Optimal. Leaf size=55 \[ -\frac{\cot ^7(a+b x)}{7 b}-\frac{3 \cot ^5(a+b x)}{5 b}-\frac{\cot ^3(a+b x)}{b}-\frac{\cot (a+b x)}{b} \]
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Rubi [A] time = 0.0165351, antiderivative size = 55, 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} \[ -\frac{\cot ^7(a+b x)}{7 b}-\frac{3 \cot ^5(a+b x)}{5 b}-\frac{\cot ^3(a+b x)}{b}-\frac{\cot (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 3767
Rubi steps
\begin{align*} \int \csc ^8(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \left (1+3 x^2+3 x^4+x^6\right ) \, dx,x,\cot (a+b x)\right )}{b}\\ &=-\frac{\cot (a+b x)}{b}-\frac{\cot ^3(a+b x)}{b}-\frac{3 \cot ^5(a+b x)}{5 b}-\frac{\cot ^7(a+b x)}{7 b}\\ \end{align*}
Mathematica [A] time = 0.0107059, size = 77, normalized size = 1.4 \[ -\frac{16 \cot (a+b x)}{35 b}-\frac{\cot (a+b x) \csc ^6(a+b x)}{7 b}-\frac{6 \cot (a+b x) \csc ^4(a+b x)}{35 b}-\frac{8 \cot (a+b x) \csc ^2(a+b x)}{35 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 43, normalized size = 0.8 \begin{align*}{\frac{\cot \left ( bx+a \right ) }{b} \left ( -{\frac{16}{35}}-{\frac{ \left ( \csc \left ( bx+a \right ) \right ) ^{6}}{7}}-{\frac{6\, \left ( \csc \left ( bx+a \right ) \right ) ^{4}}{35}}-{\frac{8\, \left ( \csc \left ( bx+a \right ) \right ) ^{2}}{35}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01128, size = 61, normalized size = 1.11 \begin{align*} -\frac{35 \, \tan \left (b x + a\right )^{6} + 35 \, \tan \left (b x + a\right )^{4} + 21 \, \tan \left (b x + a\right )^{2} + 5}{35 \, b \tan \left (b x + a\right )^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.460868, size = 223, normalized size = 4.05 \begin{align*} -\frac{16 \, \cos \left (b x + a\right )^{7} - 56 \, \cos \left (b x + a\right )^{5} + 70 \, \cos \left (b x + a\right )^{3} - 35 \, \cos \left (b x + a\right )}{35 \,{\left (b \cos \left (b x + a\right )^{6} - 3 \, b \cos \left (b x + a\right )^{4} + 3 \, 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 ^{8}{\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 = 1.22946, size = 61, normalized size = 1.11 \begin{align*} -\frac{35 \, \tan \left (b x + a\right )^{6} + 35 \, \tan \left (b x + a\right )^{4} + 21 \, \tan \left (b x + a\right )^{2} + 5}{35 \, b \tan \left (b x + a\right )^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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