Optimal. Leaf size=13 \[ -\frac {\csc (a+b x)}{4 b} \]
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Rubi [A] time = 0.03, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {4287, 2606, 8} \[ -\frac {\csc (a+b x)}{4 b} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2606
Rule 4287
Rubi steps
\begin {align*} \int \cos ^3(a+b x) \csc ^2(2 a+2 b x) \, dx &=\frac {1}{4} \int \cot (a+b x) \csc (a+b x) \, dx\\ &=-\frac {\operatorname {Subst}(\int 1 \, dx,x,\csc (a+b x))}{4 b}\\ &=-\frac {\csc (a+b x)}{4 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 13, normalized size = 1.00 \[ -\frac {\csc (a+b x)}{4 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 13, normalized size = 1.00 \[ -\frac {1}{4 \, b \sin \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 13, normalized size = 1.00 \[ -\frac {1}{4 \, b \sin \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.47, size = 14, normalized size = 1.08 \[ -\frac {1}{4 b \sin \left (b x +a \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 84, normalized size = 6.46 \[ -\frac {\cos \left (b x + a\right ) \sin \left (2 \, b x + 2 \, a\right ) - \cos \left (2 \, b x + 2 \, a\right ) \sin \left (b x + a\right ) + \sin \left (b x + a\right )}{2 \, {\left (b \cos \left (2 \, b x + 2 \, a\right )^{2} + b \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, b \cos \left (2 \, b x + 2 \, a\right ) + b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.02, size = 13, normalized size = 1.00 \[ -\frac {1}{4\,b\,\sin \left (a+b\,x\right )} \]
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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