Optimal. Leaf size=31 \[ -\frac {64 \cos ^7(a+b x)}{7 b}+\frac {64 \cos ^9(a+b x)}{9 b} \]
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Rubi [A]
time = 0.04, antiderivative size = 31, 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 = {4373, 2645, 14}
\begin {gather*} \frac {64 \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 14
Rule 2645
Rule 4373
Rubi steps
\begin {align*} \int \csc ^3(a+b x) \sin ^6(2 a+2 b x) \, dx &=64 \int \cos ^6(a+b x) \sin ^3(a+b x) \, dx\\ &=-\frac {64 \text {Subst}\left (\int x^6 \left (1-x^2\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac {64 \text {Subst}\left (\int \left (x^6-x^8\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac {64 \cos ^7(a+b x)}{7 b}+\frac {64 \cos ^9(a+b x)}{9 b}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 27, normalized size = 0.87 \begin {gather*} \frac {32 \cos ^7(a+b x) (-11+7 \cos (2 (a+b x)))}{63 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 35, normalized size = 1.13
method | result | size |
default | \(\frac {-\frac {64 \left (\sin ^{2}\left (x b +a \right )\right ) \left (\cos ^{7}\left (x b +a \right )\right )}{9}-\frac {128 \left (\cos ^{7}\left (x b +a \right )\right )}{63}}{b}\) | \(35\) |
risch | \(-\frac {3 \cos \left (x b +a \right )}{2 b}+\frac {\cos \left (9 x b +9 a \right )}{36 b}+\frac {3 \cos \left (7 x b +7 a \right )}{28 b}-\frac {2 \cos \left (3 x b +3 a \right )}{3 b}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 47, normalized size = 1.52 \begin {gather*} \frac {7 \, \cos \left (9 \, b x + 9 \, a\right ) + 27 \, \cos \left (7 \, b x + 7 \, a\right ) - 168 \, \cos \left (3 \, b x + 3 \, a\right ) - 378 \, \cos \left (b x + a\right )}{252 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.21, size = 26, normalized size = 0.84 \begin {gather*} \frac {64 \, {\left (7 \, \cos \left (b x + a\right )^{9} - 9 \, \cos \left (b x + a\right )^{7}\right )}}{63 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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. 182 vs.
\(2 (27) = 54\).
time = 0.43, size = 182, normalized size = 5.87 \begin {gather*} -\frac {256 \, {\left (\frac {9 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac {27 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac {189 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} + \frac {189 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} + \frac {315 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{5}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{5}} + \frac {105 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{6}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{6}} + \frac {63 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{7}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{7}} - 1\right )}}{63 \, b {\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1\right )}^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 26, normalized size = 0.84 \begin {gather*} -\frac {64\,\left (9\,{\cos \left (a+b\,x\right )}^7-7\,{\cos \left (a+b\,x\right )}^9\right )}{63\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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