Optimal. Leaf size=61 \[ \frac {16 \cos ^{11}(a+b x)}{11 b}-\frac {16 \cos ^9(a+b x)}{3 b}+\frac {48 \cos ^7(a+b x)}{7 b}-\frac {16 \cos ^5(a+b x)}{5 b} \]
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Rubi [A] time = 0.07, antiderivative size = 61, 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, 2565, 270} \[ \frac {16 \cos ^{11}(a+b x)}{11 b}-\frac {16 \cos ^9(a+b x)}{3 b}+\frac {48 \cos ^7(a+b x)}{7 b}-\frac {16 \cos ^5(a+b x)}{5 b} \]
Antiderivative was successfully verified.
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Rule 270
Rule 2565
Rule 4288
Rubi steps
\begin {align*} \int \sin ^3(a+b x) \sin ^4(2 a+2 b x) \, dx &=16 \int \cos ^4(a+b x) \sin ^7(a+b x) \, dx\\ &=-\frac {16 \operatorname {Subst}\left (\int x^4 \left (1-x^2\right )^3 \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac {16 \operatorname {Subst}\left (\int \left (x^4-3 x^6+3 x^8-x^{10}\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac {16 \cos ^5(a+b x)}{5 b}+\frac {48 \cos ^7(a+b x)}{7 b}-\frac {16 \cos ^9(a+b x)}{3 b}+\frac {16 \cos ^{11}(a+b x)}{11 b}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 47, normalized size = 0.77 \[ \frac {\cos ^5(a+b x) (3335 \cos (2 (a+b x))-910 \cos (4 (a+b x))+105 \cos (6 (a+b x))-3042)}{2310 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 46, normalized size = 0.75 \[ \frac {16 \, {\left (105 \, \cos \left (b x + a\right )^{11} - 385 \, \cos \left (b x + a\right )^{9} + 495 \, \cos \left (b x + a\right )^{7} - 231 \, \cos \left (b x + a\right )^{5}\right )}}{1155 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.80, size = 82, normalized size = 1.34 \[ \frac {\cos \left (11 \, b x + 11 \, a\right )}{704 \, b} - \frac {\cos \left (9 \, b x + 9 \, a\right )}{192 \, b} - \frac {\cos \left (7 \, b x + 7 \, a\right )}{448 \, b} + \frac {11 \, \cos \left (5 \, b x + 5 \, a\right )}{320 \, b} - \frac {\cos \left (3 \, b x + 3 \, a\right )}{32 \, b} - \frac {7 \, \cos \left (b x + a\right )}{32 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 83, normalized size = 1.36 \[ -\frac {7 \cos \left (b x +a \right )}{32 b}-\frac {\cos \left (3 b x +3 a \right )}{32 b}+\frac {11 \cos \left (5 b x +5 a \right )}{320 b}-\frac {\cos \left (7 b x +7 a \right )}{448 b}-\frac {\cos \left (9 b x +9 a \right )}{192 b}+\frac {\cos \left (11 b x +11 a \right )}{704 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 69, normalized size = 1.13 \[ \frac {105 \, \cos \left (11 \, b x + 11 \, a\right ) - 385 \, \cos \left (9 \, b x + 9 \, a\right ) - 165 \, \cos \left (7 \, b x + 7 \, a\right ) + 2541 \, \cos \left (5 \, b x + 5 \, a\right ) - 2310 \, \cos \left (3 \, b x + 3 \, a\right ) - 16170 \, \cos \left (b x + a\right )}{73920 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 46, normalized size = 0.75 \[ -\frac {-\frac {16\,{\cos \left (a+b\,x\right )}^{11}}{11}+\frac {16\,{\cos \left (a+b\,x\right )}^9}{3}-\frac {48\,{\cos \left (a+b\,x\right )}^7}{7}+\frac {16\,{\cos \left (a+b\,x\right )}^5}{5}}{b} \]
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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