Optimal. Leaf size=42 \[ -\frac{\cot ^5(a+b x)}{5 b}-\frac{2 \cot ^3(a+b x)}{3 b}-\frac{\cot (a+b x)}{b} \]
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Rubi [A] time = 0.0139804, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3767} \[ -\frac{\cot ^5(a+b x)}{5 b}-\frac{2 \cot ^3(a+b x)}{3 b}-\frac{\cot (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 3767
Rubi steps
\begin{align*} \int \csc ^6(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (a+b x)\right )}{b}\\ &=-\frac{\cot (a+b x)}{b}-\frac{2 \cot ^3(a+b x)}{3 b}-\frac{\cot ^5(a+b x)}{5 b}\\ \end{align*}
Mathematica [A] time = 0.0162388, size = 56, normalized size = 1.33 \[ -\frac{8 \cot (a+b x)}{15 b}-\frac{\cot (a+b x) \csc ^4(a+b x)}{5 b}-\frac{4 \cot (a+b x) \csc ^2(a+b x)}{15 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 33, normalized size = 0.8 \begin{align*}{\frac{\cot \left ( bx+a \right ) }{b} \left ( -{\frac{8}{15}}-{\frac{ \left ( \csc \left ( bx+a \right ) \right ) ^{4}}{5}}-{\frac{4\, \left ( \csc \left ( bx+a \right ) \right ) ^{2}}{15}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01335, size = 47, normalized size = 1.12 \begin{align*} -\frac{15 \, \tan \left (b x + a\right )^{4} + 10 \, \tan \left (b x + a\right )^{2} + 3}{15 \, b \tan \left (b x + a\right )^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.452196, size = 166, normalized size = 3.95 \begin{align*} -\frac{8 \, \cos \left (b x + a\right )^{5} - 20 \, \cos \left (b x + a\right )^{3} + 15 \, \cos \left (b x + a\right )}{15 \,{\left (b \cos \left (b x + a\right )^{4} - 2 \, b \cos \left (b x + a\right )^{2} + b\right )} \sin \left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \csc ^{6}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26094, size = 47, normalized size = 1.12 \begin{align*} -\frac{15 \, \tan \left (b x + a\right )^{4} + 10 \, \tan \left (b x + a\right )^{2} + 3}{15 \, b \tan \left (b x + a\right )^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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