Optimal. Leaf size=17 \[ -\frac{a \log (1-\sin (c+d x))}{d} \]
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Rubi [A] time = 0.0199603, antiderivative size = 17, 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 = {2667, 31} \[ -\frac{a \log (1-\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 31
Rubi steps
\begin{align*} \int \sec (c+d x) (a+a \sin (c+d x)) \, dx &=\frac{a \operatorname{Subst}\left (\int \frac{1}{a-x} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{a \log (1-\sin (c+d x))}{d}\\ \end{align*}
Mathematica [A] time = 0.0140097, size = 26, normalized size = 1.53 \[ \frac{a \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 16, normalized size = 0.9 \begin{align*} -{\frac{a\ln \left ( \sin \left ( dx+c \right ) -1 \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.945267, size = 20, normalized size = 1.18 \begin{align*} -\frac{a \log \left (\sin \left (d x + c\right ) - 1\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7122, size = 39, normalized size = 2.29 \begin{align*} -\frac{a \log \left (-\sin \left (d x + c\right ) + 1\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{\left (c + d x \right )}\, dx + \int \sec{\left (c + d x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.20579, size = 50, normalized size = 2.94 \begin{align*} \frac{a \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right ) - 2 \, a \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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