Optimal. Leaf size=63 \[ -\frac{a^3 \sin ^3(c+d x)}{3 d}+\frac{4 a^3 \sin (c+d x)}{d}+\frac{3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{5 a^3 x}{2} \]
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Rubi [A] time = 0.0526376, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {2645, 2637, 2635, 8, 2633} \[ -\frac{a^3 \sin ^3(c+d x)}{3 d}+\frac{4 a^3 \sin (c+d x)}{d}+\frac{3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{5 a^3 x}{2} \]
Antiderivative was successfully verified.
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Rule 2645
Rule 2637
Rule 2635
Rule 8
Rule 2633
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^3 \, dx &=\int \left (a^3+3 a^3 \cos (c+d x)+3 a^3 \cos ^2(c+d x)+a^3 \cos ^3(c+d x)\right ) \, dx\\ &=a^3 x+a^3 \int \cos ^3(c+d x) \, dx+\left (3 a^3\right ) \int \cos (c+d x) \, dx+\left (3 a^3\right ) \int \cos ^2(c+d x) \, dx\\ &=a^3 x+\frac{3 a^3 \sin (c+d x)}{d}+\frac{3 a^3 \cos (c+d x) \sin (c+d x)}{2 d}+\frac{1}{2} \left (3 a^3\right ) \int 1 \, dx-\frac{a^3 \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=\frac{5 a^3 x}{2}+\frac{4 a^3 \sin (c+d x)}{d}+\frac{3 a^3 \cos (c+d x) \sin (c+d x)}{2 d}-\frac{a^3 \sin ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.0648765, size = 44, normalized size = 0.7 \[ \frac{a^3 (45 \sin (c+d x)+9 \sin (2 (c+d x))+\sin (3 (c+d x))+30 c+30 d x)}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 74, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ({\frac{{a}^{3} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+3\,{a}^{3} \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +3\,{a}^{3}\sin \left ( dx+c \right ) +{a}^{3} \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.19853, size = 95, normalized size = 1.51 \begin{align*} a^{3} x - \frac{{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{3}}{3 \, d} + \frac{3 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3}}{4 \, d} + \frac{3 \, a^{3} \sin \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57316, size = 119, normalized size = 1.89 \begin{align*} \frac{15 \, a^{3} d x +{\left (2 \, a^{3} \cos \left (d x + c\right )^{2} + 9 \, a^{3} \cos \left (d x + c\right ) + 22 \, a^{3}\right )} \sin \left (d x + c\right )}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.622063, size = 121, normalized size = 1.92 \begin{align*} \begin{cases} \frac{3 a^{3} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac{3 a^{3} x \cos ^{2}{\left (c + d x \right )}}{2} + a^{3} x + \frac{2 a^{3} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{a^{3} \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{3 a^{3} \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} + \frac{3 a^{3} \sin{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a \cos{\left (c \right )} + a\right )^{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30949, size = 74, normalized size = 1.17 \begin{align*} \frac{5}{2} \, a^{3} x + \frac{a^{3} \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac{3 \, a^{3} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac{15 \, a^{3} \sin \left (d x + c\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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