Optimal. Leaf size=55 \[ -\frac{\tanh ^{-1}(\sin (a+b x))}{8 b}+\frac{\tan (a+b x) \sec ^3(a+b x)}{4 b}-\frac{\tan (a+b x) \sec (a+b x)}{8 b} \]
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Rubi [A] time = 0.0448754, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {2611, 3768, 3770} \[ -\frac{\tanh ^{-1}(\sin (a+b x))}{8 b}+\frac{\tan (a+b x) \sec ^3(a+b x)}{4 b}-\frac{\tan (a+b x) \sec (a+b x)}{8 b} \]
Antiderivative was successfully verified.
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Rule 2611
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int \sec ^3(a+b x) \tan ^2(a+b x) \, dx &=\frac{\sec ^3(a+b x) \tan (a+b x)}{4 b}-\frac{1}{4} \int \sec ^3(a+b x) \, dx\\ &=-\frac{\sec (a+b x) \tan (a+b x)}{8 b}+\frac{\sec ^3(a+b x) \tan (a+b x)}{4 b}-\frac{1}{8} \int \sec (a+b x) \, dx\\ &=-\frac{\tanh ^{-1}(\sin (a+b x))}{8 b}-\frac{\sec (a+b x) \tan (a+b x)}{8 b}+\frac{\sec ^3(a+b x) \tan (a+b x)}{4 b}\\ \end{align*}
Mathematica [A] time = 0.0447106, size = 55, normalized size = 1. \[ -\frac{\tanh ^{-1}(\sin (a+b x))}{8 b}+\frac{\tan (a+b x) \sec ^3(a+b x)}{4 b}-\frac{\tan (a+b x) \sec (a+b x)}{8 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 74, normalized size = 1.4 \begin{align*}{\frac{ \left ( \sin \left ( bx+a \right ) \right ) ^{3}}{4\,b \left ( \cos \left ( bx+a \right ) \right ) ^{4}}}+{\frac{ \left ( \sin \left ( bx+a \right ) \right ) ^{3}}{8\,b \left ( \cos \left ( bx+a \right ) \right ) ^{2}}}+{\frac{\sin \left ( bx+a \right ) }{8\,b}}-{\frac{\ln \left ( \sec \left ( bx+a \right ) +\tan \left ( bx+a \right ) \right ) }{8\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04326, size = 88, normalized size = 1.6 \begin{align*} \frac{\frac{2 \,{\left (\sin \left (b x + a\right )^{3} + \sin \left (b x + a\right )\right )}}{\sin \left (b x + a\right )^{4} - 2 \, \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 )}{16 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70726, size = 193, normalized size = 3.51 \begin{align*} -\frac{\cos \left (b x + a\right )^{4} \log \left (\sin \left (b x + a\right ) + 1\right ) - \cos \left (b x + a\right )^{4} \log \left (-\sin \left (b x + a\right ) + 1\right ) + 2 \,{\left (\cos \left (b x + a\right )^{2} - 2\right )} \sin \left (b x + a\right )}{16 \, b \cos \left (b x + a\right )^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25782, size = 111, normalized size = 2.02 \begin{align*} \frac{\frac{4 \,{\left (\frac{1}{\sin \left (b x + a\right )} + \sin \left (b x + a\right )\right )}}{{\left (\frac{1}{\sin \left (b x + a\right )} + \sin \left (b x + a\right )\right )}^{2} - 4} - \log \left ({\left | \frac{1}{\sin \left (b x + a\right )} + \sin \left (b x + a\right ) + 2 \right |}\right ) + \log \left ({\left | \frac{1}{\sin \left (b x + a\right )} + \sin \left (b x + a\right ) - 2 \right |}\right )}{32 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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