Optimal. Leaf size=39 \[ -\frac{\cos (a-c) \cot (b x+c)}{b}-\frac{\sin (a-c) \csc ^2(b x+c)}{2 b} \]
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Rubi [A] time = 0.0437446, antiderivative size = 39, 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 = {4582, 2606, 30, 3767, 8} \[ -\frac{\cos (a-c) \cot (b x+c)}{b}-\frac{\sin (a-c) \csc ^2(b x+c)}{2 b} \]
Antiderivative was successfully verified.
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Rule 4582
Rule 2606
Rule 30
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \csc ^3(c+b x) \sin (a+b x) \, dx &=\cos (a-c) \int \csc ^2(c+b x) \, dx+\sin (a-c) \int \cot (c+b x) \csc ^2(c+b x) \, dx\\ &=-\frac{\cos (a-c) \operatorname{Subst}(\int 1 \, dx,x,\cot (c+b x))}{b}-\frac{\sin (a-c) \operatorname{Subst}(\int x \, dx,x,\csc (c+b x))}{b}\\ &=-\frac{\cos (a-c) \cot (c+b x)}{b}-\frac{\csc ^2(c+b x) \sin (a-c)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.190225, size = 35, normalized size = 0.9 \[ \frac{\csc (c) \csc ^2(b x+c) (\cos (a)-\cos (a-c) \cos (2 b x+c))}{2 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.602, size = 120, normalized size = 3.1 \begin{align*}{\frac{1}{b} \left ( -{\frac{1}{ \left ( \cos \left ( a \right ) \cos \left ( c \right ) +\sin \left ( a \right ) \sin \left ( c \right ) \right ) ^{2} \left ( \tan \left ( bx+a \right ) \cos \left ( a \right ) \cos \left ( c \right ) +\tan \left ( bx+a \right ) \sin \left ( a \right ) \sin \left ( c \right ) +\cos \left ( a \right ) \sin \left ( c \right ) -\sin \left ( a \right ) \cos \left ( c \right ) \right ) }}-{\frac{\sin \left ( a \right ) \cos \left ( c \right ) -\cos \left ( a \right ) \sin \left ( c \right ) }{2\, \left ( \cos \left ( a \right ) \cos \left ( c \right ) +\sin \left ( a \right ) \sin \left ( c \right ) \right ) ^{2} \left ( \tan \left ( bx+a \right ) \cos \left ( a \right ) \cos \left ( c \right ) +\tan \left ( bx+a \right ) \sin \left ( a \right ) \sin \left ( c \right ) +\cos \left ( a \right ) \sin \left ( c \right ) -\sin \left ( a \right ) \cos \left ( c \right ) \right ) ^{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.19331, size = 539, normalized size = 13.82 \begin{align*} \frac{{\left (2 \, \sin \left (2 \, b x + 2 \, a + 2 \, c\right ) - \sin \left (2 \, a\right ) - \sin \left (2 \, c\right )\right )} \cos \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 )} \cos \left (2 \, b x + a + 3 \, c\right ) -{\left (\sin \left (2 \, a\right ) + \sin \left (2 \, c\right )\right )} \cos \left (a + c\right ) -{\left (2 \, \cos \left (2 \, b x + 2 \, a + 2 \, c\right ) - \cos \left (2 \, a\right ) - \cos \left (2 \, c\right )\right )} \sin \left (4 \, b x + a + 5 \, c\right ) + 2 \, \cos \left (a + c\right ) \sin \left (2 \, b x + 2 \, a + 2 \, 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 )} \sin \left (2 \, b x + a + 3 \, c\right ) +{\left (\cos \left (2 \, a\right ) + \cos \left (2 \, c\right )\right )} \sin \left (a + c\right ) - 2 \, \cos \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 = 0.469974, size = 113, normalized size = 2.9 \begin{align*} \frac{2 \, \cos \left (b x + c\right ) \cos \left (-a + c\right ) \sin \left (b x + c\right ) - \sin \left (-a + c\right )}{2 \,{\left (b \cos \left (b x + c\right )^{2} - b\right )}} \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 = 1.17347, size = 196, normalized size = 5.03 \begin{align*} -\frac{\tan \left (b x + c\right ) \tan \left (\frac{1}{2} \, a\right )^{2} \tan \left (\frac{1}{2} \, c\right )^{2} - \tan \left (b x + c\right ) \tan \left (\frac{1}{2} \, a\right )^{2} + 4 \, \tan \left (b x + c\right ) \tan \left (\frac{1}{2} \, a\right ) \tan \left (\frac{1}{2} \, c\right ) + \tan \left (\frac{1}{2} \, a\right )^{2} \tan \left (\frac{1}{2} \, c\right ) - \tan \left (b x + c\right ) \tan \left (\frac{1}{2} \, c\right )^{2} - \tan \left (\frac{1}{2} \, a\right ) \tan \left (\frac{1}{2} \, c\right )^{2} + \tan \left (b x + c\right ) + \tan \left (\frac{1}{2} \, a\right ) - \tan \left (\frac{1}{2} \, c\right )}{{\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 \tan \left (b x + c\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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