Optimal. Leaf size=53 \[ -\frac {\sin ^3(a+b x)}{3 b}+\frac {3 \sin (a+b x)}{b}-\frac {\csc ^3(a+b x)}{3 b}+\frac {3 \csc (a+b x)}{b} \]
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Rubi [A] time = 0.04, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2590, 270} \[ -\frac {\sin ^3(a+b x)}{3 b}+\frac {3 \sin (a+b x)}{b}-\frac {\csc ^3(a+b x)}{3 b}+\frac {3 \csc (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 270
Rule 2590
Rubi steps
\begin {align*} \int \cos ^3(a+b x) \cot ^4(a+b x) \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^3}{x^4} \, dx,x,-\sin (a+b x)\right )}{b}\\ &=-\frac {\operatorname {Subst}\left (\int \left (3+\frac {1}{x^4}-\frac {3}{x^2}-x^2\right ) \, dx,x,-\sin (a+b x)\right )}{b}\\ &=\frac {3 \csc (a+b x)}{b}-\frac {\csc ^3(a+b x)}{3 b}+\frac {3 \sin (a+b x)}{b}-\frac {\sin ^3(a+b x)}{3 b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 53, normalized size = 1.00 \[ -\frac {\sin ^3(a+b x)}{3 b}+\frac {3 \sin (a+b x)}{b}-\frac {\csc ^3(a+b x)}{3 b}+\frac {3 \csc (a+b x)}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 56, normalized size = 1.06 \[ -\frac {\cos \left (b x + a\right )^{6} + 6 \, \cos \left (b x + a\right )^{4} - 24 \, \cos \left (b x + a\right )^{2} + 16}{3 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )} \sin \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 41, normalized size = 0.77 \[ -\frac {{\left (\frac {1}{\sin \left (b x + a\right )} + \sin \left (b x + a\right )\right )}^{3} - \frac {12}{\sin \left (b x + a\right )} - 12 \, \sin \left (b x + a\right )}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 80, normalized size = 1.51 \[ \frac {-\frac {\cos ^{8}\left (b x +a \right )}{3 \sin \left (b x +a \right )^{3}}+\frac {5 \left (\cos ^{8}\left (b x +a \right )\right )}{3 \sin \left (b x +a \right )}+\frac {5 \left (\frac {16}{5}+\cos ^{6}\left (b x +a \right )+\frac {6 \left (\cos ^{4}\left (b x +a \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (b x +a \right )\right )}{5}\right ) \sin \left (b x +a \right )}{3}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 44, normalized size = 0.83 \[ -\frac {\sin \left (b x + a\right )^{3} - \frac {9 \, \sin \left (b x + a\right )^{2} - 1}{\sin \left (b x + a\right )^{3}} - 9 \, \sin \left (b x + a\right )}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.48, size = 45, normalized size = 0.85 \[ \frac {-{\sin \left (a+b\,x\right )}^6+9\,{\sin \left (a+b\,x\right )}^4+9\,{\sin \left (a+b\,x\right )}^2-1}{3\,b\,{\sin \left (a+b\,x\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 9.59, size = 82, normalized size = 1.55 \[ \begin {cases} \frac {16 \sin ^{3}{\left (a + b x \right )}}{3 b} + \frac {8 \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{b} + \frac {2 \cos ^{4}{\left (a + b x \right )}}{b \sin {\left (a + b x \right )}} - \frac {\cos ^{6}{\left (a + b x \right )}}{3 b \sin ^{3}{\left (a + b x \right )}} & \text {for}\: b \neq 0 \\\frac {x \cos ^{7}{\relax (a )}}{\sin ^{4}{\relax (a )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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