Optimal. Leaf size=45 \[ -\frac {\cot ^5(a+b x)}{5 b}+\frac {\cot ^3(a+b x)}{3 b}-\frac {\cot (a+b x)}{b}-x \]
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Rubi [A] time = 0.02, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3473, 8} \[ -\frac {\cot ^5(a+b x)}{5 b}+\frac {\cot ^3(a+b x)}{3 b}-\frac {\cot (a+b x)}{b}-x \]
Antiderivative was successfully verified.
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Rule 8
Rule 3473
Rubi steps
\begin {align*} \int \cot ^6(a+b x) \, dx &=-\frac {\cot ^5(a+b x)}{5 b}-\int \cot ^4(a+b x) \, dx\\ &=\frac {\cot ^3(a+b x)}{3 b}-\frac {\cot ^5(a+b x)}{5 b}+\int \cot ^2(a+b x) \, dx\\ &=-\frac {\cot (a+b x)}{b}+\frac {\cot ^3(a+b x)}{3 b}-\frac {\cot ^5(a+b x)}{5 b}-\int 1 \, dx\\ &=-x-\frac {\cot (a+b x)}{b}+\frac {\cot ^3(a+b x)}{3 b}-\frac {\cot ^5(a+b x)}{5 b}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 33, normalized size = 0.73 \[ -\frac {\cot ^5(a+b x) \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};-\tan ^2(a+b x)\right )}{5 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.61, size = 123, normalized size = 2.73 \[ -\frac {23 \, \cos \left (2 \, b x + 2 \, a\right )^{3} - \cos \left (2 \, b x + 2 \, a\right )^{2} + 15 \, {\left (b x \cos \left (2 \, b x + 2 \, a\right )^{2} - 2 \, b x \cos \left (2 \, b x + 2 \, a\right ) + b x\right )} \sin \left (2 \, b x + 2 \, a\right ) - 11 \, \cos \left (2 \, b x + 2 \, a\right ) + 13}{15 \, {\left (b \cos \left (2 \, b x + 2 \, a\right )^{2} - 2 \, b \cos \left (2 \, b x + 2 \, a\right ) + b\right )} \sin \left (2 \, b x + 2 \, a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.14, size = 91, normalized size = 2.02 \[ \frac {3 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{5} - 35 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{3} - 480 \, b x - 480 \, a - \frac {330 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} - 35 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} + 3}{\tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{5}} + 330 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )}{480 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 46, normalized size = 1.02 \[ \frac {-\frac {\left (\cot ^{5}\left (b x +a \right )\right )}{5}+\frac {\left (\cot ^{3}\left (b x +a \right )\right )}{3}-\cot \left (b x +a \right )+\frac {\pi }{2}-\mathrm {arccot}\left (\cot \left (b x +a \right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 44, normalized size = 0.98 \[ -\frac {15 \, b x + 15 \, a + \frac {15 \, \tan \left (b x + a\right )^{4} - 5 \, \tan \left (b x + a\right )^{2} + 3}{\tan \left (b x + a\right )^{5}}}{15 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 36, normalized size = 0.80 \[ -x-\frac {\frac {{\mathrm {cot}\left (a+b\,x\right )}^5}{5}-\frac {{\mathrm {cot}\left (a+b\,x\right )}^3}{3}+\mathrm {cot}\left (a+b\,x\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.41, size = 39, normalized size = 0.87 \[ \begin {cases} - x - \frac {\cot ^{5}{\left (a + b x \right )}}{5 b} + \frac {\cot ^{3}{\left (a + b x \right )}}{3 b} - \frac {\cot {\left (a + b x \right )}}{b} & \text {for}\: b \neq 0 \\x \cot ^{6}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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