Optimal. Leaf size=11 \[ \frac{\log (\tan (a+b x))}{b} \]
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Rubi [A] time = 0.0099491, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2620, 29} \[ \frac{\log (\tan (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 2620
Rule 29
Rubi steps
\begin{align*} \int \csc (a+b x) \sec (a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\tan (a+b x)\right )}{b}\\ &=\frac{\log (\tan (a+b x))}{b}\\ \end{align*}
Mathematica [B] time = 0.0214335, size = 31, normalized size = 2.82 \[ 2 \left (\frac{\log (\sin (a+b x))}{2 b}-\frac{\log (\cos (a+b x))}{2 b}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 12, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( \tan \left ( bx+a \right ) \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.977127, size = 38, normalized size = 3.45 \begin{align*} -\frac{\log \left (\sin \left (b x + a\right )^{2} - 1\right ) - \log \left (\sin \left (b x + a\right )^{2}\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.89355, size = 85, normalized size = 7.73 \begin{align*} -\frac{\log \left (\cos \left (b x + a\right )^{2}\right ) - \log \left (-\frac{1}{4} \, \cos \left (b x + a\right )^{2} + \frac{1}{4}\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{\sec{\left (a + b x \right )}}{\sin{\left (a + b x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1987, size = 76, normalized size = 6.91 \begin{align*} \frac{\log \left (\frac{{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right ) - 2 \, \log \left ({\left | -\frac{\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1 \right |}\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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