Optimal. Leaf size=15 \[ \frac{B \sin (c+d x)}{d}+C x \]
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Rubi [A] time = 0.0319256, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {4047, 2637, 8} \[ \frac{B \sin (c+d x)}{d}+C x \]
Antiderivative was successfully verified.
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Rule 4047
Rule 2637
Rule 8
Rubi steps
\begin{align*} \int \cos ^2(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=B \int \cos (c+d x) \, dx+\int C \, dx\\ &=C x+\frac{B \sin (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0087985, size = 26, normalized size = 1.73 \[ \frac{B \sin (c) \cos (d x)}{d}+\frac{B \cos (c) \sin (d x)}{d}+C x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 21, normalized size = 1.4 \begin{align*}{\frac{B\sin \left ( dx+c \right ) +C \left ( dx+c \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.935867, size = 27, normalized size = 1.8 \begin{align*} \frac{{\left (d x + c\right )} C + B \sin \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.468058, size = 38, normalized size = 2.53 \begin{align*} \frac{C d x + B \sin \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (B + C \sec{\left (c + d x \right )}\right ) \cos ^{2}{\left (c + d x \right )} \sec{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17121, size = 53, normalized size = 3.53 \begin{align*} \frac{{\left (d x + c\right )} C + \frac{2 \, B \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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