Optimal. Leaf size=44 \[ \frac {8 \sin ^{12}(a+b x)}{3 b}-\frac {32 \sin ^{10}(a+b x)}{5 b}+\frac {4 \sin ^8(a+b x)}{b} \]
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Rubi [A] time = 0.07, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4288, 2564, 266, 43} \[ \frac {8 \sin ^{12}(a+b x)}{3 b}-\frac {32 \sin ^{10}(a+b x)}{5 b}+\frac {4 \sin ^8(a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 2564
Rule 4288
Rubi steps
\begin {align*} \int \sin ^2(a+b x) \sin ^5(2 a+2 b x) \, dx &=32 \int \cos ^5(a+b x) \sin ^7(a+b x) \, dx\\ &=\frac {32 \operatorname {Subst}\left (\int x^7 \left (1-x^2\right )^2 \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac {16 \operatorname {Subst}\left (\int (1-x)^2 x^3 \, dx,x,\sin ^2(a+b x)\right )}{b}\\ &=\frac {16 \operatorname {Subst}\left (\int \left (x^3-2 x^4+x^5\right ) \, dx,x,\sin ^2(a+b x)\right )}{b}\\ &=\frac {4 \sin ^8(a+b x)}{b}-\frac {32 \sin ^{10}(a+b x)}{5 b}+\frac {8 \sin ^{12}(a+b x)}{3 b}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 68, normalized size = 1.55 \[ \frac {-600 \cos (2 (a+b x))+75 \cos (4 (a+b x))+100 \cos (6 (a+b x))-30 \cos (8 (a+b x))-12 \cos (10 (a+b x))+5 \cos (12 (a+b x))}{3840 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 46, normalized size = 1.05 \[ \frac {4 \, {\left (10 \, \cos \left (b x + a\right )^{12} - 36 \, \cos \left (b x + a\right )^{10} + 45 \, \cos \left (b x + a\right )^{8} - 20 \, \cos \left (b x + a\right )^{6}\right )}}{15 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.76, size = 85, normalized size = 1.93 \[ \frac {\cos \left (12 \, b x + 12 \, a\right )}{768 \, b} - \frac {\cos \left (10 \, b x + 10 \, a\right )}{320 \, b} - \frac {\cos \left (8 \, b x + 8 \, a\right )}{128 \, b} + \frac {5 \, \cos \left (6 \, b x + 6 \, a\right )}{192 \, b} + \frac {5 \, \cos \left (4 \, b x + 4 \, a\right )}{256 \, b} - \frac {5 \, \cos \left (2 \, b x + 2 \, a\right )}{32 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.39, size = 86, normalized size = 1.95 \[ -\frac {5 \cos \left (2 b x +2 a \right )}{32 b}+\frac {5 \cos \left (4 b x +4 a \right )}{256 b}+\frac {5 \cos \left (6 b x +6 a \right )}{192 b}-\frac {\cos \left (8 b x +8 a \right )}{128 b}-\frac {\cos \left (10 b x +10 a \right )}{320 b}+\frac {\cos \left (12 b x +12 a \right )}{768 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 72, normalized size = 1.64 \[ \frac {5 \, \cos \left (12 \, b x + 12 \, a\right ) - 12 \, \cos \left (10 \, b x + 10 \, a\right ) - 30 \, \cos \left (8 \, b x + 8 \, a\right ) + 100 \, \cos \left (6 \, b x + 6 \, a\right ) + 75 \, \cos \left (4 \, b x + 4 \, a\right ) - 600 \, \cos \left (2 \, b x + 2 \, a\right )}{3840 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 46, normalized size = 1.05 \[ -\frac {-\frac {8\,{\cos \left (a+b\,x\right )}^{12}}{3}+\frac {48\,{\cos \left (a+b\,x\right )}^{10}}{5}-12\,{\cos \left (a+b\,x\right )}^8+\frac {16\,{\cos \left (a+b\,x\right )}^6}{3}}{b} \]
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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