Optimal. Leaf size=15 \[ -x-\frac {\cot (a+b x)}{b} \]
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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 = {3554, 8}
\begin {gather*} -\frac {\cot (a+b x)}{b}-x \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3554
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] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.01, size = 29, normalized size = 1.93 \begin {gather*} -\frac {\cot (a+b x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\tan ^2(a+b x)\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 21, normalized size = 1.40
method | result | size |
derivativedivides | \(\frac {-\cot \left (b x +a \right )-b x -a}{b}\) | \(21\) |
default | \(\frac {-\cot \left (b x +a \right )-b x -a}{b}\) | \(21\) |
risch | \(-x -\frac {2 i}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}\) | \(24\) |
norman | \(\frac {-\frac {1}{2 b}+\frac {\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}{2 b}-x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{\tan \left (\frac {b x}{2}+\frac {a}{2}\right )}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 18, normalized size = 1.20 \begin {gather*} -\frac {b x + a + \frac {1}{\tan \left (b x + a\right )}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 29, normalized size = 1.93 \begin {gather*} -\frac {b x \sin \left (b x + a\right ) + \cos \left (b x + a\right )}{b \sin \left (b x + a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 29 vs.
\(2 (10) = 20\).
time = 0.31, size = 29, normalized size = 1.93 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 35 vs.
\(2 (15) = 30\).
time = 3.00, size = 35, normalized size = 2.33 \begin {gather*} -\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 {gather*}
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 \begin {gather*} -x-\frac {\mathrm {cot}\left (a+b\,x\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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