Integrand size = 26, antiderivative size = 196 \[ \int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^5 \, dx=a^2 c^5 x-\frac {19 a^2 c^5 \text {arctanh}(\sin (e+f x))}{16 f}-\frac {a^2 c^5 \tan (e+f x)}{f}+\frac {17 a^2 c^5 \sec (e+f x) \tan (e+f x)}{16 f}+\frac {a^2 c^5 \sec ^3(e+f x) \tan (e+f x)}{8 f}+\frac {a^2 c^5 \tan ^3(e+f x)}{3 f}-\frac {3 a^2 c^5 \sec (e+f x) \tan ^3(e+f x)}{4 f}-\frac {a^2 c^5 \sec ^3(e+f x) \tan ^3(e+f x)}{6 f}+\frac {3 a^2 c^5 \tan ^5(e+f x)}{5 f} \]
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Time = 0.41 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {3989, 3971, 3554, 8, 2691, 3855, 2687, 30, 3853} \[ \int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^5 \, dx=-\frac {19 a^2 c^5 \text {arctanh}(\sin (e+f x))}{16 f}+\frac {3 a^2 c^5 \tan ^5(e+f x)}{5 f}+\frac {a^2 c^5 \tan ^3(e+f x)}{3 f}-\frac {a^2 c^5 \tan (e+f x)}{f}-\frac {a^2 c^5 \tan ^3(e+f x) \sec ^3(e+f x)}{6 f}+\frac {a^2 c^5 \tan (e+f x) \sec ^3(e+f x)}{8 f}-\frac {3 a^2 c^5 \tan ^3(e+f x) \sec (e+f x)}{4 f}+\frac {17 a^2 c^5 \tan (e+f x) \sec (e+f x)}{16 f}+a^2 c^5 x \]
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Rule 8
Rule 30
Rule 2687
Rule 2691
Rule 3554
Rule 3853
Rule 3855
Rule 3971
Rule 3989
Rubi steps \begin{align*} \text {integral}& = \left (a^2 c^2\right ) \int (c-c \sec (e+f x))^3 \tan ^4(e+f x) \, dx \\ & = \left (a^2 c^2\right ) \int \left (c^3 \tan ^4(e+f x)-3 c^3 \sec (e+f x) \tan ^4(e+f x)+3 c^3 \sec ^2(e+f x) \tan ^4(e+f x)-c^3 \sec ^3(e+f x) \tan ^4(e+f x)\right ) \, dx \\ & = \left (a^2 c^5\right ) \int \tan ^4(e+f x) \, dx-\left (a^2 c^5\right ) \int \sec ^3(e+f x) \tan ^4(e+f x) \, dx-\left (3 a^2 c^5\right ) \int \sec (e+f x) \tan ^4(e+f x) \, dx+\left (3 a^2 c^5\right ) \int \sec ^2(e+f x) \tan ^4(e+f x) \, dx \\ & = \frac {a^2 c^5 \tan ^3(e+f x)}{3 f}-\frac {3 a^2 c^5 \sec (e+f x) \tan ^3(e+f x)}{4 f}-\frac {a^2 c^5 \sec ^3(e+f x) \tan ^3(e+f x)}{6 f}+\frac {1}{2} \left (a^2 c^5\right ) \int \sec ^3(e+f x) \tan ^2(e+f x) \, dx-\left (a^2 c^5\right ) \int \tan ^2(e+f x) \, dx+\frac {1}{4} \left (9 a^2 c^5\right ) \int \sec (e+f x) \tan ^2(e+f x) \, dx+\frac {\left (3 a^2 c^5\right ) \text {Subst}\left (\int x^4 \, dx,x,\tan (e+f x)\right )}{f} \\ & = -\frac {a^2 c^5 \tan (e+f x)}{f}+\frac {9 a^2 c^5 \sec (e+f x) \tan (e+f x)}{8 f}+\frac {a^2 c^5 \sec ^3(e+f x) \tan (e+f x)}{8 f}+\frac {a^2 c^5 \tan ^3(e+f x)}{3 f}-\frac {3 a^2 c^5 \sec (e+f x) \tan ^3(e+f x)}{4 f}-\frac {a^2 c^5 \sec ^3(e+f x) \tan ^3(e+f x)}{6 f}+\frac {3 a^2 c^5 \tan ^5(e+f x)}{5 f}-\frac {1}{8} \left (a^2 c^5\right ) \int \sec ^3(e+f x) \, dx+\left (a^2 c^5\right ) \int 1 \, dx-\frac {1}{8} \left (9 a^2 c^5\right ) \int \sec (e+f x) \, dx \\ & = a^2 c^5 x-\frac {9 a^2 c^5 \text {arctanh}(\sin (e+f x))}{8 f}-\frac {a^2 c^5 \tan (e+f x)}{f}+\frac {17 a^2 c^5 \sec (e+f x) \tan (e+f x)}{16 f}+\frac {a^2 c^5 \sec ^3(e+f x) \tan (e+f x)}{8 f}+\frac {a^2 c^5 \tan ^3(e+f x)}{3 f}-\frac {3 a^2 c^5 \sec (e+f x) \tan ^3(e+f x)}{4 f}-\frac {a^2 c^5 \sec ^3(e+f x) \tan ^3(e+f x)}{6 f}+\frac {3 a^2 c^5 \tan ^5(e+f x)}{5 f}-\frac {1}{16} \left (a^2 c^5\right ) \int \sec (e+f x) \, dx \\ & = a^2 c^5 x-\frac {19 a^2 c^5 \text {arctanh}(\sin (e+f x))}{16 f}-\frac {a^2 c^5 \tan (e+f x)}{f}+\frac {17 a^2 c^5 \sec (e+f x) \tan (e+f x)}{16 f}+\frac {a^2 c^5 \sec ^3(e+f x) \tan (e+f x)}{8 f}+\frac {a^2 c^5 \tan ^3(e+f x)}{3 f}-\frac {3 a^2 c^5 \sec (e+f x) \tan ^3(e+f x)}{4 f}-\frac {a^2 c^5 \sec ^3(e+f x) \tan ^3(e+f x)}{6 f}+\frac {3 a^2 c^5 \tan ^5(e+f x)}{5 f} \\ \end{align*}
Time = 1.74 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.84 \[ \int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^5 \, dx=\frac {a^2 c^5 \sec ^6(e+f x) \left (1200 e+1200 f x-4560 \text {arctanh}(\sin (e+f x)) \cos ^6(e+f x)+1800 (e+f x) \cos (2 (e+f x))+720 e \cos (4 (e+f x))+720 f x \cos (4 (e+f x))+120 e \cos (6 (e+f x))+120 f x \cos (6 (e+f x))-210 \sin (e+f x)-120 \sin (2 (e+f x))+865 \sin (3 (e+f x))-768 \sin (4 (e+f x))+435 \sin (5 (e+f x))-88 \sin (6 (e+f x))\right )}{3840 f} \]
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Result contains complex when optimal does not.
Time = 4.91 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.05
method | result | size |
risch | \(a^{2} c^{5} x -\frac {i c^{5} a^{2} \left (435 \,{\mathrm e}^{11 i \left (f x +e \right )}-240 \,{\mathrm e}^{10 i \left (f x +e \right )}+865 \,{\mathrm e}^{9 i \left (f x +e \right )}+1200 \,{\mathrm e}^{8 i \left (f x +e \right )}-210 \,{\mathrm e}^{7 i \left (f x +e \right )}+1760 \,{\mathrm e}^{6 i \left (f x +e \right )}+210 \,{\mathrm e}^{5 i \left (f x +e \right )}+1440 \,{\mathrm e}^{4 i \left (f x +e \right )}-865 \,{\mathrm e}^{3 i \left (f x +e \right )}+1296 \,{\mathrm e}^{2 i \left (f x +e \right )}-435 \,{\mathrm e}^{i \left (f x +e \right )}+176\right )}{120 f \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )^{6}}+\frac {19 c^{5} a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{16 f}-\frac {19 c^{5} a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{16 f}\) | \(206\) |
parallelrisch | \(-\frac {a^{2} c^{5} \left (\frac {19 \left (-5-\frac {15 \cos \left (2 f x +2 e \right )}{2}-3 \cos \left (4 f x +4 e \right )-\frac {\cos \left (6 f x +6 e \right )}{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{8}+\frac {19 \left (5+\frac {\cos \left (6 f x +6 e \right )}{2}+3 \cos \left (4 f x +4 e \right )+\frac {15 \cos \left (2 f x +2 e \right )}{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{8}-15 f x \cos \left (2 f x +2 e \right )-6 f x \cos \left (4 f x +4 e \right )-f x \cos \left (6 f x +6 e \right )-10 f x +\sin \left (2 f x +2 e \right )-\frac {173 \sin \left (3 f x +3 e \right )}{24}+\frac {32 \sin \left (4 f x +4 e \right )}{5}-\frac {29 \sin \left (5 f x +5 e \right )}{8}+\frac {11 \sin \left (6 f x +6 e \right )}{15}+\frac {7 \sin \left (f x +e \right )}{4}\right )}{f \left (6 \cos \left (4 f x +4 e \right )+10+15 \cos \left (2 f x +2 e \right )+\cos \left (6 f x +6 e \right )\right )}\) | \(250\) |
derivativedivides | \(\frac {-c^{5} a^{2} \left (-\left (-\frac {\sec \left (f x +e \right )^{5}}{6}-\frac {5 \sec \left (f x +e \right )^{3}}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )-3 c^{5} a^{2} \left (-\frac {8}{15}-\frac {\sec \left (f x +e \right )^{4}}{5}-\frac {4 \sec \left (f x +e \right )^{2}}{15}\right ) \tan \left (f x +e \right )-c^{5} a^{2} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )+5 c^{5} a^{2} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )+5 c^{5} a^{2} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )+c^{5} a^{2} \tan \left (f x +e \right )-3 c^{5} a^{2} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+c^{5} a^{2} \left (f x +e \right )}{f}\) | \(268\) |
default | \(\frac {-c^{5} a^{2} \left (-\left (-\frac {\sec \left (f x +e \right )^{5}}{6}-\frac {5 \sec \left (f x +e \right )^{3}}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )-3 c^{5} a^{2} \left (-\frac {8}{15}-\frac {\sec \left (f x +e \right )^{4}}{5}-\frac {4 \sec \left (f x +e \right )^{2}}{15}\right ) \tan \left (f x +e \right )-c^{5} a^{2} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )+5 c^{5} a^{2} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )+5 c^{5} a^{2} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )+c^{5} a^{2} \tan \left (f x +e \right )-3 c^{5} a^{2} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+c^{5} a^{2} \left (f x +e \right )}{f}\) | \(268\) |
parts | \(a^{2} c^{5} x +\frac {a^{2} c^{5} \tan \left (f x +e \right )}{f}-\frac {3 c^{5} a^{2} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{f}+\frac {5 c^{5} a^{2} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )}{f}+\frac {5 c^{5} a^{2} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )}{f}-\frac {c^{5} a^{2} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )}{f}-\frac {3 c^{5} a^{2} \left (-\frac {8}{15}-\frac {\sec \left (f x +e \right )^{4}}{5}-\frac {4 \sec \left (f x +e \right )^{2}}{15}\right ) \tan \left (f x +e \right )}{f}-\frac {c^{5} a^{2} \left (-\left (-\frac {\sec \left (f x +e \right )^{5}}{6}-\frac {5 \sec \left (f x +e \right )^{3}}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )}{f}\) | \(281\) |
norman | \(\frac {a^{2} c^{5} x +a^{2} c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}-6 a^{2} c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+15 a^{2} c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}-20 a^{2} c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}+15 a^{2} c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}-6 a^{2} c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}+\frac {3 c^{5} a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{8 f}-\frac {19 c^{5} a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{24 f}-\frac {61 c^{5} a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{20 f}+\frac {291 c^{5} a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{20 f}-\frac {209 c^{5} a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{8 f}+\frac {35 c^{5} a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{8 f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{6}}+\frac {19 c^{5} a^{2} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{16 f}-\frac {19 c^{5} a^{2} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{16 f}\) | \(322\) |
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Time = 0.28 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.91 \[ \int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^5 \, dx=\frac {480 \, a^{2} c^{5} f x \cos \left (f x + e\right )^{6} - 285 \, a^{2} c^{5} \cos \left (f x + e\right )^{6} \log \left (\sin \left (f x + e\right ) + 1\right ) + 285 \, a^{2} c^{5} \cos \left (f x + e\right )^{6} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (176 \, a^{2} c^{5} \cos \left (f x + e\right )^{5} - 435 \, a^{2} c^{5} \cos \left (f x + e\right )^{4} + 208 \, a^{2} c^{5} \cos \left (f x + e\right )^{3} + 110 \, a^{2} c^{5} \cos \left (f x + e\right )^{2} - 144 \, a^{2} c^{5} \cos \left (f x + e\right ) + 40 \, a^{2} c^{5}\right )} \sin \left (f x + e\right )}{480 \, f \cos \left (f x + e\right )^{6}} \]
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\[ \int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^5 \, dx=- a^{2} c^{5} \left (\int \left (-1\right )\, dx + \int 3 \sec {\left (e + f x \right )}\, dx + \int \left (- \sec ^{2}{\left (e + f x \right )}\right )\, dx + \int \left (- 5 \sec ^{3}{\left (e + f x \right )}\right )\, dx + \int 5 \sec ^{4}{\left (e + f x \right )}\, dx + \int \sec ^{5}{\left (e + f x \right )}\, dx + \int \left (- 3 \sec ^{6}{\left (e + f x \right )}\right )\, dx + \int \sec ^{7}{\left (e + f x \right )}\, dx\right ) \]
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Time = 0.23 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.70 \[ \int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^5 \, dx=\frac {96 \, {\left (3 \, \tan \left (f x + e\right )^{5} + 10 \, \tan \left (f x + e\right )^{3} + 15 \, \tan \left (f x + e\right )\right )} a^{2} c^{5} - 800 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{2} c^{5} + 480 \, {\left (f x + e\right )} a^{2} c^{5} + 5 \, a^{2} c^{5} {\left (\frac {2 \, {\left (15 \, \sin \left (f x + e\right )^{5} - 40 \, \sin \left (f x + e\right )^{3} + 33 \, \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{6} - 3 \, \sin \left (f x + e\right )^{4} + 3 \, \sin \left (f x + e\right )^{2} - 1} - 15 \, \log \left (\sin \left (f x + e\right ) + 1\right ) + 15 \, \log \left (\sin \left (f x + e\right ) - 1\right )\right )} + 30 \, a^{2} c^{5} {\left (\frac {2 \, {\left (3 \, \sin \left (f x + e\right )^{3} - 5 \, \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{4} - 2 \, \sin \left (f x + e\right )^{2} + 1} - 3 \, \log \left (\sin \left (f x + e\right ) + 1\right ) + 3 \, \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 600 \, a^{2} c^{5} {\left (\frac {2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 1440 \, a^{2} c^{5} \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) + 480 \, a^{2} c^{5} \tan \left (f x + e\right )}{480 \, f} \]
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Time = 0.40 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.97 \[ \int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^5 \, dx=\frac {240 \, {\left (f x + e\right )} a^{2} c^{5} - 285 \, a^{2} c^{5} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right ) + 285 \, a^{2} c^{5} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right ) + \frac {2 \, {\left (525 \, a^{2} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} - 3135 \, a^{2} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 1746 \, a^{2} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 366 \, a^{2} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 95 \, a^{2} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 45 \, a^{2} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{6}}}{240 \, f} \]
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Time = 15.54 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.16 \[ \int (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^5 \, dx=a^2\,c^5\,x-\frac {-\frac {35\,a^2\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}}{8}+\frac {209\,a^2\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9}{8}-\frac {291\,a^2\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7}{20}+\frac {61\,a^2\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{20}+\frac {19\,a^2\,c^5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{24}-\frac {3\,a^2\,c^5\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{8}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}-6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8-20\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}-\frac {19\,a^2\,c^5\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{8\,f} \]
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