Optimal. Leaf size=15 \[ -\frac {\cot (a+b x)}{b}-x \]
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Rubi [A] time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3473, 8} \[ -\frac {\cot (a+b x)}{b}-x \]
Antiderivative was successfully verified.
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Rule 8
Rule 3473
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*}
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Mathematica [C] time = 0.01, 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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fricas [A] time = 0.44, size = 29, normalized size = 1.93 \[ -\frac {b x \sin \left (b x + a\right ) + \cos \left (b x + a\right )}{b \sin \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 35, normalized size = 2.33 \[ -\frac {2 \, b x + 2 \, a + \frac {1}{\tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )} - \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 21, normalized size = 1.40 \[ \frac {-\cot \left (b x +a \right )-b x -a}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 18, normalized size = 1.20 \[ -\frac {b x + a + \frac {1}{\tan \left (b x + a\right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.41, size = 15, normalized size = 1.00 \[ -x-\frac {\mathrm {cot}\left (a+b\,x\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.08, size = 29, normalized size = 1.93 \[ \begin {cases} - x - \frac {\cos {\left (a + b x \right )}}{b \sin {\left (a + b x \right )}} & \text {for}\: b \neq 0 \\\frac {x \cos ^{2}{\relax (a )}}{\sin ^{2}{\relax (a )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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