Optimal. Leaf size=61 \[ -\frac{128 \sin ^{11}(a+b x)}{11 b}+\frac{128 \sin ^9(a+b x)}{3 b}-\frac{384 \sin ^7(a+b x)}{7 b}+\frac{128 \sin ^5(a+b x)}{5 b} \]
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Rubi [A] time = 0.0712247, antiderivative size = 61, 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} \[ -\frac{128 \sin ^{11}(a+b x)}{11 b}+\frac{128 \sin ^9(a+b x)}{3 b}-\frac{384 \sin ^7(a+b x)}{7 b}+\frac{128 \sin ^5(a+b x)}{5 b} \]
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 ^7(2 a+2 b x) \, dx &=128 \int \cos ^7(a+b x) \sin ^4(a+b x) \, dx\\ &=\frac{128 \operatorname{Subst}\left (\int x^4 \left (1-x^2\right )^3 \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac{128 \operatorname{Subst}\left (\int \left (x^4-3 x^6+3 x^8-x^{10}\right ) \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac{128 \sin ^5(a+b x)}{5 b}-\frac{384 \sin ^7(a+b x)}{7 b}+\frac{128 \sin ^9(a+b x)}{3 b}-\frac{128 \sin ^{11}(a+b x)}{11 b}\\ \end{align*}
Mathematica [A] time = 0.166942, size = 48, normalized size = 0.79 \[ \frac{128 \left (-105 \sin ^{11}(a+b x)+385 \sin ^9(a+b x)-495 \sin ^7(a+b x)+231 \sin ^5(a+b x)\right )}{1155 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 79, normalized size = 1.3 \begin{align*} 128\,{\frac{1}{b} \left ( -1/11\, \left ( \sin \left ( bx+a \right ) \right ) ^{3} \left ( \cos \left ( bx+a \right ) \right ) ^{8}-1/33\,\sin \left ( bx+a \right ) \left ( \cos \left ( bx+a \right ) \right ) ^{8}+{\frac{\sin \left ( bx+a \right ) }{231} \left ({\frac{16}{5}}+ \left ( \cos \left ( bx+a \right ) \right ) ^{6}+6/5\, \left ( \cos \left ( bx+a \right ) \right ) ^{4}+8/5\, \left ( \cos \left ( bx+a \right ) \right ) ^{2} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10357, size = 93, normalized size = 1.52 \begin{align*} \frac{105 \, \sin \left (11 \, b x + 11 \, a\right ) + 385 \, \sin \left (9 \, b x + 9 \, a\right ) - 165 \, \sin \left (7 \, b x + 7 \, a\right ) - 2541 \, \sin \left (5 \, b x + 5 \, a\right ) - 2310 \, \sin \left (3 \, b x + 3 \, a\right ) + 16170 \, \sin \left (b x + a\right )}{9240 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.510657, size = 176, normalized size = 2.89 \begin{align*} \frac{128 \,{\left (105 \, \cos \left (b x + a\right )^{10} - 140 \, \cos \left (b x + a\right )^{8} + 5 \, \cos \left (b x + a\right )^{6} + 6 \, \cos \left (b x + a\right )^{4} + 8 \, \cos \left (b x + a\right )^{2} + 16\right )} \sin \left (b x + a\right )}{1155 \, 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 = 1.73332, size = 62, normalized size = 1.02 \begin{align*} -\frac{128 \,{\left (105 \, \sin \left (b x + a\right )^{11} - 385 \, \sin \left (b x + a\right )^{9} + 495 \, \sin \left (b x + a\right )^{7} - 231 \, \sin \left (b x + a\right )^{5}\right )}}{1155 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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