Optimal. Leaf size=438 \[ \frac {1}{16} \left (5 a^4 B+36 a^2 b^2 B+8 b^4 B+8 a b^3 (3 A+4 C)+4 a^3 b (5 A+6 C)\right ) x+\frac {\left (336 a^3 b B+371 a b^3 B+12 a^4 (6 A+7 C)+b^4 (74 A+105 C)+3 a^2 b^2 (162 A+203 C)\right ) \sin (c+d x)}{105 d}+\frac {\left (5 a^4 B+36 a^2 b^2 B+8 b^4 B+8 a b^3 (3 A+4 C)+4 a^3 b (5 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a \left (24 A b^3+175 a^3 B+336 a b^2 B+a^2 (412 A b+504 b C)\right ) \cos ^3(c+d x) \sin (c+d x)}{840 d}+\frac {\left (4 A b^2+21 a b B+2 a^2 (6 A+7 C)\right ) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{70 d}+\frac {(4 A b+7 a B) \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{42 d}+\frac {A \cos ^6(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{7 d}-\frac {\left (4 A b^4+112 a^3 b B+91 a b^3 B+4 a^4 (6 A+7 C)+3 a^2 b^2 (50 A+63 C)\right ) \sin ^3(c+d x)}{105 d} \]
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Rubi [A]
time = 0.96, antiderivative size = 438, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 7, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {4179, 4159,
4132, 2715, 8, 4129, 3092} \begin {gather*} \frac {\sin (c+d x) \cos ^4(c+d x) \left (2 a^2 (6 A+7 C)+21 a b B+4 A b^2\right ) (a+b \sec (c+d x))^2}{70 d}+\frac {a \sin (c+d x) \cos ^3(c+d x) \left (175 a^3 B+a^2 (412 A b+504 b C)+336 a b^2 B+24 A b^3\right )}{840 d}-\frac {\sin ^3(c+d x) \left (4 a^4 (6 A+7 C)+112 a^3 b B+3 a^2 b^2 (50 A+63 C)+91 a b^3 B+4 A b^4\right )}{105 d}+\frac {\sin (c+d x) \left (12 a^4 (6 A+7 C)+336 a^3 b B+3 a^2 b^2 (162 A+203 C)+371 a b^3 B+b^4 (74 A+105 C)\right )}{105 d}+\frac {\sin (c+d x) \cos (c+d x) \left (5 a^4 B+4 a^3 b (5 A+6 C)+36 a^2 b^2 B+8 a b^3 (3 A+4 C)+8 b^4 B\right )}{16 d}+\frac {1}{16} x \left (5 a^4 B+4 a^3 b (5 A+6 C)+36 a^2 b^2 B+8 a b^3 (3 A+4 C)+8 b^4 B\right )+\frac {(7 a B+4 A b) \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{42 d}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2715
Rule 3092
Rule 4129
Rule 4132
Rule 4159
Rule 4179
Rubi steps
\begin {align*} \int \cos ^7(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+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+7 a B+(6 a A+7 b B+7 a C) \sec (c+d x)+b (2 A+7 C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {(4 A b+7 a B) \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{42 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 (3 \left (4 A b^2+21 a b B+2 a^2 (6 A+7 C)\right )+\left (68 a A b+35 a^2 B+42 b^2 B+84 a b C\right ) \sec (c+d x)+2 b (10 A b+7 a B+21 b C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {\left (4 A b^2+21 a b 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)}{70 d}+\frac {(4 A b+7 a B) \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{42 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 (24 A b^3+175 a^3 B+336 a b^2 B+a^2 (412 A b+504 b C)+\left (497 a^2 b B+210 b^3 B+24 a^3 (6 A+7 C)+2 a b^2 (244 A+315 C)\right ) \sec (c+d x)+2 b \left (98 a b B+6 a^2 (6 A+7 C)+b^2 (62 A+105 C)\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {a \left (24 A b^3+175 a^3 B+336 a b^2 B+a^2 (412 A b+504 b C)\right ) \cos ^3(c+d x) \sin (c+d x)}{840 d}+\frac {\left (4 A b^2+21 a b 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)}{70 d}+\frac {(4 A b+7 a B) \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{42 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+112 a^3 b B+91 a b^3 B+4 a^4 (6 A+7 C)+3 a^2 b^2 (50 A+63 C)\right )-105 \left (5 a^4 B+36 a^2 b^2 B+8 b^4 B+8 a b^3 (3 A+4 C)+4 a^3 b (5 A+6 C)\right ) \sec (c+d x)-8 b^2 \left (98 a b B+6 a^2 (6 A+7 C)+b^2 (62 A+105 C)\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {a \left (24 A b^3+175 a^3 B+336 a b^2 B+a^2 (412 A b+504 b C)\right ) \cos ^3(c+d x) \sin (c+d x)}{840 d}+\frac {\left (4 A b^2+21 a b 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)}{70 d}+\frac {(4 A b+7 a B) \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{42 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+112 a^3 b B+91 a b^3 B+4 a^4 (6 A+7 C)+3 a^2 b^2 (50 A+63 C)\right )-8 b^2 \left (98 a b B+6 a^2 (6 A+7 C)+b^2 (62 A+105 C)\right ) \sec ^2(c+d x)\right ) \, dx-\frac {1}{8} \left (-5 a^4 B-36 a^2 b^2 B-8 b^4 B-8 a b^3 (3 A+4 C)-4 a^3 b (5 A+6 C)\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac {\left (5 a^4 B+36 a^2 b^2 B+8 b^4 B+8 a b^3 (3 A+4 C)+4 a^3 b (5 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a \left (24 A b^3+175 a^3 B+336 a b^2 B+a^2 (412 A b+504 b C)\right ) \cos ^3(c+d x) \sin (c+d x)}{840 d}+\frac {\left (4 A b^2+21 a b 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)}{70 d}+\frac {(4 A b+7 a B) \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{42 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 (98 a b B+6 a^2 (6 A+7 C)+b^2 (62 A+105 C)\right )-24 \left (4 A b^4+112 a^3 b B+91 a b^3 B+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}{16} \left (-5 a^4 B-36 a^2 b^2 B-8 b^4 B-8 a b^3 (3 A+4 C)-4 a^3 b (5 A+6 C)\right ) \int 1 \, dx\\ &=\frac {1}{16} \left (5 a^4 B+36 a^2 b^2 B+8 b^4 B+8 a b^3 (3 A+4 C)+4 a^3 b (5 A+6 C)\right ) x+\frac {\left (5 a^4 B+36 a^2 b^2 B+8 b^4 B+8 a b^3 (3 A+4 C)+4 a^3 b (5 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a \left (24 A b^3+175 a^3 B+336 a b^2 B+a^2 (412 A b+504 b C)\right ) \cos ^3(c+d x) \sin (c+d x)}{840 d}+\frac {\left (4 A b^2+21 a b 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)}{70 d}+\frac {(4 A b+7 a B) \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{42 d}+\frac {A \cos ^6(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{7 d}+\frac {\text {Subst}\left (\int \left (-24 \left (4 A b^4+112 a^3 b B+91 a b^3 B+4 a^4 (6 A+7 C)+3 a^2 b^2 (50 A+63 C)\right )-8 b^2 \left (98 a b B+6 a^2 (6 A+7 C)+b^2 (62 A+105 C)\right )+24 \left (4 A b^4+112 a^3 b B+91 a b^3 B+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}{16} \left (5 a^4 B+36 a^2 b^2 B+8 b^4 B+8 a b^3 (3 A+4 C)+4 a^3 b (5 A+6 C)\right ) x+\frac {\left (336 a^3 b B+371 a b^3 B+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 {\left (5 a^4 B+36 a^2 b^2 B+8 b^4 B+8 a b^3 (3 A+4 C)+4 a^3 b (5 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a \left (24 A b^3+175 a^3 B+336 a b^2 B+a^2 (412 A b+504 b C)\right ) \cos ^3(c+d x) \sin (c+d x)}{840 d}+\frac {\left (4 A b^2+21 a b 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)}{70 d}+\frac {(4 A b+7 a B) \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{42 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+112 a^3 b B+91 a b^3 B+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 = 1.47, size = 528, normalized size = 1.21 \begin {gather*} \frac {8400 a^3 A b c+10080 a A b^3 c+2100 a^4 B c+15120 a^2 b^2 B c+3360 b^4 B c+10080 a^3 b c C+13440 a b^3 c C+8400 a^3 A b d x+10080 a A b^3 d x+2100 a^4 B d x+15120 a^2 b^2 B d x+3360 b^4 B d x+10080 a^3 b C d x+13440 a b^3 C d x+105 \left (160 a^3 b B+192 a b^3 B+16 b^4 (3 A+4 C)+48 a^2 b^2 (5 A+6 C)+5 a^4 (7 A+8 C)\right ) \sin (c+d x)+105 \left (15 a^4 B+96 a^2 b^2 B+16 b^4 B+64 a b^3 (A+C)+a^3 (60 A b+64 b C)\right ) \sin (2 (c+d x))+735 a^4 A \sin (3 (c+d x))+4200 a^2 A b^2 \sin (3 (c+d x))+560 A b^4 \sin (3 (c+d x))+2800 a^3 b B \sin (3 (c+d x))+2240 a b^3 B \sin (3 (c+d x))+700 a^4 C \sin (3 (c+d x))+3360 a^2 b^2 C \sin (3 (c+d x))+1260 a^3 A b \sin (4 (c+d x))+840 a A b^3 \sin (4 (c+d x))+315 a^4 B \sin (4 (c+d x))+1260 a^2 b^2 B \sin (4 (c+d x))+840 a^3 b C \sin (4 (c+d x))+147 a^4 A \sin (5 (c+d x))+504 a^2 A b^2 \sin (5 (c+d x))+336 a^3 b B \sin (5 (c+d x))+84 a^4 C \sin (5 (c+d x))+140 a^3 A b \sin (6 (c+d x))+35 a^4 B \sin (6 (c+d x))+15 a^4 A \sin (7 (c+d x))}{6720 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 505, normalized size = 1.15 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 498, normalized size = 1.14 \begin {gather*} -\frac {192 \, {\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} + 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 )} B a^{4} - 448 \, {\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} + 140 \, {\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 - 1792 \, {\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 )} B a^{3} b - 840 \, {\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 - 2688 \, {\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} - 1260 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} b^{2} + 13440 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{2} b^{2} - 840 \, {\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} + 8960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a b^{3} - 6720 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a b^{3} + 2240 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A b^{4} - 1680 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B b^{4} - 6720 \, C b^{4} \sin \left (d x + c\right )}{6720 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.74, size = 354, normalized size = 0.81 \begin {gather*} \frac {105 \, {\left (5 \, B a^{4} + 4 \, {\left (5 \, A + 6 \, C\right )} a^{3} b + 36 \, B a^{2} b^{2} + 8 \, {\left (3 \, A + 4 \, C\right )} a b^{3} + 8 \, B b^{4}\right )} d x + {\left (240 \, A a^{4} \cos \left (d x + c\right )^{6} + 280 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )^{5} + 128 \, {\left (6 \, A + 7 \, C\right )} a^{4} + 3584 \, B a^{3} b + 1344 \, {\left (4 \, A + 5 \, C\right )} a^{2} b^{2} + 4480 \, B a b^{3} + 560 \, {\left (2 \, A + 3 \, C\right )} b^{4} + 48 \, {\left ({\left (6 \, A + 7 \, C\right )} a^{4} + 28 \, B a^{3} b + 42 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{4} + 70 \, {\left (5 \, B a^{4} + 4 \, {\left (5 \, A + 6 \, C\right )} a^{3} b + 36 \, B a^{2} b^{2} + 24 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} + 16 \, {\left (4 \, {\left (6 \, A + 7 \, C\right )} a^{4} + 112 \, B a^{3} b + 42 \, {\left (4 \, A + 5 \, C\right )} a^{2} b^{2} + 140 \, B a b^{3} + 35 \, A b^{4}\right )} \cos \left (d x + c\right )^{2} + 105 \, {\left (5 \, B a^{4} + 4 \, {\left (5 \, A + 6 \, C\right )} a^{3} b + 36 \, B a^{2} b^{2} + 8 \, {\left (3 \, A + 4 \, C\right )} a b^{3} + 8 \, B b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1680 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1815 vs.
\(2 (423) = 846\).
time = 0.54, size = 1815, normalized size = 4.14 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 9.48, size = 675, normalized size = 1.54 \begin {gather*} \frac {5\,B\,a^4\,x}{16}+\frac {B\,b^4\,x}{2}+\frac {3\,A\,a\,b^3\,x}{2}+\frac {5\,A\,a^3\,b\,x}{4}+2\,C\,a\,b^3\,x+\frac {3\,C\,a^3\,b\,x}{2}+\frac {35\,A\,a^4\,\sin \left (c+d\,x\right )}{64\,d}+\frac {3\,A\,b^4\,\sin \left (c+d\,x\right )}{4\,d}+\frac {5\,C\,a^4\,\sin \left (c+d\,x\right )}{8\,d}+\frac {C\,b^4\,\sin \left (c+d\,x\right )}{d}+\frac {9\,B\,a^2\,b^2\,x}{4}+\frac {7\,A\,a^4\,\sin \left (3\,c+3\,d\,x\right )}{64\,d}+\frac {7\,A\,a^4\,\sin \left (5\,c+5\,d\,x\right )}{320\,d}+\frac {A\,a^4\,\sin \left (7\,c+7\,d\,x\right )}{448\,d}+\frac {15\,B\,a^4\,\sin \left (2\,c+2\,d\,x\right )}{64\,d}+\frac {A\,b^4\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {3\,B\,a^4\,\sin \left (4\,c+4\,d\,x\right )}{64\,d}+\frac {B\,a^4\,\sin \left (6\,c+6\,d\,x\right )}{192\,d}+\frac {B\,b^4\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {5\,C\,a^4\,\sin \left (3\,c+3\,d\,x\right )}{48\,d}+\frac {C\,a^4\,\sin \left (5\,c+5\,d\,x\right )}{80\,d}+\frac {A\,a\,b^3\,\sin \left (2\,c+2\,d\,x\right )}{d}+\frac {15\,A\,a^3\,b\,\sin \left (2\,c+2\,d\,x\right )}{16\,d}+\frac {A\,a\,b^3\,\sin \left (4\,c+4\,d\,x\right )}{8\,d}+\frac {3\,A\,a^3\,b\,\sin \left (4\,c+4\,d\,x\right )}{16\,d}+\frac {A\,a^3\,b\,\sin \left (6\,c+6\,d\,x\right )}{48\,d}+\frac {15\,A\,a^2\,b^2\,\sin \left (c+d\,x\right )}{4\,d}+\frac {B\,a\,b^3\,\sin \left (3\,c+3\,d\,x\right )}{3\,d}+\frac {5\,B\,a^3\,b\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {B\,a^3\,b\,\sin \left (5\,c+5\,d\,x\right )}{20\,d}+\frac {C\,a\,b^3\,\sin \left (2\,c+2\,d\,x\right )}{d}+\frac {C\,a^3\,b\,\sin \left (2\,c+2\,d\,x\right )}{d}+\frac {C\,a^3\,b\,\sin \left (4\,c+4\,d\,x\right )}{8\,d}+\frac {9\,C\,a^2\,b^2\,\sin \left (c+d\,x\right )}{2\,d}+\frac {5\,A\,a^2\,b^2\,\sin \left (3\,c+3\,d\,x\right )}{8\,d}+\frac {3\,A\,a^2\,b^2\,\sin \left (5\,c+5\,d\,x\right )}{40\,d}+\frac {3\,B\,a^2\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {3\,B\,a^2\,b^2\,\sin \left (4\,c+4\,d\,x\right )}{16\,d}+\frac {C\,a^2\,b^2\,\sin \left (3\,c+3\,d\,x\right )}{2\,d}+\frac {3\,B\,a\,b^3\,\sin \left (c+d\,x\right )}{d}+\frac {5\,B\,a^3\,b\,\sin \left (c+d\,x\right )}{2\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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