Optimal. Leaf size=96 \[ \frac {a (6 A-B) \tan ^5(c+d x)}{35 d}+\frac {2 a (6 A-B) \tan ^3(c+d x)}{21 d}+\frac {a (6 A-B) \tan (c+d x)}{7 d}+\frac {(A+B) \sec ^7(c+d x) (a \sin (c+d x)+a)}{7 d} \]
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Rubi [A] time = 0.08, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2855, 3767} \[ \frac {a (6 A-B) \tan ^5(c+d x)}{35 d}+\frac {2 a (6 A-B) \tan ^3(c+d x)}{21 d}+\frac {a (6 A-B) \tan (c+d x)}{7 d}+\frac {(A+B) \sec ^7(c+d x) (a \sin (c+d x)+a)}{7 d} \]
Antiderivative was successfully verified.
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Rule 2855
Rule 3767
Rubi steps
\begin {align*} \int \sec ^8(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx &=\frac {(A+B) \sec ^7(c+d x) (a+a \sin (c+d x))}{7 d}+\frac {1}{7} (a (6 A-B)) \int \sec ^6(c+d x) \, dx\\ &=\frac {(A+B) \sec ^7(c+d x) (a+a \sin (c+d x))}{7 d}-\frac {(a (6 A-B)) \operatorname {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (c+d x)\right )}{7 d}\\ &=\frac {(A+B) \sec ^7(c+d x) (a+a \sin (c+d x))}{7 d}+\frac {a (6 A-B) \tan (c+d x)}{7 d}+\frac {2 a (6 A-B) \tan ^3(c+d x)}{21 d}+\frac {a (6 A-B) \tan ^5(c+d x)}{35 d}\\ \end {align*}
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Mathematica [B] time = 2.03, size = 315, normalized size = 3.28 \[ \frac {a \sec (c) (-1500 (A+B) \cos (c+d x)+375 A \sin (2 (c+d x))+300 A \sin (4 (c+d x))+75 A \sin (6 (c+d x))+7680 A \sin (2 c+3 d x)+1536 A \sin (4 c+5 d x)-750 A \cos (3 (c+d x))-150 A \cos (5 (c+d x))+3840 A \cos (c+2 d x)+3072 A \cos (3 c+4 d x)+768 A \cos (5 c+6 d x)+15360 A \sin (d x)+375 B \sin (2 (c+d x))+300 B \sin (4 (c+d x))+75 B \sin (6 (c+d x))-1280 B \sin (2 c+3 d x)-256 B \sin (4 c+5 d x)-750 B \cos (3 (c+d x))-150 B \cos (5 (c+d x))-640 B \cos (c+2 d x)-512 B \cos (3 c+4 d x)-128 B \cos (5 c+6 d x)+8960 B \cos (c)-2560 B \sin (d x))}{53760 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^7 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 149, normalized size = 1.55 \[ -\frac {8 \, {\left (6 \, A - B\right )} a \cos \left (d x + c\right )^{6} - 4 \, {\left (6 \, A - B\right )} a \cos \left (d x + c\right )^{4} - {\left (6 \, A - B\right )} a \cos \left (d x + c\right )^{2} - 3 \, {\left (A - 6 \, B\right )} a + {\left (8 \, {\left (6 \, A - B\right )} a \cos \left (d x + c\right )^{4} + 4 \, {\left (6 \, A - B\right )} a \cos \left (d x + c\right )^{2} + 3 \, {\left (6 \, A - B\right )} a\right )} \sin \left (d x + c\right )}{105 \, {\left (d \cos \left (d x + c\right )^{5} \sin \left (d x + c\right ) - d \cos \left (d x + c\right )^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 345, normalized size = 3.59 \[ -\frac {\frac {7 \, {\left (165 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 75 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 540 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 210 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 750 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 280 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 480 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 170 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 129 \, A a - 49 \, B a\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{5}} + \frac {2205 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 525 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 10080 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1470 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 21945 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2555 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 26460 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2240 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 18963 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1407 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 7476 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 434 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1383 \, A a + 137 \, B a}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{7}}}{1680 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.60, size = 130, normalized size = 1.35 \[ \frac {\frac {a A}{7 \cos \left (d x +c \right )^{7}}+a B \left (\frac {\sin ^{3}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \left (\sin ^{3}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{3}}\right )-a A \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (d x +c \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (d x +c \right )\right )}{35}\right ) \tan \left (d x +c \right )+\frac {a B}{7 \cos \left (d x +c \right )^{7}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 107, normalized size = 1.11 \[ \frac {3 \, {\left (5 \, \tan \left (d x + c\right )^{7} + 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 35 \, \tan \left (d x + c\right )\right )} A a + {\left (15 \, \tan \left (d x + c\right )^{7} + 42 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3}\right )} B a + \frac {15 \, A a}{\cos \left (d x + c\right )^{7}} + \frac {15 \, B a}{\cos \left (d x + c\right )^{7}}}{105 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.64, size = 320, normalized size = 3.33 \[ -\frac {a\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {15\,A\,\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{8}-\frac {75\,A\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{8}-\frac {105\,A\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{16}+\frac {9\,A\,\cos \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{16}-\frac {3\,A\,\cos \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{2}-\frac {35\,B\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}+\frac {65\,B\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{8}-\frac {55\,B\,\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{8}+\frac {35\,B\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{16}-\frac {19\,B\,\cos \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{16}+\frac {B\,\cos \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{4}-\frac {843\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}+\frac {363\,A\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{16}-\frac {651\,A\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{32}+\frac {171\,A\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{32}-\frac {111\,A\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{32}+\frac {15\,A\,\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{32}+\frac {53\,B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}+\frac {27\,B\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{16}+\frac {21\,B\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{32}+\frac {59\,B\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{32}+\frac {B\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{32}+\frac {15\,B\,\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{32}\right )}{3360\,d\,{\cos \left (\frac {c}{2}-\frac {\pi }{4}+\frac {d\,x}{2}\right )}^5\,{\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d\,x}{2}\right )}^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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