Optimal. Leaf size=38 \[ \frac{\sin (a-c) \tan (b x+c)}{b}+\frac{\cos (a-c) \sec ^2(b x+c)}{2 b} \]
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Rubi [A] time = 0.0367189, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4580, 2606, 30, 3767, 8} \[ \frac{\sin (a-c) \tan (b x+c)}{b}+\frac{\cos (a-c) \sec ^2(b x+c)}{2 b} \]
Antiderivative was successfully verified.
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Rule 4580
Rule 2606
Rule 30
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \sec ^3(c+b x) \sin (a+b x) \, dx &=\cos (a-c) \int \sec ^2(c+b x) \tan (c+b x) \, dx+\sin (a-c) \int \sec ^2(c+b x) \, dx\\ &=\frac{\cos (a-c) \operatorname{Subst}(\int x \, dx,x,\sec (c+b x))}{b}-\frac{\sin (a-c) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+b x))}{b}\\ &=\frac{\cos (a-c) \sec ^2(c+b x)}{2 b}+\frac{\sin (a-c) \tan (c+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.173792, size = 34, normalized size = 0.89 \[ \frac{\sec (c) \sec ^2(b x+c) (\sin (a-c) \sin (2 b x+c)+\cos (a))}{2 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.595, size = 150, normalized size = 4. \begin{align*}{\frac{1}{b} \left ({\frac{-\cos \left ( a \right ) \cos \left ( c \right ) -\sin \left ( a \right ) \sin \left ( c \right ) }{ \left ( 2\,\cos \left ( a \right ) \sin \left ( c \right ) -2\,\sin \left ( a \right ) \cos \left ( c \right ) \right ) \left ( \sin \left ( a \right ) \cos \left ( c \right ) -\cos \left ( a \right ) \sin \left ( c \right ) \right ) \left ( -\tan \left ( bx+a \right ) \cos \left ( a \right ) \sin \left ( c \right ) +\tan \left ( bx+a \right ) \sin \left ( a \right ) \cos \left ( c \right ) +\cos \left ( a \right ) \cos \left ( c \right ) +\sin \left ( a \right ) \sin \left ( c \right ) \right ) ^{2}}}+{\frac{1}{ \left ( \cos \left ( a \right ) \sin \left ( c \right ) -\sin \left ( a \right ) \cos \left ( c \right ) \right ) \left ( \sin \left ( a \right ) \cos \left ( c \right ) -\cos \left ( a \right ) \sin \left ( c \right ) \right ) \left ( -\tan \left ( bx+a \right ) \cos \left ( a \right ) \sin \left ( c \right ) +\tan \left ( bx+a \right ) \sin \left ( a \right ) \cos \left ( c \right ) +\cos \left ( a \right ) \cos \left ( c \right ) +\sin \left ( a \right ) \sin \left ( c \right ) \right ) }} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.18113, size = 528, normalized size = 13.89 \begin{align*} \frac{{\left (2 \, \cos \left (2 \, b x + 2 \, a + 2 \, c\right ) + \cos \left (2 \, a\right ) - \cos \left (2 \, c\right )\right )} \cos \left (4 \, b x + a + 5 \, c\right ) + 2 \,{\left (2 \, \cos \left (2 \, b x + 2 \, a + 2 \, c\right ) + \cos \left (2 \, a\right ) - \cos \left (2 \, c\right )\right )} \cos \left (2 \, b x + a + 3 \, c\right ) +{\left (\cos \left (2 \, a\right ) - \cos \left (2 \, c\right )\right )} \cos \left (a + c\right ) + 2 \, \cos \left (2 \, b x + 2 \, a + 2 \, c\right ) \cos \left (a + c\right ) +{\left (2 \, \sin \left (2 \, b x + 2 \, a + 2 \, c\right ) + \sin \left (2 \, a\right ) - \sin \left (2 \, c\right )\right )} \sin \left (4 \, b x + a + 5 \, c\right ) + 2 \,{\left (2 \, \sin \left (2 \, b x + 2 \, a + 2 \, c\right ) + \sin \left (2 \, a\right ) - \sin \left (2 \, c\right )\right )} \sin \left (2 \, b x + a + 3 \, c\right ) +{\left (\sin \left (2 \, a\right ) - \sin \left (2 \, c\right )\right )} \sin \left (a + c\right ) + 2 \, \sin \left (2 \, b x + 2 \, a + 2 \, c\right ) \sin \left (a + c\right )}{b \cos \left (4 \, b x + a + 5 \, c\right )^{2} + 4 \, b \cos \left (2 \, b x + a + 3 \, c\right )^{2} + 4 \, b \cos \left (2 \, b x + a + 3 \, c\right ) \cos \left (a + c\right ) + b \cos \left (a + c\right )^{2} + b \sin \left (4 \, b x + a + 5 \, c\right )^{2} + 4 \, b \sin \left (2 \, b x + a + 3 \, c\right )^{2} + 4 \, b \sin \left (2 \, b x + a + 3 \, c\right ) \sin \left (a + c\right ) + b \sin \left (a + c\right )^{2} + 2 \,{\left (2 \, b \cos \left (2 \, b x + a + 3 \, c\right ) + b \cos \left (a + c\right )\right )} \cos \left (4 \, b x + a + 5 \, c\right ) + 2 \,{\left (2 \, b \sin \left (2 \, b x + a + 3 \, c\right ) + b \sin \left (a + c\right )\right )} \sin \left (4 \, b x + a + 5 \, c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.490323, size = 109, normalized size = 2.87 \begin{align*} -\frac{2 \, \cos \left (b x + c\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) - \cos \left (-a + c\right )}{2 \, b \cos \left (b x + c\right )^{2}} \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 [B] time = 1.19662, size = 235, normalized size = 6.18 \begin{align*} \frac{\tan \left (b x + c\right )^{2} \tan \left (\frac{1}{2} \, a\right )^{2} \tan \left (\frac{1}{2} \, c\right )^{2} - \tan \left (b x + c\right )^{2} \tan \left (\frac{1}{2} \, a\right )^{2} + 4 \, \tan \left (b x + c\right )^{2} \tan \left (\frac{1}{2} \, a\right ) \tan \left (\frac{1}{2} \, c\right ) + 4 \, \tan \left (b x + c\right ) \tan \left (\frac{1}{2} \, a\right )^{2} \tan \left (\frac{1}{2} \, c\right ) - \tan \left (b x + c\right )^{2} \tan \left (\frac{1}{2} \, c\right )^{2} - 4 \, \tan \left (b x + c\right ) \tan \left (\frac{1}{2} \, a\right ) \tan \left (\frac{1}{2} \, c\right )^{2} + \tan \left (b x + c\right )^{2} + 4 \, \tan \left (b x + c\right ) \tan \left (\frac{1}{2} \, a\right ) - 4 \, \tan \left (b x + c\right ) \tan \left (\frac{1}{2} \, c\right )}{2 \,{\left (\tan \left (\frac{1}{2} \, a\right )^{2} \tan \left (\frac{1}{2} \, c\right )^{2} + \tan \left (\frac{1}{2} \, a\right )^{2} + \tan \left (\frac{1}{2} \, c\right )^{2} + 1\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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