Integrand size = 41, antiderivative size = 254 \[ \int \frac {\sec ^5(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {(2 A-8 B+21 C) \text {arctanh}(\sin (c+d x))}{2 a^4 d}-\frac {8 (20 A-83 B+216 C) \tan (c+d x)}{105 a^4 d}+\frac {(2 A-8 B+21 C) \sec (c+d x) \tan (c+d x)}{2 a^4 d}-\frac {(10 A-52 B+129 C) \sec ^3(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {4 (20 A-83 B+216 C) \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))}-\frac {(A-B+C) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(B-2 C) \sec ^4(c+d x) \tan (c+d x)}{5 a d (a+a \sec (c+d x))^3} \]
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Time = 0.85 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {4169, 4104, 3872, 3852, 8, 3853, 3855} \[ \int \frac {\sec ^5(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {(2 A-8 B+21 C) \text {arctanh}(\sin (c+d x))}{2 a^4 d}-\frac {8 (20 A-83 B+216 C) \tan (c+d x)}{105 a^4 d}-\frac {(10 A-52 B+129 C) \tan (c+d x) \sec ^3(c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}-\frac {4 (20 A-83 B+216 C) \tan (c+d x) \sec ^2(c+d x)}{105 a^4 d (\sec (c+d x)+1)}+\frac {(2 A-8 B+21 C) \tan (c+d x) \sec (c+d x)}{2 a^4 d}-\frac {(A-B+C) \tan (c+d x) \sec ^5(c+d x)}{7 d (a \sec (c+d x)+a)^4}+\frac {(B-2 C) \tan (c+d x) \sec ^4(c+d x)}{5 a d (a \sec (c+d x)+a)^3} \]
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Rule 8
Rule 3852
Rule 3853
Rule 3855
Rule 3872
Rule 4104
Rule 4169
Rubi steps \begin{align*} \text {integral}& = -\frac {(A-B+C) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {\int \frac {\sec ^5(c+d x) (a (2 A+5 B-5 C)+a (2 A-2 B+9 C) \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx}{7 a^2} \\ & = -\frac {(A-B+C) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(B-2 C) \sec ^4(c+d x) \tan (c+d x)}{5 a d (a+a \sec (c+d x))^3}+\frac {\int \frac {\sec ^4(c+d x) \left (28 a^2 (B-2 C)+a^2 (10 A-24 B+73 C) \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{35 a^4} \\ & = -\frac {(10 A-52 B+129 C) \sec ^3(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A-B+C) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(B-2 C) \sec ^4(c+d x) \tan (c+d x)}{5 a d (a+a \sec (c+d x))^3}+\frac {\int \frac {\sec ^3(c+d x) \left (-3 a^3 (10 A-52 B+129 C)+a^3 (50 A-176 B+477 C) \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{105 a^6} \\ & = -\frac {(10 A-52 B+129 C) \sec ^3(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A-B+C) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(B-2 C) \sec ^4(c+d x) \tan (c+d x)}{5 a d (a+a \sec (c+d x))^3}-\frac {4 (20 A-83 B+216 C) \sec ^2(c+d x) \tan (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )}+\frac {\int \sec ^2(c+d x) \left (-8 a^4 (20 A-83 B+216 C)+105 a^4 (2 A-8 B+21 C) \sec (c+d x)\right ) \, dx}{105 a^8} \\ & = -\frac {(10 A-52 B+129 C) \sec ^3(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A-B+C) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(B-2 C) \sec ^4(c+d x) \tan (c+d x)}{5 a d (a+a \sec (c+d x))^3}-\frac {4 (20 A-83 B+216 C) \sec ^2(c+d x) \tan (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )}+\frac {(2 A-8 B+21 C) \int \sec ^3(c+d x) \, dx}{a^4}-\frac {(8 (20 A-83 B+216 C)) \int \sec ^2(c+d x) \, dx}{105 a^4} \\ & = \frac {(2 A-8 B+21 C) \sec (c+d x) \tan (c+d x)}{2 a^4 d}-\frac {(10 A-52 B+129 C) \sec ^3(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A-B+C) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(B-2 C) \sec ^4(c+d x) \tan (c+d x)}{5 a d (a+a \sec (c+d x))^3}-\frac {4 (20 A-83 B+216 C) \sec ^2(c+d x) \tan (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )}+\frac {(2 A-8 B+21 C) \int \sec (c+d x) \, dx}{2 a^4}+\frac {(8 (20 A-83 B+216 C)) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{105 a^4 d} \\ & = \frac {(2 A-8 B+21 C) \text {arctanh}(\sin (c+d x))}{2 a^4 d}-\frac {8 (20 A-83 B+216 C) \tan (c+d x)}{105 a^4 d}+\frac {(2 A-8 B+21 C) \sec (c+d x) \tan (c+d x)}{2 a^4 d}-\frac {(10 A-52 B+129 C) \sec ^3(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A-B+C) \sec ^5(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(B-2 C) \sec ^4(c+d x) \tan (c+d x)}{5 a d (a+a \sec (c+d x))^3}-\frac {4 (20 A-83 B+216 C) \sec ^2(c+d x) \tan (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(1322\) vs. \(2(254)=508\).
Time = 9.93 (sec) , antiderivative size = 1322, normalized size of antiderivative = 5.20 \[ \int \frac {\sec ^5(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=-\frac {16 (2 A-8 B+21 C) \cos ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^4}+\frac {16 (2 A-8 B+21 C) \cos ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^4}-\frac {4 \cos ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec \left (\frac {c}{2}\right ) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (A \sin \left (\frac {c}{2}\right )-B \sin \left (\frac {c}{2}\right )+C \sin \left (\frac {c}{2}\right )\right )}{7 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^4}-\frac {8 \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec \left (\frac {c}{2}\right ) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (10 A \sin \left (\frac {c}{2}\right )-17 B \sin \left (\frac {c}{2}\right )+24 C \sin \left (\frac {c}{2}\right )\right )}{35 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^4}-\frac {16 \cos ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec \left (\frac {c}{2}\right ) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (55 A \sin \left (\frac {c}{2}\right )-139 B \sin \left (\frac {c}{2}\right )+258 C \sin \left (\frac {c}{2}\right )\right )}{105 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^4}-\frac {4 \cos \left (\frac {c}{2}+\frac {d x}{2}\right ) \sec \left (\frac {c}{2}\right ) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (A \sin \left (\frac {d x}{2}\right )-B \sin \left (\frac {d x}{2}\right )+C \sin \left (\frac {d x}{2}\right )\right )}{7 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^4}-\frac {8 \cos ^3\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec \left (\frac {c}{2}\right ) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (10 A \sin \left (\frac {d x}{2}\right )-17 B \sin \left (\frac {d x}{2}\right )+24 C \sin \left (\frac {d x}{2}\right )\right )}{35 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^4}-\frac {16 \cos ^5\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec \left (\frac {c}{2}\right ) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (55 A \sin \left (\frac {d x}{2}\right )-139 B \sin \left (\frac {d x}{2}\right )+258 C \sin \left (\frac {d x}{2}\right )\right )}{105 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^4}-\frac {32 \cos ^7\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec \left (\frac {c}{2}\right ) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (160 A \sin \left (\frac {d x}{2}\right )-559 B \sin \left (\frac {d x}{2}\right )+1308 C \sin \left (\frac {d x}{2}\right )\right )}{105 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^4}+\frac {16 C \cos ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec (c) \sec ^4(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \sin (d x)}{d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^4}+\frac {16 \cos ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec (c) \sec ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) (C \sin (c)+2 B \sin (d x)-8 C \sin (d x))}{d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^4} \]
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Time = 0.43 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.85
method | result | size |
parallelrisch | \(\frac {-6720 \left (1+\cos \left (2 d x +2 c \right )\right ) \left (A -4 B +\frac {21 C}{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+6720 \left (1+\cos \left (2 d x +2 c \right )\right ) \left (A -4 B +\frac {21 C}{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-3280 \left (\left (\frac {159 A}{82}-\frac {342 B}{41}+\frac {3531 C}{164}\right ) \cos \left (2 d x +2 c \right )+\left (A -\frac {1733 B}{410}+\frac {4491 C}{410}\right ) \cos \left (3 d x +3 c \right )+\left (\frac {107 A}{328}-\frac {559 B}{410}+\frac {11619 C}{3280}\right ) \cos \left (4 d x +4 c \right )+\left (\frac {2 A}{41}-\frac {83 B}{410}+\frac {108 C}{205}\right ) \cos \left (5 d x +5 c \right )+\left (\frac {113 A}{41}-\frac {2497 B}{205}+\frac {12813 C}{410}\right ) \cos \left (d x +c \right )+\frac {529 A}{328}-\frac {2861 B}{410}+\frac {58161 C}{3280}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{6720 d \,a^{4} \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(216\) |
derivativedivides | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} B}{7}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} C}{7}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A +\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B}{5}-\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} C}{5}-\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}+\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B}{3}-13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C -15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A +49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B -111 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C -\frac {-36 C +8 B}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\left (84 C -32 B +8 A \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {4 C}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {-36 C +8 B}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\left (-84 C +32 B -8 A \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\frac {4 C}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}}{8 d \,a^{4}}\) | \(294\) |
default | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} B}{7}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} C}{7}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A +\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B}{5}-\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} C}{5}-\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}+\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B}{3}-13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C -15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A +49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B -111 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C -\frac {-36 C +8 B}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\left (84 C -32 B +8 A \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {4 C}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {-36 C +8 B}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\left (-84 C +32 B -8 A \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\frac {4 C}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}}{8 d \,a^{4}}\) | \(294\) |
risch | \(-\frac {i \left (3456 C +320 A -1328 B +4760 A \,{\mathrm e}^{8 i \left (d x +c \right )}+9800 A \,{\mathrm e}^{7 i \left (d x +c \right )}+17220 A \,{\mathrm e}^{5 i \left (d x +c \right )}+10920 A \,{\mathrm e}^{3 i \left (d x +c \right )}+2030 A \,{\mathrm e}^{i \left (d x +c \right )}-64384 B \,{\mathrm e}^{4 i \left (d x +c \right )}+14140 A \,{\mathrm e}^{6 i \left (d x +c \right )}+155526 C \,{\mathrm e}^{6 i \left (d x +c \right )}+15160 A \,{\mathrm e}^{4 i \left (d x +c \right )}+166668 C \,{\mathrm e}^{4 i \left (d x +c \right )}+5890 A \,{\mathrm e}^{2 i \left (d x +c \right )}+64053 C \,{\mathrm e}^{2 i \left (d x +c \right )}+102900 C \,{\mathrm e}^{7 i \left (d x +c \right )}+183162 C \,{\mathrm e}^{5 i \left (d x +c \right )}+119364 C \,{\mathrm e}^{3 i \left (d x +c \right )}-24664 B \,{\mathrm e}^{2 i \left (d x +c \right )}+21987 C \,{\mathrm e}^{i \left (d x +c \right )}+2205 C \,{\mathrm e}^{10 i \left (d x +c \right )}+210 A \,{\mathrm e}^{10 i \left (d x +c \right )}+1470 A \,{\mathrm e}^{9 i \left (d x +c \right )}+15435 C \,{\mathrm e}^{9 i \left (d x +c \right )}+49980 C \,{\mathrm e}^{8 i \left (d x +c \right )}-8456 B \,{\mathrm e}^{i \left (d x +c \right )}-39200 B \,{\mathrm e}^{7 i \left (d x +c \right )}-70896 B \,{\mathrm e}^{5 i \left (d x +c \right )}-840 B \,{\mathrm e}^{10 i \left (d x +c \right )}-5880 B \,{\mathrm e}^{9 i \left (d x +c \right )}-19040 B \,{\mathrm e}^{8 i \left (d x +c \right )}-59248 B \,{\mathrm e}^{6 i \left (d x +c \right )}-46032 B \,{\mathrm e}^{3 i \left (d x +c \right )}\right )}{105 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{a^{4} d}+\frac {4 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{a^{4} d}-\frac {21 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{2 a^{4} d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{a^{4} d}-\frac {4 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{a^{4} d}+\frac {21 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{2 a^{4} d}\) | \(538\) |
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Time = 0.28 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.57 \[ \int \frac {\sec ^5(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {105 \, {\left ({\left (2 \, A - 8 \, B + 21 \, C\right )} \cos \left (d x + c\right )^{6} + 4 \, {\left (2 \, A - 8 \, B + 21 \, C\right )} \cos \left (d x + c\right )^{5} + 6 \, {\left (2 \, A - 8 \, B + 21 \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (2 \, A - 8 \, B + 21 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (2 \, A - 8 \, B + 21 \, C\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left ({\left (2 \, A - 8 \, B + 21 \, C\right )} \cos \left (d x + c\right )^{6} + 4 \, {\left (2 \, A - 8 \, B + 21 \, C\right )} \cos \left (d x + c\right )^{5} + 6 \, {\left (2 \, A - 8 \, B + 21 \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (2 \, A - 8 \, B + 21 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (2 \, A - 8 \, B + 21 \, C\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (16 \, {\left (20 \, A - 83 \, B + 216 \, C\right )} \cos \left (d x + c\right )^{5} + {\left (1070 \, A - 4472 \, B + 11619 \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (310 \, A - 1318 \, B + 3411 \, C\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (130 \, A - 592 \, B + 1509 \, C\right )} \cos \left (d x + c\right )^{2} - 210 \, {\left (B - 2 \, C\right )} \cos \left (d x + c\right ) - 105 \, C\right )} \sin \left (d x + c\right )}{420 \, {\left (a^{4} d \cos \left (d x + c\right )^{6} + 4 \, a^{4} d \cos \left (d x + c\right )^{5} + 6 \, a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + a^{4} d \cos \left (d x + c\right )^{2}\right )}} \]
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\[ \int \frac {\sec ^5(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {\int \frac {A \sec ^{5}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {B \sec ^{6}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {C \sec ^{7}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx}{a^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 556 vs. \(2 (240) = 480\).
Time = 0.25 (sec) , antiderivative size = 556, normalized size of antiderivative = 2.19 \[ \int \frac {\sec ^5(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=-\frac {3 \, C {\left (\frac {280 \, {\left (\frac {7 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {9 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{4} - \frac {2 \, a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {3885 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {455 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {63 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {2940 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {2940 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )} - B {\left (\frac {1680 \, \sin \left (d x + c\right )}{{\left (a^{4} - \frac {a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {5145 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {805 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {147 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {3360 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {3360 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )} + 5 \, A {\left (\frac {\frac {315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {77 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {168 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {168 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )}}{840 \, d} \]
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Time = 0.38 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.33 \[ \int \frac {\sec ^5(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {\frac {420 \, {\left (2 \, A - 8 \, B + 21 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac {420 \, {\left (2 \, A - 8 \, B + 21 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{4}} - \frac {840 \, {\left (2 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 7 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2} a^{4}} - \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 105 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 147 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 189 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 385 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 805 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1365 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1575 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5145 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 11655 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]
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Time = 16.00 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.17 \[ \int \frac {\sec ^5(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,\left (A+5\,B-15\,C\right )}{8\,a^4}-\frac {3\,\left (2\,A-4\,B+6\,C\right )}{4\,a^4}-\frac {5\,\left (A-B+C\right )}{4\,a^4}+\frac {4\,A-20\,C}{8\,a^4}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (2\,B-9\,C\right )-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,B-7\,C\right )}{d\,\left (a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-2\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^4\right )}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {2\,A-4\,B+6\,C}{40\,a^4}+\frac {3\,\left (A-B+C\right )}{40\,a^4}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {2\,A-4\,B+6\,C}{8\,a^4}-\frac {A+5\,B-15\,C}{24\,a^4}+\frac {A-B+C}{4\,a^4}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (A-B+C\right )}{56\,a^4\,d}+\frac {2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (A-4\,B+\frac {21\,C}{2}\right )}{a^4\,d} \]
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