Optimal. Leaf size=61 \[ -\frac {128 \sin ^{15}(a+b x)}{15 b}+\frac {384 \sin ^{13}(a+b x)}{13 b}-\frac {384 \sin ^{11}(a+b x)}{11 b}+\frac {128 \sin ^9(a+b x)}{9 b} \]
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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 = {4288, 2564, 270} \[ -\frac {128 \sin ^{15}(a+b x)}{15 b}+\frac {384 \sin ^{13}(a+b x)}{13 b}-\frac {384 \sin ^{11}(a+b x)}{11 b}+\frac {128 \sin ^9(a+b x)}{9 b} \]
Antiderivative was successfully verified.
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Rule 270
Rule 2564
Rule 4288
Rubi steps
\begin {align*} \int \sin (a+b x) \sin ^7(2 a+2 b x) \, dx &=128 \int \cos ^7(a+b x) \sin ^8(a+b x) \, dx\\ &=\frac {128 \operatorname {Subst}\left (\int x^8 \left (1-x^2\right )^3 \, dx,x,\sin (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,\sin (a+b x)\right )}{b}\\ &=\frac {128 \sin ^9(a+b x)}{9 b}-\frac {384 \sin ^{11}(a+b x)}{11 b}+\frac {384 \sin ^{13}(a+b x)}{13 b}-\frac {128 \sin ^{15}(a+b x)}{15 b}\\ \end {align*}
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Mathematica [A] time = 0.48, size = 47, normalized size = 0.77 \[ \frac {4 \sin ^9(a+b x) (10755 \cos (2 (a+b x))+3366 \cos (4 (a+b x))+429 \cos (6 (a+b x))+8330)}{6435 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 83, normalized size = 1.36 \[ \frac {128 \, {\left (429 \, \cos \left (b x + a\right )^{14} - 1518 \, \cos \left (b x + a\right )^{12} + 1854 \, \cos \left (b x + a\right )^{10} - 800 \, \cos \left (b x + a\right )^{8} + 5 \, \cos \left (b x + a\right )^{6} + 6 \, \cos \left (b x + a\right )^{4} + 8 \, \cos \left (b x + a\right )^{2} + 16\right )} \sin \left (b x + a\right )}{6435 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.14, size = 110, normalized size = 1.80 \[ \frac {\sin \left (15 \, b x + 15 \, a\right )}{1920 \, b} - \frac {\sin \left (13 \, b x + 13 \, a\right )}{1664 \, b} - \frac {7 \, \sin \left (11 \, b x + 11 \, a\right )}{1408 \, b} + \frac {7 \, \sin \left (9 \, b x + 9 \, a\right )}{1152 \, b} + \frac {3 \, \sin \left (7 \, b x + 7 \, a\right )}{128 \, b} - \frac {21 \, \sin \left (5 \, b x + 5 \, a\right )}{640 \, b} - \frac {35 \, \sin \left (3 \, b x + 3 \, a\right )}{384 \, b} + \frac {35 \, \sin \left (b x + a\right )}{128 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.94, size = 111, normalized size = 1.82 \[ \frac {35 \sin \left (b x +a \right )}{128 b}-\frac {35 \sin \left (3 b x +3 a \right )}{384 b}-\frac {21 \sin \left (5 b x +5 a \right )}{640 b}+\frac {3 \sin \left (7 b x +7 a \right )}{128 b}+\frac {7 \sin \left (9 b x +9 a \right )}{1152 b}-\frac {7 \sin \left (11 b x +11 a \right )}{1408 b}-\frac {\sin \left (13 b x +13 a \right )}{1664 b}+\frac {\sin \left (15 b x +15 a \right )}{1920 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 91, normalized size = 1.49 \[ \frac {429 \, \sin \left (15 \, b x + 15 \, a\right ) - 495 \, \sin \left (13 \, b x + 13 \, a\right ) - 4095 \, \sin \left (11 \, b x + 11 \, a\right ) + 5005 \, \sin \left (9 \, b x + 9 \, a\right ) + 19305 \, \sin \left (7 \, b x + 7 \, a\right ) - 27027 \, \sin \left (5 \, b x + 5 \, a\right ) - 75075 \, \sin \left (3 \, b x + 3 \, a\right ) + 225225 \, \sin \left (b x + a\right )}{823680 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 45, normalized size = 0.74 \[ \frac {-\frac {128\,{\sin \left (a+b\,x\right )}^{15}}{15}+\frac {384\,{\sin \left (a+b\,x\right )}^{13}}{13}-\frac {384\,{\sin \left (a+b\,x\right )}^{11}}{11}+\frac {128\,{\sin \left (a+b\,x\right )}^9}{9}}{b} \]
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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