Optimal. Leaf size=15 \[ -\frac {\csc ^3(a+b x)}{3 b} \]
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Rubi [A] time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2606, 30} \[ -\frac {\csc ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2606
Rubi steps
\begin {align*} \int \cot (a+b x) \csc ^3(a+b x) \, dx &=-\frac {\operatorname {Subst}\left (\int x^2 \, dx,x,\csc (a+b x)\right )}{b}\\ &=-\frac {\csc ^3(a+b x)}{3 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 15, normalized size = 1.00 \[ -\frac {\csc ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 26, normalized size = 1.73 \[ \frac {1}{3 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )} \sin \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.57, size = 13, normalized size = 0.87 \[ -\frac {1}{3 \, b \sin \left (b x + a\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 14, normalized size = 0.93 \[ -\frac {1}{3 \sin \left (b x +a \right )^{3} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 13, normalized size = 0.87 \[ -\frac {1}{3 \, b \sin \left (b x + a\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.43, size = 13, normalized size = 0.87 \[ -\frac {1}{3\,b\,{\sin \left (a+b\,x\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.57, size = 24, normalized size = 1.60 \[ \begin {cases} - \frac {1}{3 b \sin ^{3}{\left (a + b x \right )}} & \text {for}\: b \neq 0 \\\frac {x \cos {\relax (a )}}{\sin ^{4}{\relax (a )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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