Optimal. Leaf size=15 \[ -\frac {16 \cos ^5(a+b x)}{5 b} \]
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Rubi [A] time = 0.04, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {4288, 2565, 30} \[ -\frac {16 \cos ^5(a+b x)}{5 b} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2565
Rule 4288
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*}
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Mathematica [A] time = 0.01, size = 15, normalized size = 1.00 \[ -\frac {16 \cos ^5(a+b x)}{5 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 13, normalized size = 0.87 \[ -\frac {16 \, \cos \left (b x + a\right )^{5}}{5 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.98, size = 74, normalized size = 4.93 \[ \frac {32 \, {\left (\frac {10 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac {5 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} + 1\right )}}{5 \, b {\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.58, size = 14, normalized size = 0.93 \[ -\frac {16 \left (\cos ^{5}\left (b x +a \right )\right )}{5 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.82, size = 34, normalized size = 2.27 \[ -\frac {\cos \left (5 \, b x + 5 \, a\right ) + 5 \, \cos \left (3 \, b x + 3 \, a\right ) + 10 \, \cos \left (b x + a\right )}{5 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 13, normalized size = 0.87 \[ -\frac {16\,{\cos \left (a+b\,x\right )}^5}{5\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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