Optimal. Leaf size=46 \[ -\frac {256 \cos ^{13}(a+b x)}{13 b}+\frac {512 \cos ^{11}(a+b x)}{11 b}-\frac {256 \cos ^9(a+b x)}{9 b} \]
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Rubi [A] time = 0.07, antiderivative size = 46, 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, 2565, 270} \[ -\frac {256 \cos ^{13}(a+b x)}{13 b}+\frac {512 \cos ^{11}(a+b x)}{11 b}-\frac {256 \cos ^9(a+b x)}{9 b} \]
Antiderivative was successfully verified.
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Rule 270
Rule 2565
Rule 4288
Rubi steps
\begin {align*} \int \csc ^3(a+b x) \sin ^8(2 a+2 b x) \, dx &=256 \int \cos ^8(a+b x) \sin ^5(a+b x) \, dx\\ &=-\frac {256 \operatorname {Subst}\left (\int x^8 \left (1-x^2\right )^2 \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac {256 \operatorname {Subst}\left (\int \left (x^8-2 x^{10}+x^{12}\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac {256 \cos ^9(a+b x)}{9 b}+\frac {512 \cos ^{11}(a+b x)}{11 b}-\frac {256 \cos ^{13}(a+b x)}{13 b}\\ \end {align*}
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Mathematica [B] time = 0.10, size = 104, normalized size = 2.26 \[ -\frac {5 \cos (a+b x)}{4 b}-\frac {25 \cos (3 (a+b x))}{48 b}+\frac {\cos (5 (a+b x))}{16 b}+\frac {\cos (7 (a+b x))}{8 b}+\frac {\cos (9 (a+b x))}{72 b}-\frac {3 \cos (11 (a+b x))}{176 b}-\frac {\cos (13 (a+b x))}{208 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 36, normalized size = 0.78 \[ -\frac {256 \, {\left (99 \, \cos \left (b x + a\right )^{13} - 234 \, \cos \left (b x + a\right )^{11} + 143 \, \cos \left (b x + a\right )^{9}\right )}}{1287 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.34, size = 248, normalized size = 5.39 \[ -\frac {4096 \, {\left (\frac {13 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac {78 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} - \frac {572 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} - \frac {3718 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} - \frac {7722 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{5}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{5}} - \frac {13728 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{6}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{6}} - \frac {12012 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{7}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{7}} - \frac {9009 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{8}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{8}} - \frac {3003 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{9}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{9}} - \frac {858 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{10}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{10}} - 1\right )}}{1287 \, b {\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1\right )}^{13}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.77, size = 53, normalized size = 1.15 \[ \frac {-\frac {256 \left (\sin ^{4}\left (b x +a \right )\right ) \left (\cos ^{9}\left (b x +a \right )\right )}{13}-\frac {1024 \left (\sin ^{2}\left (b x +a \right )\right ) \left (\cos ^{9}\left (b x +a \right )\right )}{143}-\frac {2048 \left (\cos ^{9}\left (b x +a \right )\right )}{1287}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.02, size = 80, normalized size = 1.74 \[ -\frac {99 \, \cos \left (13 \, b x + 13 \, a\right ) + 351 \, \cos \left (11 \, b x + 11 \, a\right ) - 286 \, \cos \left (9 \, b x + 9 \, a\right ) - 2574 \, \cos \left (7 \, b x + 7 \, a\right ) - 1287 \, \cos \left (5 \, b x + 5 \, a\right ) + 10725 \, \cos \left (3 \, b x + 3 \, a\right ) + 25740 \, \cos \left (b x + a\right )}{20592 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 36, normalized size = 0.78 \[ -\frac {256\,\left (99\,{\cos \left (a+b\,x\right )}^{13}-234\,{\cos \left (a+b\,x\right )}^{11}+143\,{\cos \left (a+b\,x\right )}^9\right )}{1287\,b} \]
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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