Optimal. Leaf size=46 \[ \frac {32 \sin ^{13}(a+b x)}{13 b}-\frac {64 \sin ^{11}(a+b x)}{11 b}+\frac {32 \sin ^9(a+b x)}{9 b} \]
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Rubi [A] time = 0.06, antiderivative size = 46, 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 = {4288, 2564, 270} \[ \frac {32 \sin ^{13}(a+b x)}{13 b}-\frac {64 \sin ^{11}(a+b x)}{11 b}+\frac {32 \sin ^9(a+b x)}{9 b} \]
Antiderivative was successfully verified.
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Rule 270
Rule 2564
Rule 4288
Rubi steps
\begin {align*} \int \sin ^3(a+b x) \sin ^5(2 a+2 b x) \, dx &=32 \int \cos ^5(a+b x) \sin ^8(a+b x) \, dx\\ &=\frac {32 \operatorname {Subst}\left (\int x^8 \left (1-x^2\right )^2 \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac {32 \operatorname {Subst}\left (\int \left (x^8-2 x^{10}+x^{12}\right ) \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac {32 \sin ^9(a+b x)}{9 b}-\frac {64 \sin ^{11}(a+b x)}{11 b}+\frac {32 \sin ^{13}(a+b x)}{13 b}\\ \end {align*}
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Mathematica [A] time = 0.37, size = 37, normalized size = 0.80 \[ \frac {4 \sin ^9(a+b x) (540 \cos (2 (a+b x))+99 \cos (4 (a+b x))+505)}{1287 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 73, normalized size = 1.59 \[ \frac {32 \, {\left (99 \, \cos \left (b x + a\right )^{12} - 360 \, \cos \left (b x + a\right )^{10} + 458 \, \cos \left (b x + a\right )^{8} - 212 \, \cos \left (b x + a\right )^{6} + 3 \, \cos \left (b x + a\right )^{4} + 4 \, \cos \left (b x + a\right )^{2} + 8\right )} \sin \left (b x + a\right )}{1287 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.62, size = 96, normalized size = 2.09 \[ \frac {\sin \left (13 \, b x + 13 \, a\right )}{1664 \, b} - \frac {3 \, \sin \left (11 \, b x + 11 \, a\right )}{1408 \, b} - \frac {\sin \left (9 \, b x + 9 \, a\right )}{576 \, b} + \frac {\sin \left (7 \, b x + 7 \, a\right )}{64 \, b} - \frac {\sin \left (5 \, b x + 5 \, a\right )}{128 \, b} - \frac {25 \, \sin \left (3 \, b x + 3 \, a\right )}{384 \, b} + \frac {5 \, \sin \left (b x + a\right )}{32 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.48, size = 97, normalized size = 2.11 \[ \frac {5 \sin \left (b x +a \right )}{32 b}-\frac {25 \sin \left (3 b x +3 a \right )}{384 b}-\frac {\sin \left (5 b x +5 a \right )}{128 b}+\frac {\sin \left (7 b x +7 a \right )}{64 b}-\frac {\sin \left (9 b x +9 a \right )}{576 b}-\frac {3 \sin \left (11 b x +11 a \right )}{1408 b}+\frac {\sin \left (13 b x +13 a \right )}{1664 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 80, normalized size = 1.74 \[ \frac {99 \, \sin \left (13 \, b x + 13 \, a\right ) - 351 \, \sin \left (11 \, b x + 11 \, a\right ) - 286 \, \sin \left (9 \, b x + 9 \, a\right ) + 2574 \, \sin \left (7 \, b x + 7 \, a\right ) - 1287 \, \sin \left (5 \, b x + 5 \, a\right ) - 10725 \, \sin \left (3 \, b x + 3 \, a\right ) + 25740 \, \sin \left (b x + a\right )}{164736 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 36, normalized size = 0.78 \[ \frac {32\,\left (99\,{\sin \left (a+b\,x\right )}^{13}-234\,{\sin \left (a+b\,x\right )}^{11}+143\,{\sin \left (a+b\,x\right )}^9\right )}{1287\,b} \]
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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