Optimal. Leaf size=47 \[ \frac {a (B+C) \sin (c+d x)}{d}+\frac {a B \sin (c+d x) \cos (c+d x)}{2 d}+\frac {1}{2} a x (B+2 C) \]
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Rubi [A] time = 0.13, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {4072, 3996, 3787, 2637, 8} \[ \frac {a (B+C) \sin (c+d x)}{d}+\frac {a B \sin (c+d x) \cos (c+d x)}{2 d}+\frac {1}{2} a x (B+2 C) \]
Antiderivative was successfully verified.
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Rule 8
Rule 2637
Rule 3787
Rule 3996
Rule 4072
Rubi steps
\begin {align*} \int \cos ^3(c+d x) (a+a \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\int \cos ^2(c+d x) (a+a \sec (c+d x)) (B+C \sec (c+d x)) \, dx\\ &=\frac {a B \cos (c+d x) \sin (c+d x)}{2 d}-\frac {1}{2} \int \cos (c+d x) (-2 a (B+C)-a (B+2 C) \sec (c+d x)) \, dx\\ &=\frac {a B \cos (c+d x) \sin (c+d x)}{2 d}+(a (B+C)) \int \cos (c+d x) \, dx+\frac {1}{2} (a (B+2 C)) \int 1 \, dx\\ &=\frac {1}{2} a (B+2 C) x+\frac {a (B+C) \sin (c+d x)}{d}+\frac {a B \cos (c+d x) \sin (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 44, normalized size = 0.94 \[ \frac {a (4 (B+C) \sin (c+d x)+B \sin (2 (c+d x))+2 B c+2 B d x+4 C d x)}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 38, normalized size = 0.81 \[ \frac {{\left (B + 2 \, C\right )} a d x + {\left (B a \cos \left (d x + c\right ) + 2 \, {\left (B + C\right )} a\right )} \sin \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 93, normalized size = 1.98 \[ \frac {{\left (B a + 2 \, C a\right )} {\left (d x + c\right )} + \frac {2 \, {\left (B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.80, size = 57, normalized size = 1.21 \[ \frac {a B \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a B \sin \left (d x +c \right )+a C \sin \left (d x +c \right )+a C \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 55, normalized size = 1.17 \[ \frac {{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a + 4 \, {\left (d x + c\right )} C a + 4 \, B a \sin \left (d x + c\right ) + 4 \, C a \sin \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.83, size = 50, normalized size = 1.06 \[ \frac {B\,a\,x}{2}+C\,a\,x+\frac {B\,a\,\sin \left (c+d\,x\right )}{d}+\frac {C\,a\,\sin \left (c+d\,x\right )}{d}+\frac {B\,a\,\sin \left (2\,c+2\,d\,x\right )}{4\,d} \]
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 \[ a \left (\int B \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int B \cos ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int C \cos ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int C \cos ^{3}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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