Optimal. Leaf size=102 \[ \frac {2 a^2 (2 A+3 B) \sin (c+d x)}{3 d}+\frac {a^2 (2 A+3 B) \sin (c+d x) \cos (c+d x)}{6 d}+\frac {1}{2} a^2 x (2 A+3 B)+\frac {A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^2}{3 d} \]
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Rubi [A] time = 0.15, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {4013, 3788, 2637, 4045, 8} \[ \frac {2 a^2 (2 A+3 B) \sin (c+d x)}{3 d}+\frac {a^2 (2 A+3 B) \sin (c+d x) \cos (c+d x)}{6 d}+\frac {1}{2} a^2 x (2 A+3 B)+\frac {A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^2}{3 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2637
Rule 3788
Rule 4013
Rule 4045
Rubi steps
\begin {align*} \int \cos ^3(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx &=\frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{3 d}+\frac {1}{3} (2 A+3 B) \int \cos ^2(c+d x) (a+a \sec (c+d x))^2 \, dx\\ &=\frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{3 d}+\frac {1}{3} (2 A+3 B) \int \cos ^2(c+d x) \left (a^2+a^2 \sec ^2(c+d x)\right ) \, dx+\frac {1}{3} \left (2 a^2 (2 A+3 B)\right ) \int \cos (c+d x) \, dx\\ &=\frac {2 a^2 (2 A+3 B) \sin (c+d x)}{3 d}+\frac {a^2 (2 A+3 B) \cos (c+d x) \sin (c+d x)}{6 d}+\frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{3 d}+\frac {1}{2} \left (a^2 (2 A+3 B)\right ) \int 1 \, dx\\ &=\frac {1}{2} a^2 (2 A+3 B) x+\frac {2 a^2 (2 A+3 B) \sin (c+d x)}{3 d}+\frac {a^2 (2 A+3 B) \cos (c+d x) \sin (c+d x)}{6 d}+\frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 61, normalized size = 0.60 \[ \frac {a^2 (3 (7 A+8 B) \sin (c+d x)+3 (2 A+B) \sin (2 (c+d x))+A \sin (3 (c+d x))+12 A d x+18 B d x)}{12 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 70, normalized size = 0.69 \[ \frac {3 \, {\left (2 \, A + 3 \, B\right )} a^{2} d x + {\left (2 \, A a^{2} \cos \left (d x + c\right )^{2} + 3 \, {\left (2 \, A + B\right )} a^{2} \cos \left (d x + c\right ) + 2 \, {\left (5 \, A + 6 \, B\right )} a^{2}\right )} \sin \left (d x + c\right )}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.55, size = 142, normalized size = 1.39 \[ \frac {3 \, {\left (2 \, A a^{2} + 3 \, B a^{2}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (6 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 16 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 24 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 18 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 15 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.05, size = 116, normalized size = 1.14 \[ \frac {\frac {a^{2} A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+2 a^{2} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \,a^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{2} A \sin \left (d x +c \right )+2 B \,a^{2} \sin \left (d x +c \right )+B \,a^{2} \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 110, normalized size = 1.08 \[ -\frac {4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{2} - 6 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} - 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} - 12 \, {\left (d x + c\right )} B a^{2} - 12 \, A a^{2} \sin \left (d x + c\right ) - 24 \, B a^{2} \sin \left (d x + c\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.89, size = 98, normalized size = 0.96 \[ A\,a^2\,x+\frac {3\,B\,a^2\,x}{2}+\frac {7\,A\,a^2\,\sin \left (c+d\,x\right )}{4\,d}+\frac {2\,B\,a^2\,\sin \left (c+d\,x\right )}{d}+\frac {A\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {A\,a^2\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {B\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d} \]
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 \[ a^{2} \left (\int A \cos ^{3}{\left (c + d x \right )}\, dx + \int 2 A \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int A \cos ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int B \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 2 B \cos ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int B \cos ^{3}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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