Optimal. Leaf size=27 \[ -\frac {\cot ^3(a+b x)}{3 b}+\frac {\cot (a+b x)}{b}+x \]
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Rubi [A] time = 0.02, 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 = {3473, 8} \[ -\frac {\cot ^3(a+b x)}{3 b}+\frac {\cot (a+b x)}{b}+x \]
Antiderivative was successfully verified.
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Rule 8
Rule 3473
Rubi steps
\begin {align*} \int \cot ^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] time = 0.01, size = 33, normalized size = 1.22 \[ -\frac {\cot ^3(a+b x) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-\tan ^2(a+b x)\right )}{3 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 69, normalized size = 2.56 \[ \frac {4 \, \cos \left (b x + a\right )^{3} + 3 \, {\left (b x \cos \left (b x + a\right )^{2} - b x\right )} \sin \left (b x + a\right ) - 3 \, \cos \left (b x + a\right )}{3 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )} \sin \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 62, normalized size = 2.30 \[ \frac {\tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{3} + 24 \, b x + 24 \, a + \frac {15 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} - 1}{\tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{3}} - 15 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )}{24 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 26, normalized size = 0.96 \[ \frac {-\frac {\left (\cot ^{3}\left (b x +a \right )\right )}{3}+\cot \left (b x +a \right )+b x +a}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 34, normalized size = 1.26 \[ \frac {3 \, b x + 3 \, a + \frac {3 \, \tan \left (b x + a\right )^{2} - 1}{\tan \left (b x + a\right )^{3}}}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.45, size = 24, normalized size = 0.89 \[ x+\frac {{\mathrm {tan}\left (a+b\,x\right )}^2-\frac {1}{3}}{b\,{\mathrm {tan}\left (a+b\,x\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.54, size = 48, normalized size = 1.78 \[ \begin {cases} x + \frac {\cos {\left (a + b x \right )}}{b \sin {\left (a + b x \right )}} - \frac {\cos ^{3}{\left (a + b x \right )}}{3 b \sin ^{3}{\left (a + b x \right )}} & \text {for}\: b \neq 0 \\\frac {x \cos ^{4}{\relax (a )}}{\sin ^{4}{\relax (a )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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