Optimal. Leaf size=31 \[ \frac{64 \cos ^9(a+b x)}{9 b}-\frac{64 \cos ^7(a+b x)}{7 b} \]
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Rubi [A] time = 0.0607875, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {4288, 2565, 14} \[ \frac{64 \cos ^9(a+b x)}{9 b}-\frac{64 \cos ^7(a+b x)}{7 b} \]
Antiderivative was successfully verified.
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Rule 4288
Rule 2565
Rule 14
Rubi steps
\begin{align*} \int \csc ^3(a+b x) \sin ^6(2 a+2 b x) \, dx &=64 \int \cos ^6(a+b x) \sin ^3(a+b x) \, dx\\ &=-\frac{64 \operatorname{Subst}\left (\int x^6 \left (1-x^2\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{64 \operatorname{Subst}\left (\int \left (x^6-x^8\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{64 \cos ^7(a+b x)}{7 b}+\frac{64 \cos ^9(a+b x)}{9 b}\\ \end{align*}
Mathematica [A] time = 0.145513, size = 27, normalized size = 0.87 \[ \frac{32 \cos ^7(a+b x) (7 \cos (2 (a+b x))-11)}{63 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 35, normalized size = 1.1 \begin{align*} 64\,{\frac{1}{b} \left ( -1/9\, \left ( \sin \left ( bx+a \right ) \right ) ^{2} \left ( \cos \left ( bx+a \right ) \right ) ^{7}-{\frac{2\, \left ( \cos \left ( bx+a \right ) \right ) ^{7}}{63}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04111, size = 63, normalized size = 2.03 \begin{align*} \frac{7 \, \cos \left (9 \, b x + 9 \, a\right ) + 27 \, \cos \left (7 \, b x + 7 \, a\right ) - 168 \, \cos \left (3 \, b x + 3 \, a\right ) - 378 \, \cos \left (b x + a\right )}{252 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.492255, size = 63, normalized size = 2.03 \begin{align*} \frac{64 \,{\left (7 \, \cos \left (b x + a\right )^{9} - 9 \, \cos \left (b x + a\right )^{7}\right )}}{63 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.65083, size = 246, normalized size = 7.94 \begin{align*} -\frac{256 \,{\left (\frac{9 \,{\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac{27 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac{189 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} + \frac{189 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} + \frac{315 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{5}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{5}} + \frac{105 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{6}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{6}} + \frac{63 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{7}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{7}} - 1\right )}}{63 \, b{\left (\frac{\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1\right )}^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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