Optimal. Leaf size=15 \[ -\frac{\cot (a+b x)}{b}-x \]
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Rubi [A] time = 0.0078847, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3473, 8} \[ -\frac{\cot (a+b x)}{b}-x \]
Antiderivative was successfully verified.
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Rule 3473
Rule 8
Rubi steps
\begin{align*} \int \cot ^2(a+b x) \, dx &=-\frac{\cot (a+b x)}{b}-\int 1 \, dx\\ &=-x-\frac{\cot (a+b x)}{b}\\ \end{align*}
Mathematica [C] time = 0.0132666, size = 29, normalized size = 1.93 \[ -\frac{\cot (a+b x) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};-\tan ^2(a+b x)\right )}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 21, normalized size = 1.4 \begin{align*}{\frac{-\cot \left ( bx+a \right ) -bx-a}{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47918, size = 24, normalized size = 1.6 \begin{align*} -\frac{b x + a + \frac{1}{\tan \left (b x + a\right )}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.93702, size = 72, normalized size = 4.8 \begin{align*} -\frac{b x \sin \left (b x + a\right ) + \cos \left (b x + a\right )}{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 = 0.693617, size = 29, normalized size = 1.93 \begin{align*} \begin{cases} - x - \frac{\cos{\left (a + b x \right )}}{b \sin{\left (a + b x \right )}} & \text{for}\: b \neq 0 \\\frac{x \cos ^{2}{\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 [B] time = 1.16921, size = 47, normalized size = 3.13 \begin{align*} -\frac{2 \, b x + 2 \, a + \frac{1}{\tan \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )} - \tan \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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