Optimal. Leaf size=116 \[ -\frac {a^3 (4 B+3 C) \sin ^3(c+d x)}{12 d}+\frac {a^3 (4 B+3 C) \sin (c+d x)}{d}+\frac {3 a^3 (4 B+3 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {5}{8} a^3 x (4 B+3 C)+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d} \]
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Rubi [A] time = 0.17, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {3029, 2751, 2645, 2637, 2635, 8, 2633} \[ -\frac {a^3 (4 B+3 C) \sin ^3(c+d x)}{12 d}+\frac {a^3 (4 B+3 C) \sin (c+d x)}{d}+\frac {3 a^3 (4 B+3 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {5}{8} a^3 x (4 B+3 C)+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 2637
Rule 2645
Rule 2751
Rule 3029
Rubi steps
\begin {align*} \int (a+a \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx &=\int (a+a \cos (c+d x))^3 (B+C \cos (c+d x)) \, dx\\ &=\frac {C (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {1}{4} (4 B+3 C) \int (a+a \cos (c+d x))^3 \, dx\\ &=\frac {C (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {1}{4} (4 B+3 C) \int \left (a^3+3 a^3 \cos (c+d x)+3 a^3 \cos ^2(c+d x)+a^3 \cos ^3(c+d x)\right ) \, dx\\ &=\frac {1}{4} a^3 (4 B+3 C) x+\frac {C (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {1}{4} \left (a^3 (4 B+3 C)\right ) \int \cos ^3(c+d x) \, dx+\frac {1}{4} \left (3 a^3 (4 B+3 C)\right ) \int \cos (c+d x) \, dx+\frac {1}{4} \left (3 a^3 (4 B+3 C)\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac {1}{4} a^3 (4 B+3 C) x+\frac {3 a^3 (4 B+3 C) \sin (c+d x)}{4 d}+\frac {3 a^3 (4 B+3 C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {C (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {1}{8} \left (3 a^3 (4 B+3 C)\right ) \int 1 \, dx-\frac {\left (a^3 (4 B+3 C)\right ) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{4 d}\\ &=\frac {5}{8} a^3 (4 B+3 C) x+\frac {a^3 (4 B+3 C) \sin (c+d x)}{d}+\frac {3 a^3 (4 B+3 C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {C (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}-\frac {a^3 (4 B+3 C) \sin ^3(c+d x)}{12 d}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 86, normalized size = 0.74 \[ \frac {a^3 (24 (15 B+13 C) \sin (c+d x)+24 (3 B+4 C) \sin (2 (c+d x))+8 B \sin (3 (c+d x))+240 B d x+24 C \sin (3 (c+d x))+3 C \sin (4 (c+d x))+180 C d x)}{96 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 90, normalized size = 0.78 \[ \frac {15 \, {\left (4 \, B + 3 \, C\right )} a^{3} d x + {\left (6 \, C a^{3} \cos \left (d x + c\right )^{3} + 8 \, {\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 9 \, {\left (4 \, B + 5 \, C\right )} a^{3} \cos \left (d x + c\right ) + 8 \, {\left (11 \, B + 9 \, C\right )} a^{3}\right )} \sin \left (d x + c\right )}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 176, normalized size = 1.52 \[ \frac {15 \, {\left (4 \, B a^{3} + 3 \, C a^{3}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (60 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 45 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 220 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 165 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 292 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 219 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 132 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 147 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 176, normalized size = 1.52 \[ \frac {C \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a^{3} B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+C \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+3 a^{3} B \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 C \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 a^{3} B \sin \left (d x +c \right )+C \,a^{3} \sin \left (d x +c \right )+B \left (d x +c \right ) a^{3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 167, normalized size = 1.44 \[ -\frac {32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{3} - 72 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} - 96 \, {\left (d x + c\right )} B a^{3} + 96 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{3} - 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} - 72 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} - 288 \, B a^{3} \sin \left (d x + c\right ) - 96 \, C a^{3} \sin \left (d x + c\right )}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.16, size = 134, normalized size = 1.16 \[ \frac {5\,B\,a^3\,x}{2}+\frac {15\,C\,a^3\,x}{8}+\frac {15\,B\,a^3\,\sin \left (c+d\,x\right )}{4\,d}+\frac {13\,C\,a^3\,\sin \left (c+d\,x\right )}{4\,d}+\frac {3\,B\,a^3\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {B\,a^3\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {C\,a^3\,\sin \left (2\,c+2\,d\,x\right )}{d}+\frac {C\,a^3\,\sin \left (3\,c+3\,d\,x\right )}{4\,d}+\frac {C\,a^3\,\sin \left (4\,c+4\,d\,x\right )}{32\,d} \]
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 \[ a^{3} \left (\int B \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 B \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 B \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int B \cos ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int C \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 C \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 C \cos ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int C \cos ^{5}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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