Optimal. Leaf size=113 \[ -\frac {a \sin ^{11}(c+d x)}{11 d}+\frac {a \sin ^9(c+d x)}{3 d}-\frac {3 a \sin ^7(c+d x)}{7 d}+\frac {a \sin ^5(c+d x)}{5 d}-\frac {a \cos ^{12}(c+d x)}{12 d}+\frac {a \cos ^{10}(c+d x)}{5 d}-\frac {a \cos ^8(c+d x)}{8 d} \]
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Rubi [A] time = 0.14, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2834, 2564, 270, 2565, 266, 43} \[ -\frac {a \sin ^{11}(c+d x)}{11 d}+\frac {a \sin ^9(c+d x)}{3 d}-\frac {3 a \sin ^7(c+d x)}{7 d}+\frac {a \sin ^5(c+d x)}{5 d}-\frac {a \cos ^{12}(c+d x)}{12 d}+\frac {a \cos ^{10}(c+d x)}{5 d}-\frac {a \cos ^8(c+d x)}{8 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 270
Rule 2564
Rule 2565
Rule 2834
Rubi steps
\begin {align*} \int \cos ^7(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cos ^7(c+d x) \sin ^4(c+d x) \, dx+a \int \cos ^7(c+d x) \sin ^5(c+d x) \, dx\\ &=-\frac {a \operatorname {Subst}\left (\int x^7 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{d}+\frac {a \operatorname {Subst}\left (\int x^4 \left (1-x^2\right )^3 \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac {a \operatorname {Subst}\left (\int (1-x)^2 x^3 \, dx,x,\cos ^2(c+d x)\right )}{2 d}+\frac {a \operatorname {Subst}\left (\int \left (x^4-3 x^6+3 x^8-x^{10}\right ) \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac {a \sin ^5(c+d x)}{5 d}-\frac {3 a \sin ^7(c+d x)}{7 d}+\frac {a \sin ^9(c+d x)}{3 d}-\frac {a \sin ^{11}(c+d x)}{11 d}-\frac {a \operatorname {Subst}\left (\int \left (x^3-2 x^4+x^5\right ) \, dx,x,\cos ^2(c+d x)\right )}{2 d}\\ &=-\frac {a \cos ^8(c+d x)}{8 d}+\frac {a \cos ^{10}(c+d x)}{5 d}-\frac {a \cos ^{12}(c+d x)}{12 d}+\frac {a \sin ^5(c+d x)}{5 d}-\frac {3 a \sin ^7(c+d x)}{7 d}+\frac {a \sin ^9(c+d x)}{3 d}-\frac {a \sin ^{11}(c+d x)}{11 d}\\ \end {align*}
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Mathematica [A] time = 0.56, size = 127, normalized size = 1.12 \[ -\frac {a (-129360 \sin (c+d x)+18480 \sin (3 (c+d x))+20328 \sin (5 (c+d x))+1320 \sin (7 (c+d x))-3080 \sin (9 (c+d x))-840 \sin (11 (c+d x))+46200 \cos (2 (c+d x))+5775 \cos (4 (c+d x))-7700 \cos (6 (c+d x))-2310 \cos (8 (c+d x))+924 \cos (10 (c+d x))+385 \cos (12 (c+d x)))}{9461760 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 106, normalized size = 0.94 \[ -\frac {770 \, a \cos \left (d x + c\right )^{12} - 1848 \, a \cos \left (d x + c\right )^{10} + 1155 \, a \cos \left (d x + c\right )^{8} - 8 \, {\left (105 \, a \cos \left (d x + c\right )^{10} - 140 \, a \cos \left (d x + c\right )^{8} + 5 \, a \cos \left (d x + c\right )^{6} + 6 \, a \cos \left (d x + c\right )^{4} + 8 \, a \cos \left (d x + c\right )^{2} + 16 \, a\right )} \sin \left (d x + c\right )}{9240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 178, normalized size = 1.58 \[ -\frac {a \cos \left (12 \, d x + 12 \, c\right )}{24576 \, d} - \frac {a \cos \left (10 \, d x + 10 \, c\right )}{10240 \, d} + \frac {a \cos \left (8 \, d x + 8 \, c\right )}{4096 \, d} + \frac {5 \, a \cos \left (6 \, d x + 6 \, c\right )}{6144 \, d} - \frac {5 \, a \cos \left (4 \, d x + 4 \, c\right )}{8192 \, d} - \frac {5 \, a \cos \left (2 \, d x + 2 \, c\right )}{1024 \, d} + \frac {a \sin \left (11 \, d x + 11 \, c\right )}{11264 \, d} + \frac {a \sin \left (9 \, d x + 9 \, c\right )}{3072 \, d} - \frac {a \sin \left (7 \, d x + 7 \, c\right )}{7168 \, d} - \frac {11 \, a \sin \left (5 \, d x + 5 \, c\right )}{5120 \, d} - \frac {a \sin \left (3 \, d x + 3 \, c\right )}{512 \, d} + \frac {7 \, a \sin \left (d x + c\right )}{512 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 130, normalized size = 1.15 \[ \frac {a \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{8}\left (d x +c \right )\right )}{12}-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{8}\left (d x +c \right )\right )}{30}-\frac {\left (\cos ^{8}\left (d x +c \right )\right )}{120}\right )+a \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{8}\left (d x +c \right )\right )}{11}-\frac {\sin \left (d x +c \right ) \left (\cos ^{8}\left (d x +c \right )\right )}{33}+\frac {\left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{231}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 94, normalized size = 0.83 \[ -\frac {770 \, a \sin \left (d x + c\right )^{12} + 840 \, a \sin \left (d x + c\right )^{11} - 2772 \, a \sin \left (d x + c\right )^{10} - 3080 \, a \sin \left (d x + c\right )^{9} + 3465 \, a \sin \left (d x + c\right )^{8} + 3960 \, a \sin \left (d x + c\right )^{7} - 1540 \, a \sin \left (d x + c\right )^{6} - 1848 \, a \sin \left (d x + c\right )^{5}}{9240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 93, normalized size = 0.82 \[ \frac {-\frac {a\,{\sin \left (c+d\,x\right )}^{12}}{12}-\frac {a\,{\sin \left (c+d\,x\right )}^{11}}{11}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^{10}}{10}+\frac {a\,{\sin \left (c+d\,x\right )}^9}{3}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^8}{8}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^7}{7}+\frac {a\,{\sin \left (c+d\,x\right )}^6}{6}+\frac {a\,{\sin \left (c+d\,x\right )}^5}{5}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 56.20, size = 160, normalized size = 1.42 \[ \begin {cases} \frac {16 a \sin ^{11}{\left (c + d x \right )}}{1155 d} + \frac {8 a \sin ^{9}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{105 d} + \frac {6 a \sin ^{7}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{35 d} + \frac {a \sin ^{5}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{5 d} - \frac {a \sin ^{4}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{8 d} - \frac {a \sin ^{2}{\left (c + d x \right )} \cos ^{10}{\left (c + d x \right )}}{20 d} - \frac {a \cos ^{12}{\left (c + d x \right )}}{120 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right ) \sin ^{4}{\relax (c )} \cos ^{7}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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