Optimal. Leaf size=65 \[ -\frac {a \sin ^7(c+d x)}{7 d}-\frac {a \sin ^6(c+d x)}{6 d}+\frac {a \sin ^5(c+d x)}{5 d}+\frac {a \sin ^4(c+d x)}{4 d} \]
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Rubi [A] time = 0.07, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2836, 12, 75} \[ -\frac {a \sin ^7(c+d x)}{7 d}-\frac {a \sin ^6(c+d x)}{6 d}+\frac {a \sin ^5(c+d x)}{5 d}+\frac {a \sin ^4(c+d x)}{4 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 75
Rule 2836
Rubi steps
\begin {align*} \int \cos ^3(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a-x) x^3 (a+x)^2}{a^3} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {\operatorname {Subst}\left (\int (a-x) x^3 (a+x)^2 \, dx,x,a \sin (c+d x)\right )}{a^6 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^3 x^3+a^2 x^4-a x^5-x^6\right ) \, dx,x,a \sin (c+d x)\right )}{a^6 d}\\ &=\frac {a \sin ^4(c+d x)}{4 d}+\frac {a \sin ^5(c+d x)}{5 d}-\frac {a \sin ^6(c+d x)}{6 d}-\frac {a \sin ^7(c+d x)}{7 d}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 51, normalized size = 0.78 \[ \frac {a \left (-315 \cos (2 (c+d x))+35 \cos (6 (c+d x))+96 \sin ^5(c+d x) (5 \cos (2 (c+d x))+9)\right )}{6720 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 72, normalized size = 1.11 \[ \frac {70 \, a \cos \left (d x + c\right )^{6} - 105 \, a \cos \left (d x + c\right )^{4} + 12 \, {\left (5 \, a \cos \left (d x + c\right )^{6} - 8 \, a \cos \left (d x + c\right )^{4} + a \cos \left (d x + c\right )^{2} + 2 \, a\right )} \sin \left (d x + c\right )}{420 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 50, normalized size = 0.77 \[ -\frac {60 \, a \sin \left (d x + c\right )^{7} + 70 \, a \sin \left (d x + c\right )^{6} - 84 \, a \sin \left (d x + c\right )^{5} - 105 \, a \sin \left (d x + c\right )^{4}}{420 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 92, normalized size = 1.42 \[ \frac {a \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{7}-\frac {3 \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{35}+\frac {\left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{35}\right )+a \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{6}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{12}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 50, normalized size = 0.77 \[ -\frac {60 \, a \sin \left (d x + c\right )^{7} + 70 \, a \sin \left (d x + c\right )^{6} - 84 \, a \sin \left (d x + c\right )^{5} - 105 \, a \sin \left (d x + c\right )^{4}}{420 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 49, normalized size = 0.75 \[ \frac {-\frac {a\,{\sin \left (c+d\,x\right )}^7}{7}-\frac {a\,{\sin \left (c+d\,x\right )}^6}{6}+\frac {a\,{\sin \left (c+d\,x\right )}^5}{5}+\frac {a\,{\sin \left (c+d\,x\right )}^4}{4}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.83, size = 90, normalized size = 1.38 \[ \begin {cases} \frac {2 a \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {a \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} - \frac {a \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{4 d} - \frac {a \cos ^{6}{\left (c + d x \right )}}{12 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right ) \sin ^{3}{\relax (c )} \cos ^{3}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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