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 = 0.04, antiderivative size = 39, 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 = {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 8
Rule 30
Rule 2606
Rule 3767
Rule 4582
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*}
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Mathematica [A] time = 0.20, size = 35, normalized size = 0.90 \[ \frac {\csc (c) \csc ^2(b x+c) (\cos (a)-\cos (a-c) \cos (2 b x+c))}{2 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 47, normalized size = 1.21 \[ \frac {2 \, \cos \left (b x + c\right ) \cos \left (-a + c\right ) \sin \left (b x + c\right ) - \sin \left (-a + c\right )}{2 \, {\left (b \cos \left (b x + c\right )^{2} - b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 145, normalized size = 3.72 \[ -\frac {\tan \left (b x + c\right ) \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (b x + c\right ) \tan \left (\frac {1}{2} \, a\right )^{2} + 4 \, \tan \left (b x + c\right ) \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) + \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - \tan \left (b x + c\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (b x + c\right ) + \tan \left (\frac {1}{2} \, a\right ) - \tan \left (\frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{2} + 1\right )} b \tan \left (b x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.34, size = 120, normalized size = 3.08 \[ \frac {-\frac {1}{\left (\cos \relax (a ) \cos \relax (c )+\sin \relax (a ) \sin \relax (c )\right )^{2} \left (\tan \left (b x +a \right ) \cos \relax (a ) \cos \relax (c )+\tan \left (b x +a \right ) \sin \relax (a ) \sin \relax (c )+\cos \relax (a ) \sin \relax (c )-\sin \relax (a ) \cos \relax (c )\right )}-\frac {\sin \relax (a ) \cos \relax (c )-\cos \relax (a ) \sin \relax (c )}{2 \left (\cos \relax (a ) \cos \relax (c )+\sin \relax (a ) \sin \relax (c )\right )^{2} \left (\tan \left (b x +a \right ) \cos \relax (a ) \cos \relax (c )+\tan \left (b x +a \right ) \sin \relax (a ) \sin \relax (c )+\cos \relax (a ) \sin \relax (c )-\sin \relax (a ) \cos \relax (c )\right )^{2}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 399, normalized size = 10.23 \[ \frac {{\left (2 \, \sin \left (2 \, b x + 2 \, a + 2 \, c\right ) - \sin \left (2 \, a\right ) - \sin \left (2 \, c\right )\right )} \cos \left (4 \, b x + a + 5 \, c\right ) - 2 \, {\left (2 \, \sin \left (2 \, b x + 2 \, a + 2 \, c\right ) - \sin \left (2 \, a\right ) - \sin \left (2 \, c\right )\right )} \cos \left (2 \, b x + a + 3 \, c\right ) - {\left (\sin \left (2 \, a\right ) + \sin \left (2 \, c\right )\right )} \cos \left (a + c\right ) - {\left (2 \, \cos \left (2 \, b x + 2 \, a + 2 \, c\right ) - \cos \left (2 \, a\right ) - \cos \left (2 \, c\right )\right )} \sin \left (4 \, b x + a + 5 \, c\right ) + 2 \, \cos \left (a + c\right ) \sin \left (2 \, b x + 2 \, a + 2 \, c\right ) + 2 \, {\left (2 \, \cos \left (2 \, b x + 2 \, a + 2 \, c\right ) - \cos \left (2 \, a\right ) - \cos \left (2 \, c\right )\right )} \sin \left (2 \, b x + a + 3 \, c\right ) + {\left (\cos \left (2 \, a\right ) + \cos \left (2 \, c\right )\right )} \sin \left (a + c\right ) - 2 \, \cos \left (2 \, b x + 2 \, a + 2 \, c\right ) \sin \left (a + c\right )}{b \cos \left (4 \, b x + a + 5 \, c\right )^{2} + 4 \, b \cos \left (2 \, b x + a + 3 \, c\right )^{2} - 4 \, b \cos \left (2 \, b x + a + 3 \, c\right ) \cos \left (a + c\right ) + b \cos \left (a + c\right )^{2} + b \sin \left (4 \, b x + a + 5 \, c\right )^{2} + 4 \, b \sin \left (2 \, b x + a + 3 \, c\right )^{2} - 4 \, b \sin \left (2 \, b x + a + 3 \, c\right ) \sin \left (a + c\right ) + b \sin \left (a + c\right )^{2} - 2 \, {\left (2 \, b \cos \left (2 \, b x + a + 3 \, c\right ) - b \cos \left (a + c\right )\right )} \cos \left (4 \, b x + a + 5 \, c\right ) - 2 \, {\left (2 \, b \sin \left (2 \, b x + a + 3 \, c\right ) - b \sin \left (a + c\right )\right )} \sin \left (4 \, b x + a + 5 \, c\right )} \]
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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