Optimal. Leaf size=23 \[ \frac{a \tan (c+d x)}{d}+\frac{a \sec (c+d x)}{d} \]
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Rubi [A] time = 0.0321409, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2669, 3767, 8} \[ \frac{a \tan (c+d x)}{d}+\frac{a \sec (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2669
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \sec ^2(c+d x) (a+a \sin (c+d x)) \, dx &=\frac{a \sec (c+d x)}{d}+a \int \sec ^2(c+d x) \, dx\\ &=\frac{a \sec (c+d x)}{d}-\frac{a \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=\frac{a \sec (c+d x)}{d}+\frac{a \tan (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0131422, size = 23, normalized size = 1. \[ \frac{a \tan (c+d x)}{d}+\frac{a \sec (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.069, size = 24, normalized size = 1. \begin{align*}{\frac{1}{d} \left ({\frac{a}{\cos \left ( dx+c \right ) }}+a\tan \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.946107, size = 31, normalized size = 1.35 \begin{align*} \frac{a \tan \left (d x + c\right ) + \frac{a}{\cos \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 = 1.60502, size = 104, normalized size = 4.52 \begin{align*} \frac{a \cos \left (d x + c\right ) + a \sin \left (d x + c\right ) + a}{d \cos \left (d x + c\right ) - d \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*} a \left (\int \sin{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \sec ^{2}{\left (c + d x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14352, size = 26, normalized size = 1.13 \begin{align*} -\frac{2 \, a}{d{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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