Optimal. Leaf size=15 \[ -\frac{16 \cos ^5(a+b x)}{5 b} \]
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Rubi [A] time = 0.0444241, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {4288, 2565, 30} \[ -\frac{16 \cos ^5(a+b x)}{5 b} \]
Antiderivative was successfully verified.
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Rule 4288
Rule 2565
Rule 30
Rubi steps
\begin{align*} \int \csc ^3(a+b x) \sin ^4(2 a+2 b x) \, dx &=16 \int \cos ^4(a+b x) \sin (a+b x) \, dx\\ &=-\frac{16 \operatorname{Subst}\left (\int x^4 \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{16 \cos ^5(a+b x)}{5 b}\\ \end{align*}
Mathematica [A] time = 0.0096055, size = 15, normalized size = 1. \[ -\frac{16 \cos ^5(a+b x)}{5 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 14, normalized size = 0.9 \begin{align*} -{\frac{16\, \left ( \cos \left ( bx+a \right ) \right ) ^{5}}{5\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.06807, size = 46, normalized size = 3.07 \begin{align*} -\frac{\cos \left (5 \, b x + 5 \, a\right ) + 5 \, \cos \left (3 \, b x + 3 \, a\right ) + 10 \, \cos \left (b x + a\right )}{5 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.479734, size = 32, normalized size = 2.13 \begin{align*} -\frac{16 \, \cos \left (b x + a\right )^{5}}{5 \, 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 = 1.42222, size = 100, normalized size = 6.67 \begin{align*} \frac{32 \,{\left (\frac{10 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac{5 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} + 1\right )}}{5 \, b{\left (\frac{\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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