Optimal. Leaf size=166 \[ \frac {8 (83 A-20 B) \sin (c+d x)}{105 a^4 d}-\frac {(4 A-B) \sin (c+d x)}{a^4 d (\sec (c+d x)+1)}-\frac {(88 A-25 B) \sin (c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}-\frac {x (4 A-B)}{a^4}-\frac {(12 A-5 B) \sin (c+d x)}{35 a d (a \sec (c+d x)+a)^3}-\frac {(A-B) \sin (c+d x)}{7 d (a \sec (c+d x)+a)^4} \]
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Rubi [A] time = 0.57, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {4020, 3787, 2637, 8} \[ \frac {8 (83 A-20 B) \sin (c+d x)}{105 a^4 d}-\frac {(4 A-B) \sin (c+d x)}{a^4 d (\sec (c+d x)+1)}-\frac {(88 A-25 B) \sin (c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}-\frac {x (4 A-B)}{a^4}-\frac {(12 A-5 B) \sin (c+d x)}{35 a d (a \sec (c+d x)+a)^3}-\frac {(A-B) \sin (c+d x)}{7 d (a \sec (c+d x)+a)^4} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2637
Rule 3787
Rule 4020
Rubi steps
\begin {align*} \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx &=-\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*}
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Mathematica [B] time = 1.11, size = 485, normalized size = 2.92 \[ \frac {\sec \left (\frac {c}{2}\right ) \cos \left (\frac {1}{2} (c+d x)\right ) \left (-7350 d x (4 A-B) \cos \left (c+\frac {d x}{2}\right )-7350 d x (4 A-B) \cos \left (\frac {d x}{2}\right )-46130 A \sin \left (c+\frac {d x}{2}\right )+46116 A \sin \left (c+\frac {3 d x}{2}\right )-18060 A \sin \left (2 c+\frac {3 d x}{2}\right )+19292 A \sin \left (2 c+\frac {5 d x}{2}\right )-2100 A \sin \left (3 c+\frac {5 d x}{2}\right )+3791 A \sin \left (3 c+\frac {7 d x}{2}\right )+735 A \sin \left (4 c+\frac {7 d x}{2}\right )+105 A \sin \left (4 c+\frac {9 d x}{2}\right )+105 A \sin \left (5 c+\frac {9 d x}{2}\right )-17640 A d x \cos \left (c+\frac {3 d x}{2}\right )-17640 A d x \cos \left (2 c+\frac {3 d x}{2}\right )-5880 A d x \cos \left (2 c+\frac {5 d x}{2}\right )-5880 A d x \cos \left (3 c+\frac {5 d x}{2}\right )-840 A d x \cos \left (3 c+\frac {7 d x}{2}\right )-840 A d x \cos \left (4 c+\frac {7 d x}{2}\right )+60830 A \sin \left (\frac {d x}{2}\right )+16520 B \sin \left (c+\frac {d x}{2}\right )-14280 B \sin \left (c+\frac {3 d x}{2}\right )+7560 B \sin \left (2 c+\frac {3 d x}{2}\right )-5600 B \sin \left (2 c+\frac {5 d x}{2}\right )+1680 B \sin \left (3 c+\frac {5 d x}{2}\right )-1040 B \sin \left (3 c+\frac {7 d x}{2}\right )+4410 B d x \cos \left (c+\frac {3 d x}{2}\right )+4410 B d x \cos \left (2 c+\frac {3 d x}{2}\right )+1470 B d x \cos \left (2 c+\frac {5 d x}{2}\right )+1470 B d x \cos \left (3 c+\frac {5 d x}{2}\right )+210 B d x \cos \left (3 c+\frac {7 d x}{2}\right )+210 B d x \cos \left (4 c+\frac {7 d x}{2}\right )-19880 B \sin \left (\frac {d x}{2}\right )\right )}{1680 a^4 d (\cos (c+d x)+1)^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 223, normalized size = 1.34 \[ -\frac {105 \, {\left (4 \, A - B\right )} d x \cos \left (d x + c\right )^{4} + 420 \, {\left (4 \, A - B\right )} d x \cos \left (d x + c\right )^{3} + 630 \, {\left (4 \, A - B\right )} d x \cos \left (d x + c\right )^{2} + 420 \, {\left (4 \, A - B\right )} d x \cos \left (d x + c\right ) + 105 \, {\left (4 \, A - B\right )} d x - {\left (105 \, A \cos \left (d x + c\right )^{4} + 4 \, {\left (296 \, A - 65 \, B\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (659 \, A - 155 \, B\right )} \cos \left (d x + c\right )^{2} + {\left (2236 \, A - 535 \, B\right )} \cos \left (d x + c\right ) + 664 \, A - 160 \, B\right )} \sin \left (d x + c\right )}{105 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.70, size = 190, normalized size = 1.14 \[ -\frac {\frac {840 \, {\left (d x + c\right )} {\left (4 \, A - B\right )}}{a^{4}} - \frac {1680 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a^{4}} + \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 147 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 105 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 805 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 385 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5145 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1575 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.23, size = 229, normalized size = 1.38 \[ -\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{56 d \,a^{4}}+\frac {B \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 d \,a^{4}}+\frac {7 A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 d \,a^{4}}-\frac {B \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}-\frac {23 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{24 d \,a^{4}}+\frac {11 B \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d \,a^{4}}+\frac {49 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}-\frac {15 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}+\frac {2 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {8 A \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{4}}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{d \,a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.71, size = 271, normalized size = 1.63 \[ \frac {A {\left (\frac {1680 \, \sin \left (d x + c\right )}{{\left (a^{4} + \frac {a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {5145 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {805 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {147 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {6720 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )} - 5 \, B {\left (\frac {\frac {315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {77 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {336 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )}}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.06, size = 202, normalized size = 1.22 \[ \frac {\left (\frac {764\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{105}-\frac {52\,B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{21}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\left (\frac {16\,B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{21}-\frac {143\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{105}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (\frac {8\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{35}-\frac {5\,B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{28}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\frac {A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{56}+\frac {B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{56}}{a^4\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}-\frac {4\,A\,d\,x-B\,d\,x}{a^4\,d}+\frac {2\,A\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a^4\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {A \cos {\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {B \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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