Optimal. Leaf size=40 \[ -\frac{3 \cot (a+b x)}{2 b}+\frac{\cos ^2(a+b x) \cot (a+b x)}{2 b}-\frac{3 x}{2} \]
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Rubi [A] time = 0.0367128, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {2591, 288, 321, 203} \[ -\frac{3 \cot (a+b x)}{2 b}+\frac{\cos ^2(a+b x) \cot (a+b x)}{2 b}-\frac{3 x}{2} \]
Antiderivative was successfully verified.
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Rule 2591
Rule 288
Rule 321
Rule 203
Rubi steps
\begin{align*} \int \cos ^2(a+b x) \cot ^2(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x^4}{\left (1+x^2\right )^2} \, dx,x,\cot (a+b x)\right )}{b}\\ &=\frac{\cos ^2(a+b x) \cot (a+b x)}{2 b}-\frac{3 \operatorname{Subst}\left (\int \frac{x^2}{1+x^2} \, dx,x,\cot (a+b x)\right )}{2 b}\\ &=-\frac{3 \cot (a+b x)}{2 b}+\frac{\cos ^2(a+b x) \cot (a+b x)}{2 b}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\cot (a+b x)\right )}{2 b}\\ &=-\frac{3 x}{2}-\frac{3 \cot (a+b x)}{2 b}+\frac{\cos ^2(a+b x) \cot (a+b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.134405, size = 31, normalized size = 0.78 \[ -\frac{6 (a+b x)+\sin (2 (a+b x))+4 \cot (a+b x)}{4 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 56, normalized size = 1.4 \begin{align*}{\frac{1}{b} \left ( -{\frac{ \left ( \cos \left ( bx+a \right ) \right ) ^{5}}{\sin \left ( bx+a \right ) }}- \left ( \left ( \cos \left ( bx+a \right ) \right ) ^{3}+{\frac{3\,\cos \left ( bx+a \right ) }{2}} \right ) \sin \left ( bx+a \right ) -{\frac{3\,bx}{2}}-{\frac{3\,a}{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48251, size = 58, normalized size = 1.45 \begin{align*} -\frac{3 \, b x + 3 \, a + \frac{3 \, \tan \left (b x + a\right )^{2} + 2}{\tan \left (b x + a\right )^{3} + \tan \left (b x + a\right )}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82469, size = 104, normalized size = 2.6 \begin{align*} \frac{\cos \left (b x + a\right )^{3} - 3 \, b x \sin \left (b x + a\right ) - 3 \, \cos \left (b x + a\right )}{2 \, b \sin \left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.63634, size = 75, normalized size = 1.88 \begin{align*} \begin{cases} - \frac{3 x \sin ^{2}{\left (a + b x \right )}}{2} - \frac{3 x \cos ^{2}{\left (a + b x \right )}}{2} - \frac{3 \sin{\left (a + b x \right )} \cos{\left (a + b x \right )}}{2 b} - \frac{\cos ^{3}{\left (a + b x \right )}}{b \sin{\left (a + b x \right )}} & \text{for}\: b \neq 0 \\\frac{x \cos ^{4}{\left (a \right )}}{\sin ^{2}{\left (a \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15988, size = 58, normalized size = 1.45 \begin{align*} -\frac{3 \, b x + 3 \, a + \frac{3 \, \tan \left (b x + a\right )^{2} + 2}{\tan \left (b x + a\right )^{3} + \tan \left (b x + a\right )}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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