Optimal. Leaf size=16 \[ B x+\frac{C \tanh ^{-1}(\sin (c+d x))}{d} \]
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Rubi [A] time = 0.0294041, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {4047, 8, 12, 3770} \[ B x+\frac{C \tanh ^{-1}(\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 4047
Rule 8
Rule 12
Rule 3770
Rubi steps
\begin{align*} \int \cos (c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=B \int 1 \, dx+\int C \sec (c+d x) \, dx\\ &=B x+C \int \sec (c+d x) \, dx\\ &=B x+\frac{C \tanh ^{-1}(\sin (c+d x))}{d}\\ \end{align*}
Mathematica [A] time = 0.0040731, size = 16, normalized size = 1. \[ B x+\frac{C \tanh ^{-1}(\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 30, normalized size = 1.9 \begin{align*} Bx+{\frac{Bc}{d}}+{\frac{C\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.932997, size = 50, normalized size = 3.12 \begin{align*} \frac{2 \,{\left (d x + c\right )} B + C{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.49889, size = 95, normalized size = 5.94 \begin{align*} \frac{2 \, B d x + C \log \left (\sin \left (d x + c\right ) + 1\right ) - C \log \left (-\sin \left (d x + c\right ) + 1\right )}{2 \, 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{\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 = 1.20795, size = 58, normalized size = 3.62 \begin{align*} \frac{{\left (d x + c\right )} B + C \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - C \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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