Optimal. Leaf size=113 \[ -\frac {a \sin ^{13}(c+d x)}{13 d}+\frac {3 a \sin ^{11}(c+d x)}{11 d}-\frac {a \sin ^9(c+d x)}{3 d}+\frac {a \sin ^7(c+d x)}{7 d}-\frac {a \cos ^{12}(c+d x)}{12 d}+\frac {a \cos ^{10}(c+d x)}{5 d}-\frac {a \cos ^8(c+d x)}{8 d} \]
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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, 2565, 266, 43, 2564, 270} \[ -\frac {a \sin ^{13}(c+d x)}{13 d}+\frac {3 a \sin ^{11}(c+d x)}{11 d}-\frac {a \sin ^9(c+d x)}{3 d}+\frac {a \sin ^7(c+d x)}{7 d}-\frac {a \cos ^{12}(c+d x)}{12 d}+\frac {a \cos ^{10}(c+d x)}{5 d}-\frac {a \cos ^8(c+d x)}{8 d} \]
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 ^5(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cos ^7(c+d x) \sin ^5(c+d x) \, dx+a \int \cos ^7(c+d x) \sin ^6(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^6 \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^6-3 x^8+3 x^{10}-x^{12}\right ) \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac {a \sin ^7(c+d x)}{7 d}-\frac {a \sin ^9(c+d x)}{3 d}+\frac {3 a \sin ^{11}(c+d x)}{11 d}-\frac {a \sin ^{13}(c+d x)}{13 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 ^7(c+d x)}{7 d}-\frac {a \sin ^9(c+d x)}{3 d}+\frac {3 a \sin ^{11}(c+d x)}{11 d}-\frac {a \sin ^{13}(c+d x)}{13 d}\\ \end {align*}
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Mathematica [A] time = 0.73, size = 137, normalized size = 1.21 \[ -\frac {a (-600600 \sin (c+d x)+150150 \sin (3 (c+d x))+90090 \sin (5 (c+d x))-25740 \sin (7 (c+d x))-20020 \sin (9 (c+d x))+2730 \sin (11 (c+d x))+2310 \sin (13 (c+d x))+600600 \cos (2 (c+d x))+75075 \cos (4 (c+d x))-100100 \cos (6 (c+d x))-30030 \cos (8 (c+d x))+12012 \cos (10 (c+d x))+5005 \cos (12 (c+d x)))}{123002880 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 117, normalized size = 1.04 \[ -\frac {10010 \, a \cos \left (d x + c\right )^{12} - 24024 \, a \cos \left (d x + c\right )^{10} + 15015 \, a \cos \left (d x + c\right )^{8} + 40 \, {\left (231 \, a \cos \left (d x + c\right )^{12} - 567 \, a \cos \left (d x + c\right )^{10} + 371 \, a \cos \left (d x + c\right )^{8} - 5 \, a \cos \left (d x + c\right )^{6} - 6 \, a \cos \left (d x + c\right )^{4} - 8 \, a \cos \left (d x + c\right )^{2} - 16 \, a\right )} \sin \left (d x + c\right )}{120120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 193, normalized size = 1.71 \[ -\frac {a \cos \left (12 \, d x + 12 \, c\right )}{24576 \, d} - \frac {a \cos \left (10 \, d x + 10 \, c\right )}{10240 \, d} + \frac {a \cos \left (8 \, d x + 8 \, c\right )}{4096 \, d} + \frac {5 \, a \cos \left (6 \, d x + 6 \, c\right )}{6144 \, d} - \frac {5 \, a \cos \left (4 \, d x + 4 \, c\right )}{8192 \, d} - \frac {5 \, a \cos \left (2 \, d x + 2 \, c\right )}{1024 \, d} - \frac {a \sin \left (13 \, d x + 13 \, c\right )}{53248 \, d} - \frac {a \sin \left (11 \, d x + 11 \, c\right )}{45056 \, d} + \frac {a \sin \left (9 \, d x + 9 \, c\right )}{6144 \, d} + \frac {3 \, a \sin \left (7 \, d x + 7 \, c\right )}{14336 \, d} - \frac {3 \, a \sin \left (5 \, d x + 5 \, c\right )}{4096 \, d} - \frac {5 \, a \sin \left (3 \, d x + 3 \, c\right )}{4096 \, d} + \frac {5 \, a \sin \left (d x + c\right )}{1024 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 148, normalized size = 1.31 \[ \frac {a \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{8}\left (d x +c \right )\right )}{13}-\frac {5 \left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{8}\left (d x +c \right )\right )}{143}-\frac {5 \sin \left (d x +c \right ) \left (\cos ^{8}\left (d x +c \right )\right )}{429}+\frac {5 \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{3003}\right )+a \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{8}\left (d x +c \right )\right )}{12}-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{8}\left (d x +c \right )\right )}{30}-\frac {\left (\cos ^{8}\left (d x +c \right )\right )}{120}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 94, normalized size = 0.83 \[ -\frac {9240 \, a \sin \left (d x + c\right )^{13} + 10010 \, a \sin \left (d x + c\right )^{12} - 32760 \, a \sin \left (d x + c\right )^{11} - 36036 \, a \sin \left (d x + c\right )^{10} + 40040 \, a \sin \left (d x + c\right )^{9} + 45045 \, a \sin \left (d x + c\right )^{8} - 17160 \, a \sin \left (d x + c\right )^{7} - 20020 \, a \sin \left (d x + c\right )^{6}}{120120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.82, size = 93, normalized size = 0.82 \[ \frac {-\frac {a\,{\sin \left (c+d\,x\right )}^{13}}{13}-\frac {a\,{\sin \left (c+d\,x\right )}^{12}}{12}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^{11}}{11}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^{10}}{10}-\frac {a\,{\sin \left (c+d\,x\right )}^9}{3}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^8}{8}+\frac {a\,{\sin \left (c+d\,x\right )}^7}{7}+\frac {a\,{\sin \left (c+d\,x\right )}^6}{6}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 83.46, size = 160, normalized size = 1.42 \[ \begin {cases} \frac {16 a \sin ^{13}{\left (c + d x \right )}}{3003 d} + \frac {8 a \sin ^{11}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{231 d} + \frac {2 a \sin ^{9}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{21 d} + \frac {a \sin ^{7}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{7 d} - \frac {a \sin ^{4}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{8 d} - \frac {a \sin ^{2}{\left (c + d x \right )} \cos ^{10}{\left (c + d x \right )}}{20 d} - \frac {a \cos ^{12}{\left (c + d x \right )}}{120 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right ) \sin ^{5}{\relax (c )} \cos ^{7}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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