Optimal. Leaf size=46 \[ -\frac {64 \cos ^7(a+b x)}{7 b}+\frac {128 \cos ^9(a+b x)}{9 b}-\frac {64 \cos ^{11}(a+b x)}{11 b} \]
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Rubi [A]
time = 0.04, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4373, 2645,
276} \begin {gather*} -\frac {64 \cos ^{11}(a+b x)}{11 b}+\frac {128 \cos ^9(a+b x)}{9 b}-\frac {64 \cos ^7(a+b x)}{7 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 276
Rule 2645
Rule 4373
Rubi steps
\begin {align*} \int \csc (a+b x) \sin ^6(2 a+2 b x) \, dx &=64 \int \cos ^6(a+b x) \sin ^5(a+b x) \, dx\\ &=-\frac {64 \text {Subst}\left (\int x^6 \left (1-x^2\right )^2 \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac {64 \text {Subst}\left (\int \left (x^6-2 x^8+x^{10}\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac {64 \cos ^7(a+b x)}{7 b}+\frac {128 \cos ^9(a+b x)}{9 b}-\frac {64 \cos ^{11}(a+b x)}{11 b}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 89, normalized size = 1.93 \begin {gather*} -\frac {5 \cos (a+b x)}{8 b}-\frac {5 \cos (3 (a+b x))}{24 b}+\frac {\cos (5 (a+b x))}{16 b}+\frac {5 \cos (7 (a+b x))}{112 b}-\frac {\cos (9 (a+b x))}{144 b}-\frac {\cos (11 (a+b x))}{176 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 53, normalized size = 1.15
method | result | size |
default | \(\frac {-\frac {64 \left (\sin ^{4}\left (x b +a \right )\right ) \left (\cos ^{7}\left (x b +a \right )\right )}{11}-\frac {256 \left (\sin ^{2}\left (x b +a \right )\right ) \left (\cos ^{7}\left (x b +a \right )\right )}{99}-\frac {512 \left (\cos ^{7}\left (x b +a \right )\right )}{693}}{b}\) | \(53\) |
risch | \(-\frac {5 \cos \left (x b +a \right )}{8 b}-\frac {\cos \left (11 x b +11 a \right )}{176 b}-\frac {\cos \left (9 x b +9 a \right )}{144 b}+\frac {5 \cos \left (7 x b +7 a \right )}{112 b}+\frac {\cos \left (5 x b +5 a \right )}{16 b}-\frac {5 \cos \left (3 x b +3 a \right )}{24 b}\) | \(83\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 69, normalized size = 1.50 \begin {gather*} -\frac {63 \, \cos \left (11 \, b x + 11 \, a\right ) + 77 \, \cos \left (9 \, b x + 9 \, a\right ) - 495 \, \cos \left (7 \, b x + 7 \, a\right ) - 693 \, \cos \left (5 \, b x + 5 \, a\right ) + 2310 \, \cos \left (3 \, b x + 3 \, a\right ) + 6930 \, \cos \left (b x + a\right )}{11088 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.36, size = 36, normalized size = 0.78 \begin {gather*} -\frac {64 \, {\left (63 \, \cos \left (b x + a\right )^{11} - 154 \, \cos \left (b x + a\right )^{9} + 99 \, \cos \left (b x + a\right )^{7}\right )}}{693 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 204 vs.
\(2 (40) = 80\).
time = 0.43, size = 204, normalized size = 4.43 \begin {gather*} -\frac {1024 \, {\left (\frac {11 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac {55 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} - \frac {297 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} - \frac {1485 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} - \frac {2079 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{5}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{5}} - \frac {2541 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{6}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{6}} - \frac {1155 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{7}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{7}} - \frac {462 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{8}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{8}} - 1\right )}}{693 \, b {\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1\right )}^{11}} \end {gather*}
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 \begin {gather*} -\frac {64\,\left (63\,{\cos \left (a+b\,x\right )}^{11}-154\,{\cos \left (a+b\,x\right )}^9+99\,{\cos \left (a+b\,x\right )}^7\right )}{693\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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