Optimal. Leaf size=34 \[ \frac{\tan (a+b x) \sec (a+b x)}{2 b}-\frac{\tanh ^{-1}(\sin (a+b x))}{2 b} \]
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Rubi [A] time = 0.0230989, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2611, 3770} \[ \frac{\tan (a+b x) \sec (a+b x)}{2 b}-\frac{\tanh ^{-1}(\sin (a+b x))}{2 b} \]
Antiderivative was successfully verified.
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Rule 2611
Rule 3770
Rubi steps
\begin{align*} \int \sec (a+b x) \tan ^2(a+b x) \, dx &=\frac{\sec (a+b x) \tan (a+b x)}{2 b}-\frac{1}{2} \int \sec (a+b x) \, dx\\ &=-\frac{\tanh ^{-1}(\sin (a+b x))}{2 b}+\frac{\sec (a+b x) \tan (a+b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0141463, size = 34, normalized size = 1. \[ \frac{\tan (a+b x) \sec (a+b x)}{2 b}-\frac{\tanh ^{-1}(\sin (a+b x))}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 53, normalized size = 1.6 \begin{align*}{\frac{ \left ( \sin \left ( bx+a \right ) \right ) ^{3}}{2\,b \left ( \cos \left ( bx+a \right ) \right ) ^{2}}}+{\frac{\sin \left ( bx+a \right ) }{2\,b}}-{\frac{\ln \left ( \sec \left ( bx+a \right ) +\tan \left ( bx+a \right ) \right ) }{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.979486, size = 62, normalized size = 1.82 \begin{align*} -\frac{\frac{2 \, \sin \left (b x + a\right )}{\sin \left (b x + a\right )^{2} - 1} + \log \left (\sin \left (b x + a\right ) + 1\right ) - \log \left (\sin \left (b x + a\right ) - 1\right )}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.63399, size = 163, normalized size = 4.79 \begin{align*} -\frac{\cos \left (b x + a\right )^{2} \log \left (\sin \left (b x + a\right ) + 1\right ) - \cos \left (b x + a\right )^{2} \log \left (-\sin \left (b x + a\right ) + 1\right ) - 2 \, \sin \left (b x + a\right )}{4 \, b \cos \left (b x + a\right )^{2}} \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 \sin ^{2}{\left (a + b x \right )} \sec ^{3}{\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 = 1.29343, size = 65, normalized size = 1.91 \begin{align*} -\frac{\frac{2 \, \sin \left (b x + a\right )}{\sin \left (b x + a\right )^{2} - 1} + \log \left ({\left | \sin \left (b x + a\right ) + 1 \right |}\right ) - \log \left ({\left | \sin \left (b x + a\right ) - 1 \right |}\right )}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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