Optimal. Leaf size=24 \[ \frac{a \sin (c+d x)}{d}-\frac{b \cos (c+d x)}{d} \]
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Rubi [A] time = 0.0196113, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {3486, 2637} \[ \frac{a \sin (c+d x)}{d}-\frac{b \cos (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3486
Rule 2637
Rubi steps
\begin{align*} \int \cos (c+d x) (a+b \tan (c+d x)) \, dx &=-\frac{b \cos (c+d x)}{d}+a \int \cos (c+d x) \, dx\\ &=-\frac{b \cos (c+d x)}{d}+\frac{a \sin (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0187987, size = 46, normalized size = 1.92 \[ \frac{a \sin (c) \cos (d x)}{d}+\frac{a \cos (c) \sin (d x)}{d}+\frac{b \sin (c) \sin (d x)}{d}-\frac{b \cos (c) \cos (d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 23, normalized size = 1. \begin{align*}{\frac{-b\cos \left ( dx+c \right ) +a\sin \left ( dx+c \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.18442, size = 31, normalized size = 1.29 \begin{align*} -\frac{b \cos \left (d x + c\right ) - 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 = 1.84741, size = 51, normalized size = 2.12 \begin{align*} -\frac{b \cos \left (d x + c\right ) - 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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \tan{\left (c + d x \right )}\right ) \cos{\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 = 1.24499, size = 174, normalized size = 7.25 \begin{align*} -\frac{b \tan \left (\frac{1}{2} \, d x\right )^{2} \tan \left (\frac{1}{2} \, c\right )^{2} + 2 \, a \tan \left (\frac{1}{2} \, d x\right )^{2} \tan \left (\frac{1}{2} \, c\right ) + 2 \, a \tan \left (\frac{1}{2} \, d x\right ) \tan \left (\frac{1}{2} \, c\right )^{2} - b \tan \left (\frac{1}{2} \, d x\right )^{2} - 4 \, b \tan \left (\frac{1}{2} \, d x\right ) \tan \left (\frac{1}{2} \, c\right ) - b \tan \left (\frac{1}{2} \, c\right )^{2} - 2 \, a \tan \left (\frac{1}{2} \, d x\right ) - 2 \, a \tan \left (\frac{1}{2} \, c\right ) + b}{d \tan \left (\frac{1}{2} \, d x\right )^{2} \tan \left (\frac{1}{2} \, c\right )^{2} + d \tan \left (\frac{1}{2} \, d x\right )^{2} + d \tan \left (\frac{1}{2} \, c\right )^{2} + d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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