Optimal. Leaf size=15 \[ B x+\frac {C \sin (c+d x)}{d} \]
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Rubi [A] time = 0.03, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {3010, 2637} \[ B x+\frac {C \sin (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3010
Rubi steps
\begin {align*} \int \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx &=\int (B+C \cos (c+d x)) \, dx\\ &=B x+C \int \cos (c+d x) \, dx\\ &=B x+\frac {C \sin (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 1.73 \[ B x+\frac {C \sin (c) \cos (d x)}{d}+\frac {C \cos (c) \sin (d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 17, normalized size = 1.13 \[ \frac {B d x + C \sin \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.35, size = 39, normalized size = 2.60 \[ \frac {{\left (d x + c\right )} B + \frac {2 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 21, normalized size = 1.40 \[ \frac {C \sin \left (d x +c \right )+B \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 20, normalized size = 1.33 \[ \frac {{\left (d x + c\right )} B + C \sin \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.98, size = 17, normalized size = 1.13 \[ \frac {C\,\sin \left (c+d\,x\right )+B\,d\,x}{d} \]
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 \[ \int \left (B + C \cos {\left (c + d x \right )}\right ) \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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