Optimal. Leaf size=19 \[ -\frac{\log (\sin (a+b x)+\cos (a+b x))}{b} \]
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Rubi [A] time = 0.31409, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {801, 260} \[ -\frac{\log (\sin (a+b x)+\cos (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 801
Rule 260
Rubi steps
\begin{align*} \int \frac{-\csc (a+b x)+\sec (a+b x)}{\csc (a+b x)+\sec (a+b x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{-1+x}{(1+x) \left (1+x^2\right )} \, dx,x,\tan (a+b x)\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{-1-x}+\frac{x}{1+x^2}\right ) \, dx,x,\tan (a+b x)\right )}{b}\\ &=-\frac{\log (1+\tan (a+b x))}{b}+\frac{\operatorname{Subst}\left (\int \frac{x}{1+x^2} \, dx,x,\tan (a+b x)\right )}{b}\\ &=-\frac{\log (\cos (a+b x))}{b}-\frac{\log (1+\tan (a+b x))}{b}\\ \end{align*}
Mathematica [A] time = 0.0578265, size = 19, normalized size = 1. \[ -\frac{\log (\sin (a+b x)+\cos (a+b x))}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.131, size = 32, normalized size = 1.7 \begin{align*} -{\frac{\ln \left ( 1+\tan \left ( bx+a \right ) \right ) }{b}}+{\frac{\ln \left ( 1+ \left ( \tan \left ( bx+a \right ) \right ) ^{2} \right ) }{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.4379, size = 95, normalized size = 5. \begin{align*} -\frac{\log \left (-\frac{2 \, \sin \left (b x + a\right )}{\cos \left (b x + a\right ) + 1} + \frac{\sin \left (b x + a\right )^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} - 1\right ) - \log \left (\frac{\sin \left (b x + a\right )^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + 1\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.12089, size = 61, normalized size = 3.21 \begin{align*} -\frac{\log \left (2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 1\right )}{2 \, b} \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 \frac{\csc{\left (a + b x \right )}}{\csc{\left (a + b x \right )} + \sec{\left (a + b x \right )}}\, dx - \int - \frac{\sec{\left (a + b x \right )}}{\csc{\left (a + b x \right )} + \sec{\left (a + b x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15848, size = 39, normalized size = 2.05 \begin{align*} \frac{\log \left (\tan \left (b x + a\right )^{2} + 1\right ) - 2 \, \log \left ({\left | \tan \left (b x + a\right ) + 1 \right |}\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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