Optimal. Leaf size=46 \[ \frac{32 \sin ^7(a+b x)}{7 b}-\frac{64 \sin ^5(a+b x)}{5 b}+\frac{32 \sin ^3(a+b x)}{3 b} \]
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Rubi [A] time = 0.0656032, antiderivative size = 46, 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, 2564, 270} \[ \frac{32 \sin ^7(a+b x)}{7 b}-\frac{64 \sin ^5(a+b x)}{5 b}+\frac{32 \sin ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 4288
Rule 2564
Rule 270
Rubi steps
\begin{align*} \int \csc ^3(a+b x) \sin ^5(2 a+2 b x) \, dx &=32 \int \cos ^5(a+b x) \sin ^2(a+b x) \, dx\\ &=\frac{32 \operatorname{Subst}\left (\int x^2 \left (1-x^2\right )^2 \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac{32 \operatorname{Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac{32 \sin ^3(a+b x)}{3 b}-\frac{64 \sin ^5(a+b x)}{5 b}+\frac{32 \sin ^7(a+b x)}{7 b}\\ \end{align*}
Mathematica [A] time = 0.108067, size = 37, normalized size = 0.8 \[ \frac{4 \sin ^3(a+b x) (108 \cos (2 (a+b x))+15 \cos (4 (a+b x))+157)}{105 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 51, normalized size = 1.1 \begin{align*} 32\,{\frac{-1/7\,\sin \left ( bx+a \right ) \left ( \cos \left ( bx+a \right ) \right ) ^{6}+1/35\, \left ( 8/3+ \left ( \cos \left ( bx+a \right ) \right ) ^{4}+4/3\, \left ( \cos \left ( bx+a \right ) \right ) ^{2} \right ) \sin \left ( bx+a \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03676, size = 63, normalized size = 1.37 \begin{align*} -\frac{15 \, \sin \left (7 \, b x + 7 \, a\right ) + 63 \, \sin \left (5 \, b x + 5 \, a\right ) + 35 \, \sin \left (3 \, b x + 3 \, a\right ) - 525 \, \sin \left (b x + a\right )}{210 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.484142, size = 116, normalized size = 2.52 \begin{align*} -\frac{32 \,{\left (15 \, \cos \left (b x + a\right )^{6} - 3 \, \cos \left (b x + a\right )^{4} - 4 \, \cos \left (b x + a\right )^{2} - 8\right )} \sin \left (b x + a\right )}{105 \, 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 [A] time = 1.36548, size = 49, normalized size = 1.07 \begin{align*} \frac{32 \,{\left (15 \, \sin \left (b x + a\right )^{7} - 42 \, \sin \left (b x + a\right )^{5} + 35 \, \sin \left (b x + a\right )^{3}\right )}}{105 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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