Optimal. Leaf size=33 \[ \frac{\sec ^2(x)}{2 a}+\frac{\tanh ^{-1}(\sin (x))}{2 a}-\frac{\tan (x) \sec (x)}{2 a} \]
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Rubi [A] time = 0.072992, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546, Rules used = {3872, 2706, 2606, 30, 2611, 3770} \[ \frac{\sec ^2(x)}{2 a}+\frac{\tanh ^{-1}(\sin (x))}{2 a}-\frac{\tan (x) \sec (x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2706
Rule 2606
Rule 30
Rule 2611
Rule 3770
Rubi steps
\begin{align*} \int \frac{\sec (x)}{a+a \csc (x)} \, dx &=\int \frac{\tan (x)}{a+a \sin (x)} \, dx\\ &=\frac{\int \sec ^2(x) \tan (x) \, dx}{a}-\frac{\int \sec (x) \tan ^2(x) \, dx}{a}\\ &=-\frac{\sec (x) \tan (x)}{2 a}+\frac{\int \sec (x) \, dx}{2 a}+\frac{\operatorname{Subst}(\int x \, dx,x,\sec (x))}{a}\\ &=\frac{\tanh ^{-1}(\sin (x))}{2 a}+\frac{\sec ^2(x)}{2 a}-\frac{\sec (x) \tan (x)}{2 a}\\ \end{align*}
Mathematica [A] time = 0.0249201, size = 17, normalized size = 0.52 \[ \frac{\frac{1}{\sin (x)+1}+\tanh ^{-1}(\sin (x))}{2 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 33, normalized size = 1. \begin{align*}{\frac{1}{2\,a \left ( \sin \left ( x \right ) +1 \right ) }}+{\frac{\ln \left ( \sin \left ( x \right ) +1 \right ) }{4\,a}}-{\frac{\ln \left ( \sin \left ( x \right ) -1 \right ) }{4\,a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04997, size = 42, normalized size = 1.27 \begin{align*} \frac{\log \left (\sin \left (x\right ) + 1\right )}{4 \, a} - \frac{\log \left (\sin \left (x\right ) - 1\right )}{4 \, a} + \frac{1}{2 \,{\left (a \sin \left (x\right ) + a\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.483032, size = 117, normalized size = 3.55 \begin{align*} \frac{{\left (\sin \left (x\right ) + 1\right )} \log \left (\sin \left (x\right ) + 1\right ) -{\left (\sin \left (x\right ) + 1\right )} \log \left (-\sin \left (x\right ) + 1\right ) + 2}{4 \,{\left (a \sin \left (x\right ) + a\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*} \frac{\int \frac{\sec{\left (x \right )}}{\csc{\left (x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32379, size = 46, normalized size = 1.39 \begin{align*} \frac{\log \left (\sin \left (x\right ) + 1\right )}{4 \, a} - \frac{\log \left (-\sin \left (x\right ) + 1\right )}{4 \, a} + \frac{1}{2 \, a{\left (\sin \left (x\right ) + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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