Optimal. Leaf size=154 \[ \frac {a^2 (7 A-2 B) \tan ^7(c+d x)}{63 d}+\frac {a^2 (7 A-2 B) \tan ^5(c+d x)}{15 d}+\frac {a^2 (7 A-2 B) \tan ^3(c+d x)}{9 d}+\frac {a^2 (7 A-2 B) \tan (c+d x)}{9 d}+\frac {a^2 (7 A-2 B) \sec ^7(c+d x)}{63 d}+\frac {(A+B) \sec ^9(c+d x) (a \sin (c+d x)+a)^2}{9 d} \]
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Rubi [A] time = 0.14, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {2855, 2669, 3767} \[ \frac {a^2 (7 A-2 B) \tan ^7(c+d x)}{63 d}+\frac {a^2 (7 A-2 B) \tan ^5(c+d x)}{15 d}+\frac {a^2 (7 A-2 B) \tan ^3(c+d x)}{9 d}+\frac {a^2 (7 A-2 B) \tan (c+d x)}{9 d}+\frac {a^2 (7 A-2 B) \sec ^7(c+d x)}{63 d}+\frac {(A+B) \sec ^9(c+d x) (a \sin (c+d x)+a)^2}{9 d} \]
Antiderivative was successfully verified.
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Rule 2669
Rule 2855
Rule 3767
Rubi steps
\begin {align*} \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx &=\frac {(A+B) \sec ^9(c+d x) (a+a \sin (c+d x))^2}{9 d}+\frac {1}{9} (a (7 A-2 B)) \int \sec ^8(c+d x) (a+a \sin (c+d x)) \, dx\\ &=\frac {a^2 (7 A-2 B) \sec ^7(c+d x)}{63 d}+\frac {(A+B) \sec ^9(c+d x) (a+a \sin (c+d x))^2}{9 d}+\frac {1}{9} \left (a^2 (7 A-2 B)\right ) \int \sec ^8(c+d x) \, dx\\ &=\frac {a^2 (7 A-2 B) \sec ^7(c+d x)}{63 d}+\frac {(A+B) \sec ^9(c+d x) (a+a \sin (c+d x))^2}{9 d}-\frac {\left (a^2 (7 A-2 B)\right ) \operatorname {Subst}\left (\int \left (1+3 x^2+3 x^4+x^6\right ) \, dx,x,-\tan (c+d x)\right )}{9 d}\\ &=\frac {a^2 (7 A-2 B) \sec ^7(c+d x)}{63 d}+\frac {(A+B) \sec ^9(c+d x) (a+a \sin (c+d x))^2}{9 d}+\frac {a^2 (7 A-2 B) \tan (c+d x)}{9 d}+\frac {a^2 (7 A-2 B) \tan ^3(c+d x)}{9 d}+\frac {a^2 (7 A-2 B) \tan ^5(c+d x)}{15 d}+\frac {a^2 (7 A-2 B) \tan ^7(c+d x)}{63 d}\\ \end {align*}
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Mathematica [A] time = 0.45, size = 156, normalized size = 1.01 \[ \frac {a^2 \left (16 (7 A-2 B) \tan ^9(c+d x)+5 (14 A+5 B) \sec ^9(c+d x)-105 (7 A-2 B) \tan ^3(c+d x) \sec ^6(c+d x)+126 (7 A-2 B) \tan ^5(c+d x) \sec ^4(c+d x)-72 (7 A-2 B) \tan ^7(c+d x) \sec ^2(c+d x)+315 A \tan (c+d x) \sec ^8(c+d x)+45 B \tan ^2(c+d x) \sec ^7(c+d x)\right )}{315 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 197, normalized size = 1.28 \[ -\frac {32 \, {\left (7 \, A - 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{6} - 16 \, {\left (7 \, A - 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{4} - 4 \, {\left (7 \, A - 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} - 7 \, {\left (2 \, A - 7 \, B\right )} a^{2} - {\left (16 \, {\left (7 \, A - 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{6} - 24 \, {\left (7 \, A - 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{4} - 10 \, {\left (7 \, A - 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} - 7 \, {\left (7 \, A - 2 \, B\right )} a^{2}\right )} \sin \left (d x + c\right )}{315 \, {\left (d \cos \left (d x + c\right )^{7} + 2 \, d \cos \left (d x + c\right )^{5} \sin \left (d x + c\right ) - 2 \, d \cos \left (d x + c\right )^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 461, normalized size = 2.99 \[ -\frac {\frac {21 \, {\left (435 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 225 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1470 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 690 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2060 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 940 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1330 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 590 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 353 \, A a^{2} - 163 \, B a^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{5}} + \frac {31185 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 4725 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 185220 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 11340 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 546840 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 15120 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 961380 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3780 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1101618 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 24318 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 828492 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 33852 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 404208 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 19368 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 116172 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6732 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 16373 \, A a^{2} - 223 \, B a^{2}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{9}}}{20160 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.71, size = 359, normalized size = 2.33 \[ \frac {a^{2} A \left (\frac {\sin ^{3}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{21 \cos \left (d x +c \right )^{7}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{5}}+\frac {16 \left (\sin ^{3}\left (d x +c \right )\right )}{315 \cos \left (d x +c \right )^{3}}\right )+B \,a^{2} \left (\frac {\sin ^{4}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {5 \left (\sin ^{4}\left (d x +c \right )\right )}{63 \cos \left (d x +c \right )^{7}}+\frac {\sin ^{4}\left (d x +c \right )}{21 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{63 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{63 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{63}\right )+\frac {2 a^{2} A}{9 \cos \left (d x +c \right )^{9}}+2 B \,a^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{21 \cos \left (d x +c \right )^{7}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{5}}+\frac {16 \left (\sin ^{3}\left (d x +c \right )\right )}{315 \cos \left (d x +c \right )^{3}}\right )-a^{2} A \left (-\frac {128}{315}-\frac {\left (\sec ^{8}\left (d x +c \right )\right )}{9}-\frac {8 \left (\sec ^{6}\left (d x +c \right )\right )}{63}-\frac {16 \left (\sec ^{4}\left (d x +c \right )\right )}{105}-\frac {64 \left (\sec ^{2}\left (d x +c \right )\right )}{315}\right ) \tan \left (d x +c \right )+\frac {B \,a^{2}}{9 \cos \left (d x +c \right )^{9}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 207, normalized size = 1.34 \[ \frac {{\left (35 \, \tan \left (d x + c\right )^{9} + 180 \, \tan \left (d x + c\right )^{7} + 378 \, \tan \left (d x + c\right )^{5} + 420 \, \tan \left (d x + c\right )^{3} + 315 \, \tan \left (d x + c\right )\right )} A a^{2} + {\left (35 \, \tan \left (d x + c\right )^{9} + 135 \, \tan \left (d x + c\right )^{7} + 189 \, \tan \left (d x + c\right )^{5} + 105 \, \tan \left (d x + c\right )^{3}\right )} A a^{2} + 2 \, {\left (35 \, \tan \left (d x + c\right )^{9} + 135 \, \tan \left (d x + c\right )^{7} + 189 \, \tan \left (d x + c\right )^{5} + 105 \, \tan \left (d x + c\right )^{3}\right )} B a^{2} - \frac {5 \, {\left (9 \, \cos \left (d x + c\right )^{2} - 7\right )} B a^{2}}{\cos \left (d x + c\right )^{9}} + \frac {70 \, A a^{2}}{\cos \left (d x + c\right )^{9}} + \frac {35 \, B a^{2}}{\cos \left (d x + c\right )^{9}}}{315 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.95, size = 370, normalized size = 2.40 \[ -\frac {a^2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {455\,A\,\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{32}-\frac {1575\,A\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{32}-35\,A\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )+7\,A\,\cos \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )-\frac {259\,A\,\cos \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{32}+\frac {35\,A\,\cos \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )}{32}-45\,B\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {1755\,B\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{64}-\frac {1115\,B\,\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{64}+10\,B\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )-2\,B\,\cos \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )+\frac {103\,B\,\cos \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{64}+\frac {25\,B\,\cos \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )}{64}-\frac {623\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}+77\,A\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )-\frac {441\,A\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{8}+\frac {175\,A\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{8}-\frac {35\,A\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{8}+\frac {21\,A\,\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{8}+\frac {7\,A\,\sin \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )}{4}+\frac {131\,B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {49\,B\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{8}+\frac {27\,B\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{16}+\frac {125\,B\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{16}-\frac {25\,B\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{16}+\frac {33\,B\,\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{16}-\frac {B\,\sin \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )}{2}\right )}{20160\,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 )}^9} \]
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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