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 = 0.07, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, 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 270
Rule 2564
Rule 4288
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*}
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Mathematica [A] time = 0.17, 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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fricas [A] time = 0.43, size = 63, normalized size = 1.03 \[ \frac {128 \, {\left (105 \, \cos \left (b x + a\right )^{10} - 140 \, \cos \left (b x + a\right )^{8} + 5 \, \cos \left (b x + a\right )^{6} + 6 \, \cos \left (b x + a\right )^{4} + 8 \, \cos \left (b x + a\right )^{2} + 16\right )} \sin \left (b x + a\right )}{1155 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.34, size = 46, normalized size = 0.75 \[ -\frac {128 \, {\left (105 \, \sin \left (b x + a\right )^{11} - 385 \, \sin \left (b x + a\right )^{9} + 495 \, \sin \left (b x + a\right )^{7} - 231 \, \sin \left (b x + a\right )^{5}\right )}}{1155 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.99, size = 79, normalized size = 1.30 \[ \frac {-\frac {128 \left (\sin ^{3}\left (b x +a \right )\right ) \left (\cos ^{8}\left (b x +a \right )\right )}{11}-\frac {128 \sin \left (b x +a \right ) \left (\cos ^{8}\left (b x +a \right )\right )}{33}+\frac {128 \left (\frac {16}{5}+\cos ^{6}\left (b x +a \right )+\frac {6 \left (\cos ^{4}\left (b x +a \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (b x +a \right )\right )}{5}\right ) \sin \left (b x +a \right )}{231}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.75, size = 69, normalized size = 1.13 \[ \frac {105 \, \sin \left (11 \, b x + 11 \, a\right ) + 385 \, \sin \left (9 \, b x + 9 \, a\right ) - 165 \, \sin \left (7 \, b x + 7 \, a\right ) - 2541 \, \sin \left (5 \, b x + 5 \, a\right ) - 2310 \, \sin \left (3 \, b x + 3 \, a\right ) + 16170 \, \sin \left (b x + a\right )}{9240 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 45, normalized size = 0.74 \[ \frac {-\frac {128\,{\sin \left (a+b\,x\right )}^{11}}{11}+\frac {128\,{\sin \left (a+b\,x\right )}^9}{3}-\frac {384\,{\sin \left (a+b\,x\right )}^7}{7}+\frac {128\,{\sin \left (a+b\,x\right )}^5}{5}}{b} \]
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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