Optimal. Leaf size=15 \[ \frac{a \sin (c+d x)}{d}+b x \]
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Rubi [A] time = 0.0231894, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {3787, 2637, 8} \[ \frac{a \sin (c+d x)}{d}+b x \]
Antiderivative was successfully verified.
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Rule 3787
Rule 2637
Rule 8
Rubi steps
\begin{align*} \int \cos (c+d x) (a+b \sec (c+d x)) \, dx &=a \int \cos (c+d x) \, dx+b \int 1 \, dx\\ &=b x+\frac{a \sin (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0092314, size = 26, normalized size = 1.73 \[ \frac{a \sin (c) \cos (d x)}{d}+\frac{a \cos (c) \sin (d x)}{d}+b x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 21, normalized size = 1.4 \begin{align*}{\frac{\sin \left ( dx+c \right ) a+ \left ( dx+c \right ) b}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16739, size = 27, normalized size = 1.8 \begin{align*} \frac{{\left (d x + c\right )} b + a \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 = 1.7471, size = 38, normalized size = 2.53 \begin{align*} \frac{b d x + a \sin \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.44176, size = 15, normalized size = 1. \begin{align*} a \left (\begin{cases} \sin{\left (c \right )} & \text{for}\: d = 0 \\\frac{\sin{\left (c + d x \right )}}{d} & \text{otherwise} \end{cases}\right ) + b x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.23696, size = 53, normalized size = 3.53 \begin{align*} \frac{{\left (d x + c\right )} b + \frac{2 \, a \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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