Optimal. Leaf size=43 \[ \frac{(a-b) \log (\sin (c+d x)+1)}{2 d}-\frac{(a+b) \log (1-\sin (c+d x))}{2 d} \]
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Rubi [A] time = 0.0400481, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {2668, 633, 31} \[ \frac{(a-b) \log (\sin (c+d x)+1)}{2 d}-\frac{(a+b) \log (1-\sin (c+d x))}{2 d} \]
Antiderivative was successfully verified.
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Rule 2668
Rule 633
Rule 31
Rubi steps
\begin{align*} \int \sec (c+d x) (a+b \sin (c+d x)) \, dx &=\frac{b \operatorname{Subst}\left (\int \frac{a+x}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac{(a-b) \operatorname{Subst}\left (\int \frac{1}{-b-x} \, dx,x,b \sin (c+d x)\right )}{2 d}+\frac{(a+b) \operatorname{Subst}\left (\int \frac{1}{b-x} \, dx,x,b \sin (c+d x)\right )}{2 d}\\ &=-\frac{(a+b) \log (1-\sin (c+d x))}{2 d}+\frac{(a-b) \log (1+\sin (c+d x))}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0108662, size = 26, normalized size = 0.6 \[ \frac{a \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 34, normalized size = 0.8 \begin{align*} -{\frac{b\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+{\frac{a\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 [A] time = 0.950124, size = 47, normalized size = 1.09 \begin{align*} \frac{{\left (a - b\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (a + b\right )} \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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Fricas [A] time = 2.29831, size = 97, normalized size = 2.26 \begin{align*} \frac{{\left (a - b\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (a + b\right )} \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 (a + b \sin{\left (c + d x \right )}\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 [A] time = 1.10903, size = 50, normalized size = 1.16 \begin{align*} \frac{{\left (a - b\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) -{\left (a + b\right )} \log \left ({\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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