Optimal. Leaf size=38 \[ \frac {\sin (a-c) \cot (b x+c)}{b}-\frac {\cos (a-c) \csc ^2(b x+c)}{2 b} \]
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Rubi [A] time = 0.04, antiderivative size = 38, normalized size of antiderivative = 1.00, 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 = {4581, 2606, 30, 3767, 8} \[ \frac {\sin (a-c) \cot (b x+c)}{b}-\frac {\cos (a-c) \csc ^2(b x+c)}{2 b} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 2606
Rule 3767
Rule 4581
Rubi steps
\begin {align*} \int \cos (a+b x) \csc ^3(c+b x) \, dx &=\cos (a-c) \int \cot (c+b x) \csc ^2(c+b x) \, dx-\sin (a-c) \int \csc ^2(c+b x) \, dx\\ &=-\frac {\cos (a-c) \operatorname {Subst}(\int x \, dx,x,\csc (c+b x))}{b}+\frac {\sin (a-c) \operatorname {Subst}(\int 1 \, dx,x,\cot (c+b x))}{b}\\ &=-\frac {\cos (a-c) \csc ^2(c+b x)}{2 b}+\frac {\cot (c+b x) \sin (a-c)}{b}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 35, normalized size = 0.92 \[ -\frac {\csc (c) \csc ^2(b x+c) (\sin (a)-\sin (a-c) \cos (2 b x+c))}{2 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 45, normalized size = 1.18 \[ \frac {2 \, \cos \left (b x + c\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) + \cos \left (-a + c\right )}{2 \, {\left (b \cos \left (b x + c\right )^{2} - b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.29, size = 327, normalized size = 8.61 \[ -\frac {\tan \left (\frac {1}{2} \, a\right )^{6} \tan \left (\frac {1}{2} \, c\right )^{6} + 3 \, \tan \left (\frac {1}{2} \, a\right )^{6} \tan \left (\frac {1}{2} \, c\right )^{4} + 3 \, \tan \left (\frac {1}{2} \, a\right )^{4} \tan \left (\frac {1}{2} \, c\right )^{6} + 3 \, \tan \left (\frac {1}{2} \, a\right )^{6} \tan \left (\frac {1}{2} \, c\right )^{2} + 9 \, \tan \left (\frac {1}{2} \, a\right )^{4} \tan \left (\frac {1}{2} \, c\right )^{4} + 3 \, \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{6} + \tan \left (\frac {1}{2} \, a\right )^{6} + 9 \, \tan \left (\frac {1}{2} \, a\right )^{4} \tan \left (\frac {1}{2} \, c\right )^{2} + 9 \, \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{4} + \tan \left (\frac {1}{2} \, c\right )^{6} + 3 \, \tan \left (\frac {1}{2} \, a\right )^{4} + 9 \, \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 3 \, \tan \left (\frac {1}{2} \, c\right )^{4} + 3 \, \tan \left (\frac {1}{2} \, a\right )^{2} + 3 \, \tan \left (\frac {1}{2} \, c\right )^{2} + 1}{2 \, {\left (\tan \left (b x + a\right ) \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (b x + a\right ) \tan \left (\frac {1}{2} \, a\right )^{2} + 4 \, \tan \left (b x + a\right ) \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) - 2 \, \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - \tan \left (b x + a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (b x + a\right ) - 2 \, \tan \left (\frac {1}{2} \, a\right ) + 2 \, \tan \left (\frac {1}{2} \, c\right )\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} + 4 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, c\right )^{2} + 1\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 12.39, size = 55, normalized size = 1.45 \[ -\frac {1}{2 b \left (\cos \relax (a ) \cos \relax (c )+\sin \relax (a ) \sin \relax (c )\right ) \left (\tan \left (b x +a \right ) \cos \relax (a ) \cos \relax (c )+\tan \left (b x +a \right ) \sin \relax (a ) \sin \relax (c )+\cos \relax (a ) \sin \relax (c )-\sin \relax (a ) \cos \relax (c )\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 395, normalized size = 10.39 \[ \frac {{\left (2 \, \cos \left (2 \, b x + 2 \, a + 2 \, c\right ) - \cos \left (2 \, a\right ) + \cos \left (2 \, c\right )\right )} \cos \left (4 \, b x + a + 5 \, c\right ) - 2 \, {\left (2 \, \cos \left (2 \, b x + 2 \, a + 2 \, c\right ) - \cos \left (2 \, a\right ) + \cos \left (2 \, c\right )\right )} \cos \left (2 \, b x + a + 3 \, c\right ) - {\left (\cos \left (2 \, a\right ) - \cos \left (2 \, c\right )\right )} \cos \left (a + c\right ) + 2 \, \cos \left (2 \, b x + 2 \, a + 2 \, c\right ) \cos \left (a + c\right ) + {\left (2 \, \sin \left (2 \, b x + 2 \, a + 2 \, c\right ) - \sin \left (2 \, a\right ) + \sin \left (2 \, c\right )\right )} \sin \left (4 \, b x + a + 5 \, c\right ) - 2 \, {\left (2 \, \sin \left (2 \, b x + 2 \, a + 2 \, c\right ) - \sin \left (2 \, a\right ) + \sin \left (2 \, c\right )\right )} \sin \left (2 \, b x + a + 3 \, c\right ) - {\left (\sin \left (2 \, a\right ) - \sin \left (2 \, c\right )\right )} \sin \left (a + c\right ) + 2 \, \sin \left (2 \, b x + 2 \, a + 2 \, c\right ) \sin \left (a + c\right )}{b \cos \left (4 \, b x + a + 5 \, c\right )^{2} + 4 \, b \cos \left (2 \, b x + a + 3 \, c\right )^{2} - 4 \, b \cos \left (2 \, b x + a + 3 \, c\right ) \cos \left (a + c\right ) + b \cos \left (a + c\right )^{2} + b \sin \left (4 \, b x + a + 5 \, c\right )^{2} + 4 \, b \sin \left (2 \, b x + a + 3 \, c\right )^{2} - 4 \, b \sin \left (2 \, b x + a + 3 \, c\right ) \sin \left (a + c\right ) + b \sin \left (a + c\right )^{2} - 2 \, {\left (2 \, b \cos \left (2 \, b x + a + 3 \, c\right ) - b \cos \left (a + c\right )\right )} \cos \left (4 \, b x + a + 5 \, c\right ) - 2 \, {\left (2 \, b \sin \left (2 \, b x + a + 3 \, c\right ) - b \sin \left (a + c\right )\right )} \sin \left (4 \, b x + a + 5 \, c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.03 \[ \text {Hanged} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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