Optimal. Leaf size=61 \[ \frac {64 \cos ^{13}(a+b x)}{13 b}-\frac {192 \cos ^{11}(a+b x)}{11 b}+\frac {64 \cos ^9(a+b x)}{3 b}-\frac {64 \cos ^7(a+b x)}{7 b} \]
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Rubi [A] time = 0.06, antiderivative size = 61, 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 = {4288, 2565, 270} \[ \frac {64 \cos ^{13}(a+b x)}{13 b}-\frac {192 \cos ^{11}(a+b x)}{11 b}+\frac {64 \cos ^9(a+b x)}{3 b}-\frac {64 \cos ^7(a+b x)}{7 b} \]
Antiderivative was successfully verified.
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Rule 270
Rule 2565
Rule 4288
Rubi steps
\begin {align*} \int \sin (a+b x) \sin ^6(2 a+2 b x) \, dx &=64 \int \cos ^6(a+b x) \sin ^7(a+b x) \, dx\\ &=-\frac {64 \operatorname {Subst}\left (\int x^6 \left (1-x^2\right )^3 \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac {64 \operatorname {Subst}\left (\int \left (x^6-3 x^8+3 x^{10}-x^{12}\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)}{3 b}-\frac {192 \cos ^{11}(a+b x)}{11 b}+\frac {64 \cos ^{13}(a+b x)}{13 b}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 47, normalized size = 0.77 \[ \frac {2 \cos ^7(a+b x) (6377 \cos (2 (a+b x))-1890 \cos (4 (a+b x))+231 \cos (6 (a+b x))-5230)}{3003 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 46, normalized size = 0.75 \[ \frac {64 \, {\left (231 \, \cos \left (b x + a\right )^{13} - 819 \, \cos \left (b x + a\right )^{11} + 1001 \, \cos \left (b x + a\right )^{9} - 429 \, \cos \left (b x + a\right )^{7}\right )}}{3003 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.91, size = 96, normalized size = 1.57 \[ \frac {\cos \left (13 \, b x + 13 \, a\right )}{832 \, b} - \frac {\cos \left (11 \, b x + 11 \, a\right )}{704 \, b} - \frac {\cos \left (9 \, b x + 9 \, a\right )}{96 \, b} + \frac {3 \, \cos \left (7 \, b x + 7 \, a\right )}{224 \, b} + \frac {3 \, \cos \left (5 \, b x + 5 \, a\right )}{64 \, b} - \frac {5 \, \cos \left (3 \, b x + 3 \, a\right )}{64 \, b} - \frac {5 \, \cos \left (b x + a\right )}{16 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 97, normalized size = 1.59 \[ -\frac {5 \cos \left (b x +a \right )}{16 b}-\frac {5 \cos \left (3 b x +3 a \right )}{64 b}+\frac {3 \cos \left (5 b x +5 a \right )}{64 b}+\frac {3 \cos \left (7 b x +7 a \right )}{224 b}-\frac {\cos \left (9 b x +9 a \right )}{96 b}-\frac {\cos \left (11 b x +11 a \right )}{704 b}+\frac {\cos \left (13 b x +13 a \right )}{832 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 80, normalized size = 1.31 \[ \frac {231 \, \cos \left (13 \, b x + 13 \, a\right ) - 273 \, \cos \left (11 \, b x + 11 \, a\right ) - 2002 \, \cos \left (9 \, b x + 9 \, a\right ) + 2574 \, \cos \left (7 \, b x + 7 \, a\right ) + 9009 \, \cos \left (5 \, b x + 5 \, a\right ) - 15015 \, \cos \left (3 \, b x + 3 \, a\right ) - 60060 \, \cos \left (b x + a\right )}{192192 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 46, normalized size = 0.75 \[ -\frac {-\frac {64\,{\cos \left (a+b\,x\right )}^{13}}{13}+\frac {192\,{\cos \left (a+b\,x\right )}^{11}}{11}-\frac {64\,{\cos \left (a+b\,x\right )}^9}{3}+\frac {64\,{\cos \left (a+b\,x\right )}^7}{7}}{b} \]
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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