Optimal. Leaf size=31 \[ \frac {8 \sin ^3(a+b x)}{3 b}-\frac {8 \sin ^5(a+b x)}{5 b} \]
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Rubi [A] time = 0.05, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4288, 2564, 14} \[ \frac {8 \sin ^3(a+b x)}{3 b}-\frac {8 \sin ^5(a+b x)}{5 b} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2564
Rule 4288
Rubi steps
\begin {align*} \int \csc (a+b x) \sin ^3(2 a+2 b x) \, dx &=8 \int \cos ^3(a+b x) \sin ^2(a+b x) \, dx\\ &=\frac {8 \operatorname {Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac {8 \operatorname {Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac {8 \sin ^3(a+b x)}{3 b}-\frac {8 \sin ^5(a+b x)}{5 b}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 28, normalized size = 0.90 \[ \frac {8 \left (5 \sin ^3(a+b x)-3 \sin ^5(a+b x)\right )}{15 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 33, normalized size = 1.06 \[ -\frac {8 \, {\left (3 \, \cos \left (b x + a\right )^{4} - \cos \left (b x + a\right )^{2} - 2\right )} \sin \left (b x + a\right )}{15 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 26, normalized size = 0.84 \[ -\frac {8 \, {\left (3 \, \sin \left (b x + a\right )^{5} - 5 \, \sin \left (b x + a\right )^{3}\right )}}{15 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.05, size = 41, normalized size = 1.32 \[ \frac {-\frac {8 \sin \left (b x +a \right ) \left (\cos ^{4}\left (b x +a \right )\right )}{5}+\frac {8 \left (2+\cos ^{2}\left (b x +a \right )\right ) \sin \left (b x +a \right )}{15}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 36, normalized size = 1.16 \[ -\frac {3 \, \sin \left (5 \, b x + 5 \, a\right ) + 5 \, \sin \left (3 \, b x + 3 \, a\right ) - 30 \, \sin \left (b x + a\right )}{30 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 26, normalized size = 0.84 \[ \frac {8\,\left (5\,{\sin \left (a+b\,x\right )}^3-3\,{\sin \left (a+b\,x\right )}^5\right )}{15\,b} \]
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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