Optimal. Leaf size=15 \[ \frac{A \tan (c+d x)}{d}+C x \]
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Rubi [A] time = 0.0244259, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3012, 8} \[ \frac{A \tan (c+d x)}{d}+C x \]
Antiderivative was successfully verified.
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Rule 3012
Rule 8
Rubi steps
\begin{align*} \int \left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx &=\frac{A \tan (c+d x)}{d}+C \int 1 \, dx\\ &=C x+\frac{A \tan (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0086647, size = 15, normalized size = 1. \[ \frac{A \tan (c+d x)}{d}+C x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.064, size = 21, normalized size = 1.4 \begin{align*}{\frac{A\tan \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 = 1.49507, size = 27, normalized size = 1.8 \begin{align*} \frac{{\left (d x + c\right )} C + A \tan \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.59324, size = 76, normalized size = 5.07 \begin{align*} \frac{C d x \cos \left (d x + c\right ) + A \sin \left (d x + c\right )}{d \cos \left (d x + c\right )} \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 + C \cos ^{2}{\left (c + d x \right )}\right ) \sec ^{2}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16327, size = 27, normalized size = 1.8 \begin{align*} \frac{{\left (d x + c\right )} C + A \tan \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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