Optimal. Leaf size=33 \[ \frac{\csc (a+c) \log (\cos (c-b x))}{b}-\frac{\csc (a+c) \log (\cos (a+b x))}{b} \]
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Rubi [A] time = 0.018378, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4610, 3475} \[ \frac{\csc (a+c) \log (\cos (c-b x))}{b}-\frac{\csc (a+c) \log (\cos (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 4610
Rule 3475
Rubi steps
\begin{align*} \int \sec (c-b x) \sec (a+b x) \, dx &=\csc (a+c) \int \tan (c-b x) \, dx+\csc (a+c) \int \tan (a+b x) \, dx\\ &=\frac{\csc (a+c) \log (\cos (c-b x))}{b}-\frac{\csc (a+c) \log (\cos (a+b x))}{b}\\ \end{align*}
Mathematica [A] time = 0.228994, size = 26, normalized size = 0.79 \[ \frac{\csc (a+c) (\log (\cos (c-b x))-\log (\cos (a+b x)))}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.221, size = 53, normalized size = 1.6 \begin{align*}{\frac{\ln \left ( \tan \left ( bx+a \right ) \cos \left ( a \right ) \sin \left ( c \right ) +\tan \left ( bx+a \right ) \sin \left ( a \right ) \cos \left ( c \right ) +\cos \left ( a \right ) \cos \left ( c \right ) -\sin \left ( a \right ) \sin \left ( c \right ) \right ) }{b \left ( \sin \left ( a \right ) \cos \left ( c \right ) +\cos \left ( a \right ) \sin \left ( c \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.09119, size = 435, normalized size = 13.18 \begin{align*} \frac{2 \,{\left (\cos \left (2 \, a + 2 \, c\right ) \cos \left (a + c\right ) + \sin \left (2 \, a + 2 \, c\right ) \sin \left (a + c\right ) - \cos \left (a + c\right )\right )} \arctan \left (\sin \left (2 \, b x\right ) - \sin \left (2 \, a\right ), \cos \left (2 \, b x\right ) + \cos \left (2 \, a\right )\right ) - 2 \,{\left (\cos \left (2 \, a + 2 \, c\right ) \cos \left (a + c\right ) + \sin \left (2 \, a + 2 \, c\right ) \sin \left (a + c\right ) - \cos \left (a + c\right )\right )} \arctan \left (\sin \left (2 \, b x\right ) + \sin \left (2 \, c\right ), \cos \left (2 \, b x\right ) + \cos \left (2 \, c\right )\right ) -{\left (\cos \left (a + c\right ) \sin \left (2 \, a + 2 \, c\right ) - \cos \left (2 \, a + 2 \, c\right ) \sin \left (a + c\right ) + \sin \left (a + c\right )\right )} \log \left (\cos \left (2 \, b x\right )^{2} + 2 \, \cos \left (2 \, b x\right ) \cos \left (2 \, a\right ) + \cos \left (2 \, a\right )^{2} + \sin \left (2 \, b x\right )^{2} - 2 \, \sin \left (2 \, b x\right ) \sin \left (2 \, a\right ) + \sin \left (2 \, a\right )^{2}\right ) +{\left (\cos \left (a + c\right ) \sin \left (2 \, a + 2 \, c\right ) - \cos \left (2 \, a + 2 \, c\right ) \sin \left (a + c\right ) + \sin \left (a + c\right )\right )} \log \left (\cos \left (2 \, b x\right )^{2} + 2 \, \cos \left (2 \, b x\right ) \cos \left (2 \, c\right ) + \cos \left (2 \, c\right )^{2} + \sin \left (2 \, b x\right )^{2} + 2 \, \sin \left (2 \, b x\right ) \sin \left (2 \, c\right ) + \sin \left (2 \, c\right )^{2}\right )}{b \cos \left (2 \, a + 2 \, c\right )^{2} + b \sin \left (2 \, a + 2 \, c\right )^{2} - 2 \, b \cos \left (2 \, a + 2 \, c\right ) + b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.52903, size = 263, normalized size = 7.97 \begin{align*} -\frac{\log \left (\cos \left (b x + a\right )^{2}\right ) - \log \left (\frac{4 \,{\left (2 \, \cos \left (b x + a\right ) \cos \left (a + c\right ) \sin \left (b x + a\right ) \sin \left (a + c\right ) +{\left (2 \, \cos \left (a + c\right )^{2} - 1\right )} \cos \left (b x + a\right )^{2} - \cos \left (a + c\right )^{2} + 1\right )}}{\cos \left (a + c\right )^{2} + 2 \, \cos \left (a + c\right ) + 1}\right )}{2 \, b \sin \left (a + 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 \sec{\left (a + b x \right )} \sec{\left (b x - c \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.24222, size = 228, normalized size = 6.91 \begin{align*} -\frac{{\left (\tan \left (\frac{1}{2} \, a\right )^{2} \tan \left (\frac{1}{2} \, c\right )^{2} + \tan \left (\frac{1}{2} \, a\right )^{2} + \tan \left (\frac{1}{2} \, c\right )^{2} + 1\right )} \log \left ({\left | 2 \, \tan \left (b x + a\right ) \tan \left (\frac{1}{2} \, a\right )^{2} \tan \left (\frac{1}{2} \, c\right ) + 2 \, \tan \left (b x + a\right ) \tan \left (\frac{1}{2} \, a\right ) \tan \left (\frac{1}{2} \, c\right )^{2} - \tan \left (\frac{1}{2} \, a\right )^{2} \tan \left (\frac{1}{2} \, c\right )^{2} - 2 \, \tan \left (b x + a\right ) \tan \left (\frac{1}{2} \, a\right ) + \tan \left (\frac{1}{2} \, a\right )^{2} - 2 \, \tan \left (b x + a\right ) \tan \left (\frac{1}{2} \, c\right ) + 4 \, \tan \left (\frac{1}{2} \, a\right ) \tan \left (\frac{1}{2} \, c\right ) + \tan \left (\frac{1}{2} \, c\right )^{2} - 1 \right |}\right )}{2 \,{\left (\tan \left (\frac{1}{2} \, a\right )^{2} \tan \left (\frac{1}{2} \, c\right ) + \tan \left (\frac{1}{2} \, a\right ) \tan \left (\frac{1}{2} \, c\right )^{2} - \tan \left (\frac{1}{2} \, a\right ) - \tan \left (\frac{1}{2} \, c\right )\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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