Optimal. Leaf size=61 \[ \frac {128 \cos ^{15}(a+b x)}{15 b}-\frac {384 \cos ^{13}(a+b x)}{13 b}+\frac {384 \cos ^{11}(a+b x)}{11 b}-\frac {128 \cos ^9(a+b x)}{9 b} \]
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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 = {4287, 2565, 270} \[ \frac {128 \cos ^{15}(a+b x)}{15 b}-\frac {384 \cos ^{13}(a+b x)}{13 b}+\frac {384 \cos ^{11}(a+b x)}{11 b}-\frac {128 \cos ^9(a+b x)}{9 b} \]
Antiderivative was successfully verified.
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Rule 270
Rule 2565
Rule 4287
Rubi steps
\begin {align*} \int \cos (a+b x) \sin ^7(2 a+2 b x) \, dx &=128 \int \cos ^8(a+b x) \sin ^7(a+b x) \, dx\\ &=-\frac {128 \operatorname {Subst}\left (\int x^8 \left (1-x^2\right )^3 \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac {128 \operatorname {Subst}\left (\int \left (x^8-3 x^{10}+3 x^{12}-x^{14}\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac {128 \cos ^9(a+b x)}{9 b}+\frac {384 \cos ^{11}(a+b x)}{11 b}-\frac {384 \cos ^{13}(a+b x)}{13 b}+\frac {128 \cos ^{15}(a+b x)}{15 b}\\ \end {align*}
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Mathematica [A] time = 0.44, size = 47, normalized size = 0.77 \[ \frac {4 \cos ^9(a+b x) (10755 \cos (2 (a+b x))-3366 \cos (4 (a+b x))+429 \cos (6 (a+b x))-8330)}{6435 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 46, normalized size = 0.75 \[ \frac {128 \, {\left (429 \, \cos \left (b x + a\right )^{15} - 1485 \, \cos \left (b x + a\right )^{13} + 1755 \, \cos \left (b x + a\right )^{11} - 715 \, \cos \left (b x + a\right )^{9}\right )}}{6435 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.90, size = 110, normalized size = 1.80 \[ \frac {\cos \left (15 \, b x + 15 \, a\right )}{1920 \, b} + \frac {\cos \left (13 \, b x + 13 \, a\right )}{1664 \, b} - \frac {7 \, \cos \left (11 \, b x + 11 \, a\right )}{1408 \, b} - \frac {7 \, \cos \left (9 \, b x + 9 \, a\right )}{1152 \, b} + \frac {3 \, \cos \left (7 \, b x + 7 \, a\right )}{128 \, b} + \frac {21 \, \cos \left (5 \, b x + 5 \, a\right )}{640 \, b} - \frac {35 \, \cos \left (3 \, b x + 3 \, a\right )}{384 \, b} - \frac {35 \, \cos \left (b x + a\right )}{128 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.41, size = 111, normalized size = 1.82 \[ -\frac {35 \cos \left (b x +a \right )}{128 b}-\frac {35 \cos \left (3 b x +3 a \right )}{384 b}+\frac {21 \cos \left (5 b x +5 a \right )}{640 b}+\frac {3 \cos \left (7 b x +7 a \right )}{128 b}-\frac {7 \cos \left (9 b x +9 a \right )}{1152 b}-\frac {7 \cos \left (11 b x +11 a \right )}{1408 b}+\frac {\cos \left (13 b x +13 a \right )}{1664 b}+\frac {\cos \left (15 b x +15 a \right )}{1920 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 91, normalized size = 1.49 \[ \frac {429 \, \cos \left (15 \, b x + 15 \, a\right ) + 495 \, \cos \left (13 \, b x + 13 \, a\right ) - 4095 \, \cos \left (11 \, b x + 11 \, a\right ) - 5005 \, \cos \left (9 \, b x + 9 \, a\right ) + 19305 \, \cos \left (7 \, b x + 7 \, a\right ) + 27027 \, \cos \left (5 \, b x + 5 \, a\right ) - 75075 \, \cos \left (3 \, b x + 3 \, a\right ) - 225225 \, \cos \left (b x + a\right )}{823680 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 46, normalized size = 0.75 \[ -\frac {-\frac {128\,{\cos \left (a+b\,x\right )}^{15}}{15}+\frac {384\,{\cos \left (a+b\,x\right )}^{13}}{13}-\frac {384\,{\cos \left (a+b\,x\right )}^{11}}{11}+\frac {128\,{\cos \left (a+b\,x\right )}^9}{9}}{b} \]
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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