Optimal. Leaf size=11 \[ -\frac {2 \csc (a+b x)}{b} \]
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Rubi [A] time = 0.03, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4288, 2606, 8} \[ -\frac {2 \csc (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2606
Rule 4288
Rubi steps
\begin {align*} \int \csc ^3(a+b x) \sin (2 a+2 b x) \, dx &=2 \int \cot (a+b x) \csc (a+b x) \, dx\\ &=-\frac {2 \operatorname {Subst}(\int 1 \, dx,x,\csc (a+b x))}{b}\\ &=-\frac {2 \csc (a+b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 11, normalized size = 1.00 \[ -\frac {2 \csc (a+b x)}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 13, normalized size = 1.18 \[ -\frac {2}{b \sin \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 13, normalized size = 1.18 \[ -\frac {2}{b \sin \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 14, normalized size = 1.27 \[ -\frac {2}{b \sin \left (b x +a \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 84, normalized size = 7.64 \[ -\frac {4 \, {\left (\cos \left (b x + a\right ) \sin \left (2 \, b x + 2 \, a\right ) - \cos \left (2 \, b x + 2 \, a\right ) \sin \left (b x + a\right ) + \sin \left (b x + a\right )\right )}}{b \cos \left (2 \, b x + 2 \, a\right )^{2} + b \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, b \cos \left (2 \, b x + 2 \, a\right ) + b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 13, normalized size = 1.18 \[ -\frac {2}{b\,\sin \left (a+b\,x\right )} \]
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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