Optimal. Leaf size=339 \[ \frac {a b \left (a^2 (103 A+126 C)+6 A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{210 d}+\frac {a b \left (a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right ) \sin (c+d x) \cos (c+d x)}{4 d}+\frac {\left (a^2 (6 A+7 C)+2 A b^2\right ) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{35 d}+\frac {1}{4} a b x \left (a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right )-\frac {\left (4 a^4 (6 A+7 C)+3 a^2 b^2 (50 A+63 C)+4 A b^4\right ) \sin ^3(c+d x)}{105 d}+\frac {\left (12 a^4 (6 A+7 C)+3 a^2 b^2 (162 A+203 C)+b^4 (74 A+105 C)\right ) \sin (c+d x)}{105 d}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d}+\frac {2 A b \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{21 d} \]
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Rubi [A] time = 1.15, antiderivative size = 339, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {4095, 4094, 4074, 4047, 2635, 8, 4044, 3013} \[ -\frac {\left (3 a^2 b^2 (50 A+63 C)+4 a^4 (6 A+7 C)+4 A b^4\right ) \sin ^3(c+d x)}{105 d}+\frac {\left (3 a^2 b^2 (162 A+203 C)+12 a^4 (6 A+7 C)+b^4 (74 A+105 C)\right ) \sin (c+d x)}{105 d}+\frac {a b \left (a^2 (103 A+126 C)+6 A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{210 d}+\frac {a b \left (a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right ) \sin (c+d x) \cos (c+d x)}{4 d}+\frac {\left (a^2 (6 A+7 C)+2 A b^2\right ) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{35 d}+\frac {1}{4} a b x \left (a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right )+\frac {A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d}+\frac {2 A b \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{21 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 3013
Rule 4044
Rule 4047
Rule 4074
Rule 4094
Rule 4095
Rubi steps
\begin {align*} \int \cos ^7(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {A \cos ^6(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{7 d}+\frac {1}{7} \int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left (4 A b+a (6 A+7 C) \sec (c+d x)+b (2 A+7 C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {2 A b \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{21 d}+\frac {A \cos ^6(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{7 d}+\frac {1}{42} \int \cos ^5(c+d x) (a+b \sec (c+d x))^2 \left (6 \left (2 A b^2+a^2 (6 A+7 C)\right )+4 a b (17 A+21 C) \sec (c+d x)+2 b^2 (10 A+21 C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {\left (2 A b^2+a^2 (6 A+7 C)\right ) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{35 d}+\frac {2 A b \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{21 d}+\frac {A \cos ^6(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{7 d}+\frac {1}{210} \int \cos ^4(c+d x) (a+b \sec (c+d x)) \left (4 b \left (6 A b^2+a^2 (103 A+126 C)\right )+2 a \left (12 a^2 (6 A+7 C)+b^2 (244 A+315 C)\right ) \sec (c+d x)+2 b \left (6 a^2 (6 A+7 C)+b^2 (62 A+105 C)\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {a b \left (6 A b^2+a^2 (103 A+126 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{210 d}+\frac {\left (2 A b^2+a^2 (6 A+7 C)\right ) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{35 d}+\frac {2 A b \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{21 d}+\frac {A \cos ^6(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{7 d}-\frac {1}{840} \int \cos ^3(c+d x) \left (-24 \left (4 A b^4+4 a^4 (6 A+7 C)+3 a^2 b^2 (50 A+63 C)\right )-420 a b \left (2 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right ) \sec (c+d x)-8 b^2 \left (6 a^2 (6 A+7 C)+b^2 (62 A+105 C)\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {a b \left (6 A b^2+a^2 (103 A+126 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{210 d}+\frac {\left (2 A b^2+a^2 (6 A+7 C)\right ) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{35 d}+\frac {2 A b \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{21 d}+\frac {A \cos ^6(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{7 d}-\frac {1}{840} \int \cos ^3(c+d x) \left (-24 \left (4 A b^4+4 a^4 (6 A+7 C)+3 a^2 b^2 (50 A+63 C)\right )-8 b^2 \left (6 a^2 (6 A+7 C)+b^2 (62 A+105 C)\right ) \sec ^2(c+d x)\right ) \, dx+\frac {1}{2} \left (a b \left (2 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right )\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac {a b \left (2 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{4 d}+\frac {a b \left (6 A b^2+a^2 (103 A+126 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{210 d}+\frac {\left (2 A b^2+a^2 (6 A+7 C)\right ) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{35 d}+\frac {2 A b \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{21 d}+\frac {A \cos ^6(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{7 d}-\frac {1}{840} \int \cos (c+d x) \left (-8 b^2 \left (6 a^2 (6 A+7 C)+b^2 (62 A+105 C)\right )-24 \left (4 A b^4+4 a^4 (6 A+7 C)+3 a^2 b^2 (50 A+63 C)\right ) \cos ^2(c+d x)\right ) \, dx+\frac {1}{4} \left (a b \left (2 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right )\right ) \int 1 \, dx\\ &=\frac {1}{4} a b \left (2 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right ) x+\frac {a b \left (2 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{4 d}+\frac {a b \left (6 A b^2+a^2 (103 A+126 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{210 d}+\frac {\left (2 A b^2+a^2 (6 A+7 C)\right ) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{35 d}+\frac {2 A b \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{21 d}+\frac {A \cos ^6(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{7 d}+\frac {\operatorname {Subst}\left (\int \left (-24 \left (4 A b^4+4 a^4 (6 A+7 C)+3 a^2 b^2 (50 A+63 C)\right )-8 b^2 \left (6 a^2 (6 A+7 C)+b^2 (62 A+105 C)\right )+24 \left (4 A b^4+4 a^4 (6 A+7 C)+3 a^2 b^2 (50 A+63 C)\right ) x^2\right ) \, dx,x,-\sin (c+d x)\right )}{840 d}\\ &=\frac {1}{4} a b \left (2 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right ) x+\frac {\left (12 a^4 (6 A+7 C)+b^4 (74 A+105 C)+3 a^2 b^2 (162 A+203 C)\right ) \sin (c+d x)}{105 d}+\frac {a b \left (2 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{4 d}+\frac {a b \left (6 A b^2+a^2 (103 A+126 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{210 d}+\frac {\left (2 A b^2+a^2 (6 A+7 C)\right ) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{35 d}+\frac {2 A b \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{21 d}+\frac {A \cos ^6(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{7 d}-\frac {\left (4 A b^4+4 a^4 (6 A+7 C)+3 a^2 b^2 (50 A+63 C)\right ) \sin ^3(c+d x)}{105 d}\\ \end {align*}
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Mathematica [A] time = 0.86, size = 351, normalized size = 1.04 \[ \frac {735 a^4 A \sin (3 (c+d x))+147 a^4 A \sin (5 (c+d x))+15 a^4 A \sin (7 (c+d x))+700 a^4 C \sin (3 (c+d x))+84 a^4 C \sin (5 (c+d x))+1260 a^3 A b \sin (4 (c+d x))+140 a^3 A b \sin (6 (c+d x))+8400 a^3 A b c+8400 a^3 A b d x+840 a^3 b C \sin (4 (c+d x))+10080 a^3 b c C+10080 a^3 b C d x+420 a b \left (a^2 (15 A+16 C)+16 b^2 (A+C)\right ) \sin (2 (c+d x))+4200 a^2 A b^2 \sin (3 (c+d x))+504 a^2 A b^2 \sin (5 (c+d x))+3360 a^2 b^2 C \sin (3 (c+d x))+105 \left (5 a^4 (7 A+8 C)+48 a^2 b^2 (5 A+6 C)+16 b^4 (3 A+4 C)\right ) \sin (c+d x)+840 a A b^3 \sin (4 (c+d x))+10080 a A b^3 c+10080 a A b^3 d x+13440 a b^3 c C+13440 a b^3 C d x+560 A b^4 \sin (3 (c+d x))}{6720 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 251, normalized size = 0.74 \[ \frac {105 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{3} b + 2 \, {\left (3 \, A + 4 \, C\right )} a b^{3}\right )} d x + {\left (60 \, A a^{4} \cos \left (d x + c\right )^{6} + 280 \, A a^{3} b \cos \left (d x + c\right )^{5} + 32 \, {\left (6 \, A + 7 \, C\right )} a^{4} + 336 \, {\left (4 \, A + 5 \, C\right )} a^{2} b^{2} + 140 \, {\left (2 \, A + 3 \, C\right )} b^{4} + 12 \, {\left ({\left (6 \, A + 7 \, C\right )} a^{4} + 42 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{4} + 70 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{3} b + 6 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (4 \, {\left (6 \, A + 7 \, C\right )} a^{4} + 42 \, {\left (4 \, A + 5 \, C\right )} a^{2} b^{2} + 35 \, A b^{4}\right )} \cos \left (d x + c\right )^{2} + 105 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{3} b + 2 \, {\left (3 \, A + 4 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{420 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.33, size = 1228, normalized size = 3.62 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.65, size = 332, normalized size = 0.98 \[ \frac {\frac {A \,a^{4} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}+\frac {a^{4} C \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+4 A \,a^{3} b \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+4 a^{3} b C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {6 A \,a^{2} b^{2} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+2 C \,a^{2} b^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+4 a A \,b^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 C a \,b^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {A \,b^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+C \,b^{4} \sin \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 329, normalized size = 0.97 \[ -\frac {48 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} A a^{4} - 112 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a^{4} + 35 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} b - 210 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} b - 672 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{2} b^{2} + 3360 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{2} b^{2} - 210 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a b^{3} - 1680 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a b^{3} + 560 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A b^{4} - 1680 \, C b^{4} \sin \left (d x + c\right )}{1680 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.80, size = 751, normalized size = 2.22 \[ \frac {\left (2\,A\,a^4+2\,A\,b^4+2\,C\,a^4+2\,C\,b^4+12\,A\,a^2\,b^2+12\,C\,a^2\,b^2-5\,A\,a\,b^3-\frac {11\,A\,a^3\,b}{2}-4\,C\,a\,b^3-5\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (4\,A\,a^4+\frac {28\,A\,b^4}{3}+\frac {20\,C\,a^4}{3}+12\,C\,b^4+40\,A\,a^2\,b^2+56\,C\,a^2\,b^2-12\,A\,a\,b^3-\frac {14\,A\,a^3\,b}{3}-16\,C\,a\,b^3-12\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {86\,A\,a^4}{5}+\frac {58\,A\,b^4}{3}+\frac {226\,C\,a^4}{15}+30\,C\,b^4+\frac {452\,A\,a^2\,b^2}{5}+116\,C\,a^2\,b^2-9\,A\,a\,b^3-\frac {85\,A\,a^3\,b}{6}-20\,C\,a\,b^3-9\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {424\,A\,a^4}{35}+24\,A\,b^4+\frac {104\,C\,a^4}{5}+40\,C\,b^4+\frac {624\,A\,a^2\,b^2}{5}+144\,C\,a^2\,b^2\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {86\,A\,a^4}{5}+\frac {58\,A\,b^4}{3}+\frac {226\,C\,a^4}{15}+30\,C\,b^4+\frac {452\,A\,a^2\,b^2}{5}+116\,C\,a^2\,b^2+9\,A\,a\,b^3+\frac {85\,A\,a^3\,b}{6}+20\,C\,a\,b^3+9\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (4\,A\,a^4+\frac {28\,A\,b^4}{3}+\frac {20\,C\,a^4}{3}+12\,C\,b^4+40\,A\,a^2\,b^2+56\,C\,a^2\,b^2+12\,A\,a\,b^3+\frac {14\,A\,a^3\,b}{3}+16\,C\,a\,b^3+12\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,A\,a^4+2\,A\,b^4+2\,C\,a^4+2\,C\,b^4+12\,A\,a^2\,b^2+12\,C\,a^2\,b^2+5\,A\,a\,b^3+\frac {11\,A\,a^3\,b}{2}+4\,C\,a\,b^3+5\,C\,a^3\,b\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a\,b\,\mathrm {atan}\left (\frac {a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (5\,A\,a^2+6\,A\,b^2+6\,C\,a^2+8\,C\,b^2\right )}{2\,\left (3\,A\,a\,b^3+\frac {5\,A\,a^3\,b}{2}+4\,C\,a\,b^3+3\,C\,a^3\,b\right )}\right )\,\left (5\,A\,a^2+6\,A\,b^2+6\,C\,a^2+8\,C\,b^2\right )}{2\,d} \]
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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