Optimal. Leaf size=45 \[ \frac {2 (a \sin (c+d x)+a)^5}{5 a^2 d}-\frac {(a \sin (c+d x)+a)^6}{6 a^3 d} \]
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Rubi [A] time = 0.05, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2667, 43} \[ \frac {2 (a \sin (c+d x)+a)^5}{5 a^2 d}-\frac {(a \sin (c+d x)+a)^6}{6 a^3 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2667
Rubi steps
\begin {align*} \int \cos ^3(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac {\operatorname {Subst}\left (\int (a-x) (a+x)^4 \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (2 a (a+x)^4-(a+x)^5\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {2 (a+a \sin (c+d x))^5}{5 a^2 d}-\frac {(a+a \sin (c+d x))^6}{6 a^3 d}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 43, normalized size = 0.96 \[ -\frac {a^3 (5 \sin (c+d x)-7) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^{10}}{30 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 72, normalized size = 1.60 \[ \frac {5 \, a^{3} \cos \left (d x + c\right )^{6} - 30 \, a^{3} \cos \left (d x + c\right )^{4} - 2 \, {\left (9 \, a^{3} \cos \left (d x + c\right )^{4} - 8 \, a^{3} \cos \left (d x + c\right )^{2} - 16 \, a^{3}\right )} \sin \left (d x + c\right )}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.71, size = 82, normalized size = 1.82 \[ -\frac {5 \, a^{3} \sin \left (d x + c\right )^{6} + 18 \, a^{3} \sin \left (d x + c\right )^{5} + 15 \, a^{3} \sin \left (d x + c\right )^{4} - 20 \, a^{3} \sin \left (d x + c\right )^{3} - 45 \, a^{3} \sin \left (d x + c\right )^{2} - 30 \, a^{3} \sin \left (d x + c\right )}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.17, size = 113, normalized size = 2.51 \[ \frac {a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{6}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{12}\right )+3 a^{3} \left (-\frac {\left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{5}+\frac {\left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{15}\right )-\frac {3 \left (\cos ^{4}\left (d x +c \right )\right ) a^{3}}{4}+\frac {a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 82, normalized size = 1.82 \[ -\frac {5 \, a^{3} \sin \left (d x + c\right )^{6} + 18 \, a^{3} \sin \left (d x + c\right )^{5} + 15 \, a^{3} \sin \left (d x + c\right )^{4} - 20 \, a^{3} \sin \left (d x + c\right )^{3} - 45 \, a^{3} \sin \left (d x + c\right )^{2} - 30 \, a^{3} \sin \left (d x + c\right )}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.47, size = 80, normalized size = 1.78 \[ \frac {-\frac {a^3\,{\sin \left (c+d\,x\right )}^6}{6}-\frac {3\,a^3\,{\sin \left (c+d\,x\right )}^5}{5}-\frac {a^3\,{\sin \left (c+d\,x\right )}^4}{2}+\frac {2\,a^3\,{\sin \left (c+d\,x\right )}^3}{3}+\frac {3\,a^3\,{\sin \left (c+d\,x\right )}^2}{2}+a^3\,\sin \left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.78, size = 146, normalized size = 3.24 \[ \begin {cases} \frac {2 a^{3} \sin ^{5}{\left (c + d x \right )}}{5 d} + \frac {a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {2 a^{3} \sin ^{3}{\left (c + d x \right )}}{3 d} - \frac {a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{4 d} + \frac {a^{3} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} - \frac {a^{3} \cos ^{6}{\left (c + d x \right )}}{12 d} - \frac {3 a^{3} \cos ^{4}{\left (c + d x \right )}}{4 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right )^{3} \cos ^{3}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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