Integrand size = 29, antiderivative size = 166 \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=-\frac {(4 A-B) x}{a^4}+\frac {8 (83 A-20 B) \sin (c+d x)}{105 a^4 d}-\frac {(88 A-25 B) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(4 A-B) \sin (c+d x)}{a^4 d (1+\sec (c+d x))}-\frac {(A-B) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {(12 A-5 B) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3} \]
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Time = 0.58 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {4105, 3872, 2717, 8} \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {8 (83 A-20 B) \sin (c+d x)}{105 a^4 d}-\frac {(4 A-B) \sin (c+d x)}{a^4 d (\sec (c+d x)+1)}-\frac {(88 A-25 B) \sin (c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}-\frac {x (4 A-B)}{a^4}-\frac {(12 A-5 B) \sin (c+d x)}{35 a d (a \sec (c+d x)+a)^3}-\frac {(A-B) \sin (c+d x)}{7 d (a \sec (c+d x)+a)^4} \]
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Rule 8
Rule 2717
Rule 3872
Rule 4105
Rubi steps \begin{align*} \text {integral}& = -\frac {(A-B) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {\int \frac {\cos (c+d x) (a (8 A-B)-4 a (A-B) \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx}{7 a^2} \\ & = -\frac {(A-B) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {(12 A-5 B) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac {\int \frac {\cos (c+d x) \left (2 a^2 (26 A-5 B)-3 a^2 (12 A-5 B) \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{35 a^4} \\ & = -\frac {(88 A-25 B) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A-B) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {(12 A-5 B) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac {\int \frac {\cos (c+d x) \left (a^3 (244 A-55 B)-2 a^3 (88 A-25 B) \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{105 a^6} \\ & = -\frac {(88 A-25 B) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A-B) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {(12 A-5 B) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {(4 A-B) \sin (c+d x)}{d \left (a^4+a^4 \sec (c+d x)\right )}+\frac {\int \cos (c+d x) \left (8 a^4 (83 A-20 B)-105 a^4 (4 A-B) \sec (c+d x)\right ) \, dx}{105 a^8} \\ & = -\frac {(88 A-25 B) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A-B) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {(12 A-5 B) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {(4 A-B) \sin (c+d x)}{d \left (a^4+a^4 \sec (c+d x)\right )}+\frac {(8 (83 A-20 B)) \int \cos (c+d x) \, dx}{105 a^4}-\frac {(4 A-B) \int 1 \, dx}{a^4} \\ & = -\frac {(4 A-B) x}{a^4}+\frac {8 (83 A-20 B) \sin (c+d x)}{105 a^4 d}-\frac {(88 A-25 B) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A-B) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {(12 A-5 B) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {(4 A-B) \sin (c+d x)}{d \left (a^4+a^4 \sec (c+d x)\right )} \\ \end{align*}
Time = 5.25 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.84 \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {\sin (c+d x) \left (244 A-55 B-\frac {15 (A-B)}{(1+\sec (c+d x))^4}+\frac {-36 A+15 B}{(1+\sec (c+d x))^3}+\frac {-88 A+25 B}{(1+\sec (c+d x))^2}+\frac {105 (4 A-B) \left (\arcsin (\cos (c+d x)) (1+\cos (c+d x))+\sqrt {\sin ^2(c+d x)}\right )}{\sqrt {1-\cos (c+d x)} (1+\cos (c+d x))^{3/2}}\right )}{105 a^4 d} \]
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Time = 1.08 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.64
method | result | size |
parallelrisch | \(\frac {105 \left (\left (\frac {10964 A}{105}-\frac {496 B}{21}\right ) \cos \left (2 d x +2 c \right )+\left (\frac {2368 A}{105}-\frac {104 B}{21}\right ) \cos \left (3 d x +3 c \right )+A \cos \left (4 d x +4 c \right )+\left (\frac {24992 A}{105}-\frac {1168 B}{21}\right ) \cos \left (d x +c \right )+\frac {16171 A}{105}-\frac {752 B}{21}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-26880 d x \left (A -\frac {B}{4}\right )}{6720 a^{4} d}\) | \(107\) |
derivativedivides | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} B}{7}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B -\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B}{3}+49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A -15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +\frac {16 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-16 \left (4 A -B \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\) | \(164\) |
default | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} B}{7}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B -\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B}{3}+49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A -15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +\frac {16 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-16 \left (4 A -B \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\) | \(164\) |
norman | \(\frac {-\frac {\left (4 A -B \right ) x}{a}-\frac {\left (A -B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{56 a d}-\frac {\left (4 A -B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a}+\frac {5 \left (13 A -3 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}+\frac {\left (22 A -15 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{140 a d}-\frac {\left (47 A -20 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{60 a d}+\frac {\left (62 A -17 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 a d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) a^{3}}\) | \(184\) |
risch | \(-\frac {4 A x}{a^{4}}+\frac {x B}{a^{4}}-\frac {i A \,{\mathrm e}^{i \left (d x +c \right )}}{2 a^{4} d}+\frac {i A \,{\mathrm e}^{-i \left (d x +c \right )}}{2 a^{4} d}+\frac {4 i \left (525 A \,{\mathrm e}^{6 i \left (d x +c \right )}-210 B \,{\mathrm e}^{6 i \left (d x +c \right )}+2625 A \,{\mathrm e}^{5 i \left (d x +c \right )}-945 B \,{\mathrm e}^{5 i \left (d x +c \right )}+5950 A \,{\mathrm e}^{4 i \left (d x +c \right )}-2065 B \,{\mathrm e}^{4 i \left (d x +c \right )}+7420 A \,{\mathrm e}^{3 i \left (d x +c \right )}-2485 B \,{\mathrm e}^{3 i \left (d x +c \right )}+5397 A \,{\mathrm e}^{2 i \left (d x +c \right )}-1785 B \,{\mathrm e}^{2 i \left (d x +c \right )}+2149 \,{\mathrm e}^{i \left (d x +c \right )} A -700 B \,{\mathrm e}^{i \left (d x +c \right )}+382 A -130 B \right )}{105 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}\) | \(226\) |
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Time = 0.26 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.34 \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=-\frac {105 \, {\left (4 \, A - B\right )} d x \cos \left (d x + c\right )^{4} + 420 \, {\left (4 \, A - B\right )} d x \cos \left (d x + c\right )^{3} + 630 \, {\left (4 \, A - B\right )} d x \cos \left (d x + c\right )^{2} + 420 \, {\left (4 \, A - B\right )} d x \cos \left (d x + c\right ) + 105 \, {\left (4 \, A - B\right )} d x - {\left (105 \, A \cos \left (d x + c\right )^{4} + 4 \, {\left (296 \, A - 65 \, B\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (659 \, A - 155 \, B\right )} \cos \left (d x + c\right )^{2} + {\left (2236 \, A - 535 \, B\right )} \cos \left (d x + c\right ) + 664 \, A - 160 \, B\right )} \sin \left (d x + c\right )}{105 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
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\[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {\int \frac {A \cos {\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {B \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx}{a^{4}} \]
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Time = 0.31 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.63 \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {A {\left (\frac {1680 \, \sin \left (d x + c\right )}{{\left (a^{4} + \frac {a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {5145 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {805 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {147 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {6720 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )} - 5 \, B {\left (\frac {\frac {315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {77 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {336 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )}}{840 \, d} \]
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Time = 0.32 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.14 \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=-\frac {\frac {840 \, {\left (d x + c\right )} {\left (4 \, A - B\right )}}{a^{4}} - \frac {1680 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a^{4}} + \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 147 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 105 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 805 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 385 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5145 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1575 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]
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Time = 14.20 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.22 \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {\left (\frac {764\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{105}-\frac {52\,B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{21}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\left (\frac {16\,B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{21}-\frac {143\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{105}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (\frac {8\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{35}-\frac {5\,B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{28}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\frac {A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{56}+\frac {B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{56}}{a^4\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}-\frac {4\,A\,d\,x-B\,d\,x}{a^4\,d}+\frac {2\,A\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a^4\,d} \]
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