Optimal. Leaf size=22 \[ \frac{(a \sin (c+d x)+a)^4}{4 a d} \]
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Rubi [A] time = 0.0253023, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2667, 32} \[ \frac{(a \sin (c+d x)+a)^4}{4 a d} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 32
Rubi steps
\begin{align*} \int \cos (c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac{\operatorname{Subst}\left (\int (a+x)^3 \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{(a+a \sin (c+d x))^4}{4 a d}\\ \end{align*}
Mathematica [B] time = 0.027618, size = 65, normalized size = 2.95 \[ \frac{a^3 \sin ^4(c+d x)}{4 d}+\frac{a^3 \sin ^3(c+d x)}{d}+\frac{3 a^3 \sin ^2(c+d x)}{2 d}+\frac{a^3 \sin (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 21, normalized size = 1. \begin{align*}{\frac{ \left ( a+a\sin \left ( dx+c \right ) \right ) ^{4}}{4\,da}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.94555, size = 27, normalized size = 1.23 \begin{align*} \frac{{\left (a \sin \left (d x + c\right ) + a\right )}^{4}}{4 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.68935, size = 131, normalized size = 5.95 \begin{align*} \frac{a^{3} \cos \left (d x + c\right )^{4} - 8 \, a^{3} \cos \left (d x + c\right )^{2} - 4 \,{\left (a^{3} \cos \left (d x + c\right )^{2} - 2 \, a^{3}\right )} \sin \left (d x + c\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.3255, size = 94, normalized size = 4.27 \begin{align*} \begin{cases} \frac{a^{3} \sin ^{3}{\left (c + d x \right )}}{d} - \frac{a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2 d} + \frac{3 a^{3} \sin ^{2}{\left (c + d x \right )}}{2 d} + \frac{a^{3} \sin{\left (c + d x \right )}}{d} - \frac{a^{3} \cos ^{4}{\left (c + d x \right )}}{4 d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right )^{3} \cos{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15015, size = 27, normalized size = 1.23 \begin{align*} \frac{{\left (a \sin \left (d x + c\right ) + a\right )}^{4}}{4 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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