Optimal. Leaf size=27 \[ x+\frac {\cot (a+b x)}{b}-\frac {\cot ^3(a+b x)}{3 b} \]
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Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 3, 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 ^3(a+b x)}{3 b}+\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 ^4(a+b x) \, dx &=-\frac {\cot ^3(a+b x)}{3 b}-\int \cot ^2(a+b x) \, dx\\ &=\frac {\cot (a+b x)}{b}-\frac {\cot ^3(a+b x)}{3 b}+\int 1 \, dx\\ &=x+\frac {\cot (a+b x)}{b}-\frac {\cot ^3(a+b x)}{3 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 = 0.02, size = 33, normalized size = 1.22 \begin {gather*} -\frac {\cot ^3(a+b x) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-\tan ^2(a+b x)\right )}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 32, normalized size = 1.19
method | result | size |
derivativedivides | \(\frac {-\frac {\left (\cot ^{3}\left (b x +a \right )\right )}{3}+\cot \left (b x +a \right )-\frac {\pi }{2}+\mathrm {arccot}\left (\cot \left (b x +a \right )\right )}{b}\) | \(32\) |
default | \(\frac {-\frac {\left (\cot ^{3}\left (b x +a \right )\right )}{3}+\cot \left (b x +a \right )-\frac {\pi }{2}+\mathrm {arccot}\left (\cot \left (b x +a \right )\right )}{b}\) | \(32\) |
norman | \(\frac {\frac {\tan ^{2}\left (b x +a \right )}{b}+x \left (\tan ^{3}\left (b x +a \right )\right )-\frac {1}{3 b}}{\tan \left (b x +a \right )^{3}}\) | \(38\) |
risch | \(x +\frac {4 i \left (3 \,{\mathrm e}^{4 i \left (b x +a \right )}-3 \,{\mathrm e}^{2 i \left (b x +a \right )}+2\right )}{3 b \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3}}\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 34, normalized size = 1.26 \begin {gather*} \frac {3 \, b x + 3 \, a + \frac {3 \, \tan \left (b x + a\right )^{2} - 1}{\tan \left (b x + a\right )^{3}}}{3 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 84 vs.
\(2 (25) = 50\).
time = 2.41, size = 84, normalized size = 3.11 \begin {gather*} \frac {4 \, \cos \left (2 \, b x + 2 \, a\right )^{2} + 3 \, {\left (b x \cos \left (2 \, b x + 2 \, a\right ) - b x\right )} \sin \left (2 \, b x + 2 \, a\right ) + 2 \, \cos \left (2 \, b x + 2 \, a\right ) - 2}{3 \, {\left (b \cos \left (2 \, b x + 2 \, a\right ) - b\right )} \sin \left (2 \, b x + 2 \, a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 27, normalized size = 1.00 \begin {gather*} \begin {cases} x - \frac {\cot ^{3}{\left (a + b x \right )}}{3 b} + \frac {\cot {\left (a + b x \right )}}{b} & \text {for}\: b \neq 0 \\x \cot ^{4}{\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. 62 vs.
\(2 (25) = 50\).
time = 0.44, size = 62, normalized size = 2.30 \begin {gather*} \frac {\tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{3} + 24 \, b x + 24 \, a + \frac {15 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} - 1}{\tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{3}} - 15 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )}{24 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.12, size = 23, normalized size = 0.85 \begin {gather*} x+\frac {\mathrm {cot}\left (a+b\,x\right )-\frac {{\mathrm {cot}\left (a+b\,x\right )}^3}{3}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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