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 = 0.0244749, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, 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 3473
Rule 8
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*}
Mathematica [C] time = 0.0202553, size = 33, normalized size = 0.73 \[ -\frac{\cot ^5(a+b x) \text{Hypergeometric2F1}\left (-\frac{5}{2},1,-\frac{3}{2},-\tan ^2(a+b x)\right )}{5 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 46, normalized size = 1. \begin{align*}{\frac{1}{b} \left ( -{\frac{ \left ( \cot \left ( bx+a \right ) \right ) ^{5}}{5}}+{\frac{ \left ( \cot \left ( bx+a \right ) \right ) ^{3}}{3}}-\cot \left ( bx+a \right ) +{\frac{\pi }{2}}-{\rm arccot} \left (\cot \left ( bx+a \right ) \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.52765, size = 59, normalized size = 1.31 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.35594, size = 305, normalized size = 6.78 \begin{align*} -\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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.492814, size = 39, normalized size = 0.87 \begin{align*} \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}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13303, size = 123, normalized size = 2.73 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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