+f x) \csc (e+f x) \, dx}{a^3}+\frac {\left (8 c^5\right ) \int \csc ^6(e+f x) \sec (e+f x) \, dx}{a^3}-\frac {\left (28 c^5\right ) \int \cot ^4(e+f x) \csc ^2(e+f x) \, dx}{a^3}-\frac {\left (28 c^5\right ) \int \csc ^6(e+f x) \, dx}{a^3}+\frac {\left (56 c^5\right ) \int \cot ^3(e+f x) \csc ^3(e+f x) \, dx}{a^3}+\frac {\left (56 c^5\right ) \int \cot (e+f x) \csc ^5(e+f x) \, dx}{a^3}-\frac {\left (70 c^5\right ) \int \cot ^2(e+f x) \csc ^4(e+f x) \, dx}{a^3} \\ & = \frac {c^5 \cot ^5(e+f x)}{5 a^3 f}+\frac {c^5 \int \cot ^4(e+f x) \, dx}{a^3}-\frac {c^5 \text {Subst}\left (\int \frac {\left (1+x^2\right )^3}{x^6} \, dx,x,\tan (e+f x)\right )}{a^3 f}-\frac {\left (8 c^5\right ) \text {Subst}\left (\int \frac {x^6}{-1+x^2} \, dx,x,\csc (e+f x)\right )}{a^3 f}-\frac {\left (8 c^5\right ) \text {Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,\csc (e+f x)\right )}{a^3 f}-\frac {\left (28 c^5\right ) \text {Subst}\left (\int x^4 \, dx,x,-\cot (e+f x)\right )}{a^3 f}+\frac {\left (28 c^5\right ) \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (e+f x)\right )}{a^3 f}-\frac {\left (56 c^5\right ) \text {Subst}\left (\int x^4 \, dx,x,\csc (e+f x)\right )}{a^3 f}-\frac {\left (56 c^5\right ) \text {Subst}\left (\int x^2 \left (-1+x^2\right ) \, dx,x,\csc (e+f x)\right )}{a^3 f}-\frac {\left (70 c^5\right ) \text {Subst}\left (\int x^2 \left (1+x^2\right ) \, dx,x,-\cot (e+f x)\right )}{a^3 f} \\ & = \frac {28 c^5 \cot (e+f x)}{a^3 f}+\frac {55 c^5 \cot ^3(e+f x)}{3 a^3 f}+\frac {57 c^5 \cot ^5(e+f x)}{5 a^3 f}-\frac {56 c^5 \csc ^5(e+f x)}{5 a^3 f}-\frac {c^5 \int \cot ^2(e+f x) \, dx}{a^3}-\frac {c^5 \text {Subst}\left (\int \left (1+\frac {1}{x^6}+\frac {3}{x^4}+\frac {3}{x^2}\right ) \, dx,x,\tan (e+f x)\right )}{a^3 f}-\frac {\left (8 c^5\right ) \text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\csc (e+f x)\right )}{a^3 f}-\frac {\left (8 c^5\right ) \text {Subst}\left (\int \left (1+x^2+x^4+\frac {1}{-1+x^2}\right ) \, dx,x,\csc (e+f x)\right )}{a^3 f}-\frac {\left (56 c^5\right ) \text {Subst}\left (\int \left (-x^2+x^4\right ) \, dx,x,\csc (e+f x)\right )}{a^3 f}-\frac {\left (70 c^5\right ) \text {Subst}\left (\int \left (x^2+x^4\right ) \, dx,x,-\cot (e+f x)\right )}{a^3 f} \\ & = \frac {32 c^5 \cot (e+f x)}{a^3 f}+\frac {128 c^5 \cot ^3(e+f x)}{3 a^3 f}+\frac {128 c^5 \cot ^5(e+f x)}{5 a^3 f}-\frac {16 c^5 \csc (e+f x)}{a^3 f}+\frac {64 c^5 \csc ^3(e+f x)}{3 a^3 f}-\frac {128 c^5 \csc ^5(e+f x)}{5 a^3 f}-\frac {c^5 \tan (e+f x)}{a^3 f}+\frac {c^5 \int 1 \, dx}{a^3}-\frac {\left (8 c^5\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (e+f x)\right )}{a^3 f} \\ & = \frac {c^5 x}{a^3}+\frac {8 c^5 \text {arctanh}(\sin (e+f x))}{a^3 f}+\frac {32 c^5 \cot (e+f x)}{a^3 f}+\frac {128 c^5 \cot ^3(e+f x)}{3 a^3 f}+\frac {128 c^5 \cot ^5(e+f x)}{5 a^3 f}-\frac {16 c^5 \csc (e+f x)}{a^3 f}+\frac {64 c^5 \csc ^3(e+f x)}{3 a^3 f}-\frac {128 c^5 \csc ^5(e+f x)}{5 a^3 f}-\frac {c^5 \tan (e+f x)}{a^3 f} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 5.29 (sec) , antiderivative size = 372, normalized size of antiderivative = 2.30 \[ \int \frac {(c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^3} \, dx=-\frac {c^{9/2} \tan (e+f x) \left (-46 \sqrt {a} \sqrt {c}-48 \sqrt {2} \sqrt {a} \sqrt {c} \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},-\frac {5}{2},-\frac {3}{2},\frac {1}{2} (1+\sec (e+f x))\right ) \sqrt {1-\sec (e+f x)}-24 \sqrt {2} \sqrt {a} \sqrt {c} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},-\frac {5}{2},-\frac {3}{2},\frac {1}{2} (1+\sec (e+f x))\right ) \sqrt {1-\sec (e+f x)}-2 \sqrt {a} \sqrt {c} \sec (e+f x)+22 \sqrt {a} \sqrt {c} \sec ^2(e+f x)+26 \sqrt {a} \sqrt {c} \sec ^3(e+f x)+15 \text {arctanh}\left (\frac {\sqrt {-a c \tan ^2(e+f x)}}{\sqrt {a} \sqrt {c}}\right ) \sqrt {-a c \tan ^2(e+f x)}+30 \text {arctanh}\left (\frac {\sqrt {-a c \tan ^2(e+f x)}}{\sqrt {a} \sqrt {c}}\right ) \sec (e+f x) \sqrt {-a c \tan ^2(e+f x)}+15 \text {arctanh}\left (\frac {\sqrt {-a c \tan ^2(e+f x)}}{\sqrt {a} \sqrt {c}}\right ) \sec ^2(e+f x) \sqrt {-a c \tan ^2(e+f x)}\right )}{15 a^{7/2} f (-1+\sec (e+f x)) (1+\sec (e+f x))^3} \]
[In]
[Out]
Time = 0.77 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.73
method | result | size |
derivativedivides | \(\frac {8 c^{5} \left (-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{5}-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}-2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+\frac {\arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4}+\ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )+\frac {1}{8 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+8}+\frac {1}{8 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-8}-\ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )\right )}{f \,a^{3}}\) | \(118\) |
default | \(\frac {8 c^{5} \left (-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{5}-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}-2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+\frac {\arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4}+\ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )+\frac {1}{8 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+8}+\frac {1}{8 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-8}-\ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )\right )}{f \,a^{3}}\) | \(118\) |
parallelrisch | \(-\frac {2113 \left (\frac {1920 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right ) \cos \left (f x +e \right )}{2113}-\frac {1920 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right ) \cos \left (f x +e \right )}{2113}+\tan \left (\frac {f x}{2}+\frac {e}{2}\right ) \left (\cos \left (f x +e \right )+\frac {954 \cos \left (2 f x +2 e \right )}{2113}+\frac {239 \cos \left (3 f x +3 e \right )}{2113}+\frac {1014}{2113}\right ) \sec \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}-\frac {240 f x \cos \left (f x +e \right )}{2113}\right ) c^{5}}{240 \cos \left (f x +e \right ) a^{3} f}\) | \(122\) |
risch | \(\frac {c^{5} x}{a^{3}}-\frac {2 i c^{5} \left (240 \,{\mathrm e}^{6 i \left (f x +e \right )}+735 \,{\mathrm e}^{5 i \left (f x +e \right )}+1835 \,{\mathrm e}^{4 i \left (f x +e \right )}+1750 \,{\mathrm e}^{3 i \left (f x +e \right )}+1894 \,{\mathrm e}^{2 i \left (f x +e \right )}+955 \,{\mathrm e}^{i \left (f x +e \right )}+239\right )}{15 f \,a^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{5} \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )}+\frac {8 c^{5} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{a^{3} f}-\frac {8 c^{5} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{a^{3} f}\) | \(164\) |
norman | \(\frac {\frac {c^{5} x}{a}+\frac {c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{a}-\frac {4 c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{a}+\frac {6 c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{a}-\frac {4 c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{a}-\frac {18 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a f}+\frac {202 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3 a f}-\frac {1394 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{15 a f}+\frac {282 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{5 a f}-\frac {224 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{15 a f}+\frac {56 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{15 a f}-\frac {8 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{13}}{5 a f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{4} a^{2}}-\frac {8 c^{5} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{a^{3} f}+\frac {8 c^{5} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{a^{3} f}\) | \(307\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.78 \[ \int \frac {(c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^3} \, dx=\frac {15 \, c^{5} f x \cos \left (f x + e\right )^{4} + 45 \, c^{5} f x \cos \left (f x + e\right )^{3} + 45 \, c^{5} f x \cos \left (f x + e\right )^{2} + 15 \, c^{5} f x \cos \left (f x + e\right ) + 60 \, {\left (c^{5} \cos \left (f x + e\right )^{4} + 3 \, c^{5} \cos \left (f x + e\right )^{3} + 3 \, c^{5} \cos \left (f x + e\right )^{2} + c^{5} \cos \left (f x + e\right )\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) - 60 \, {\left (c^{5} \cos \left (f x + e\right )^{4} + 3 \, c^{5} \cos \left (f x + e\right )^{3} + 3 \, c^{5} \cos \left (f x + e\right )^{2} + c^{5} \cos \left (f x + e\right )\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) - {\left (239 \, c^{5} \cos \left (f x + e\right )^{3} + 477 \, c^{5} \cos \left (f x + e\right )^{2} + 349 \, c^{5} \cos \left (f x + e\right ) + 15 \, c^{5}\right )} \sin \left (f x + e\right )}{15 \, {\left (a^{3} f \cos \left (f x + e\right )^{4} + 3 \, a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} + a^{3} f \cos \left (f x + e\right )\right )}} \]
[In]
[Out]
\[ \int \frac {(c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^3} \, dx=- \frac {c^{5} \left (\int \frac {5 \sec {\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx + \int \left (- \frac {10 \sec ^{2}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\right )\, dx + \int \frac {10 \sec ^{3}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx + \int \left (- \frac {5 \sec ^{4}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\right )\, dx + \int \frac {\sec ^{5}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\, dx + \int \left (- \frac {1}{\sec ^{3}{\left (e + f x \right )} + 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} + 1}\right )\, dx\right )}{a^{3}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 562 vs. \(2 (154) = 308\).
Time = 0.31 (sec) , antiderivative size = 562, normalized size of antiderivative = 3.47 \[ \int \frac {(c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^3} \, dx=-\frac {3 \, c^{5} {\left (\frac {40 \, \sin \left (f x + e\right )}{{\left (a^{3} - \frac {a^{3} \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 )}} + \frac {\frac {85 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {\sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}}{a^{3}} - \frac {60 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a^{3}} + \frac {60 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a^{3}}\right )} + 5 \, c^{5} {\left (\frac {\frac {105 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {20 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}}{a^{3}} - \frac {60 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a^{3}} + \frac {60 \, \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a^{3}}\right )} + c^{5} {\left (\frac {\frac {105 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {20 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}}{a^{3}} - \frac {120 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{3}}\right )} + \frac {10 \, c^{5} {\left (\frac {15 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3}} + \frac {5 \, c^{5} {\left (\frac {15 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {10 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3}} - \frac {30 \, c^{5} {\left (\frac {5 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {\sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3}}}{60 \, f} \]
[In]
[Out]
none
Time = 0.41 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.95 \[ \int \frac {(c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^3} \, dx=\frac {\frac {15 \, {\left (f x + e\right )} c^{5}}{a^{3}} + \frac {120 \, c^{5} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{a^{3}} - \frac {120 \, c^{5} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{a^{3}} + \frac {30 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )} a^{3}} - \frac {8 \, {\left (3 \, a^{12} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 5 \, a^{12} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 30 \, a^{12} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{15}}}{15 \, f} \]
[In]
[Out]
Time = 14.37 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.83 \[ \int \frac {(c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^3} \, dx=\frac {c^5\,x}{a^3}-\frac {16\,c^5\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{a^3\,f}-\frac {8\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{3\,a^3\,f}-\frac {8\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{5\,a^3\,f}+\frac {16\,c^5\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{a^3\,f}+\frac {2\,c^5\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,\left (a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-a^3\right )} \]
[In]
[Out]