Optimal. Leaf size=76 \[ \frac {512 \sin ^{15}(a+b x)}{15 b}-\frac {2048 \sin ^{13}(a+b x)}{13 b}+\frac {3072 \sin ^{11}(a+b x)}{11 b}-\frac {2048 \sin ^9(a+b x)}{9 b}+\frac {512 \sin ^7(a+b x)}{7 b} \]
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Rubi [A] time = 0.08, antiderivative size = 76, 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} \[ \frac {512 \sin ^{15}(a+b x)}{15 b}-\frac {2048 \sin ^{13}(a+b x)}{13 b}+\frac {3072 \sin ^{11}(a+b x)}{11 b}-\frac {2048 \sin ^9(a+b x)}{9 b}+\frac {512 \sin ^7(a+b x)}{7 b} \]
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 ^9(2 a+2 b x) \, dx &=512 \int \cos ^9(a+b x) \sin ^6(a+b x) \, dx\\ &=\frac {512 \operatorname {Subst}\left (\int x^6 \left (1-x^2\right )^4 \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac {512 \operatorname {Subst}\left (\int \left (x^6-4 x^8+6 x^{10}-4 x^{12}+x^{14}\right ) \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac {512 \sin ^7(a+b x)}{7 b}-\frac {2048 \sin ^9(a+b x)}{9 b}+\frac {3072 \sin ^{11}(a+b x)}{11 b}-\frac {2048 \sin ^{13}(a+b x)}{13 b}+\frac {512 \sin ^{15}(a+b x)}{15 b}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 58, normalized size = 0.76 \[ \frac {512 \left (3003 \sin ^{15}(a+b x)-13860 \sin ^{13}(a+b x)+24570 \sin ^{11}(a+b x)-20020 \sin ^9(a+b x)+6435 \sin ^7(a+b x)\right )}{45045 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 83, normalized size = 1.09 \[ -\frac {512 \, {\left (3003 \, \cos \left (b x + a\right )^{14} - 7161 \, \cos \left (b x + a\right )^{12} + 4473 \, \cos \left (b x + a\right )^{10} - 35 \, \cos \left (b x + a\right )^{8} - 40 \, \cos \left (b x + a\right )^{6} - 48 \, \cos \left (b x + a\right )^{4} - 64 \, \cos \left (b x + a\right )^{2} - 128\right )} \sin \left (b x + a\right )}{45045 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.93, size = 56, normalized size = 0.74 \[ \frac {512 \, {\left (3003 \, \sin \left (b x + a\right )^{15} - 13860 \, \sin \left (b x + a\right )^{13} + 24570 \, \sin \left (b x + a\right )^{11} - 20020 \, \sin \left (b x + a\right )^{9} + 6435 \, \sin \left (b x + a\right )^{7}\right )}}{45045 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.33, size = 107, normalized size = 1.41 \[ \frac {-\frac {512 \left (\sin ^{5}\left (b x +a \right )\right ) \left (\cos ^{10}\left (b x +a \right )\right )}{15}-\frac {512 \left (\sin ^{3}\left (b x +a \right )\right ) \left (\cos ^{10}\left (b x +a \right )\right )}{39}-\frac {512 \sin \left (b x +a \right ) \left (\cos ^{10}\left (b x +a \right )\right )}{143}+\frac {512 \left (\frac {128}{35}+\cos ^{8}\left (b x +a \right )+\frac {8 \left (\cos ^{6}\left (b x +a \right )\right )}{7}+\frac {48 \left (\cos ^{4}\left (b x +a \right )\right )}{35}+\frac {64 \left (\cos ^{2}\left (b x +a \right )\right )}{35}\right ) \sin \left (b x +a \right )}{1287}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.83, size = 91, normalized size = 1.20 \[ -\frac {3003 \, \sin \left (15 \, b x + 15 \, a\right ) + 10395 \, \sin \left (13 \, b x + 13 \, a\right ) - 12285 \, \sin \left (11 \, b x + 11 \, a\right ) - 85085 \, \sin \left (9 \, b x + 9 \, a\right ) - 19305 \, \sin \left (7 \, b x + 7 \, a\right ) + 351351 \, \sin \left (5 \, b x + 5 \, a\right ) + 375375 \, \sin \left (3 \, b x + 3 \, a\right ) - 2027025 \, \sin \left (b x + a\right )}{1441440 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 55, normalized size = 0.72 \[ \frac {\frac {512\,{\sin \left (a+b\,x\right )}^{15}}{15}-\frac {2048\,{\sin \left (a+b\,x\right )}^{13}}{13}+\frac {3072\,{\sin \left (a+b\,x\right )}^{11}}{11}-\frac {2048\,{\sin \left (a+b\,x\right )}^9}{9}+\frac {512\,{\sin \left (a+b\,x\right )}^7}{7}}{b} \]
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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