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 = 0.05, antiderivative size = 63, normalized size of antiderivative = 1.00, 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 8
Rule 2633
Rule 2635
Rule 2637
Rule 2645
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*}
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Mathematica [A] time = 0.07, size = 44, normalized size = 0.70 \[ \frac {a^3 (45 \sin (c+d x)+9 \sin (2 (c+d x))+\sin (3 (c+d x))+30 c+30 d x)}{12 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.97, size = 50, normalized size = 0.79 \[ \frac {15 \, a^{3} d x + {\left (2 \, a^{3} \cos \left (d x + c\right )^{2} + 9 \, a^{3} \cos \left (d x + c\right ) + 22 \, a^{3}\right )} \sin \left (d x + c\right )}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 55, normalized size = 0.87 \[ \frac {5}{2} \, a^{3} x + \frac {a^{3} \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac {3 \, a^{3} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {15 \, a^{3} \sin \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 74, normalized size = 1.17 \[ \frac {\frac {a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+3 a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 a^{3} \sin \left (d x +c \right )+a^{3} \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 70, normalized size = 1.11 \[ a^{3} x - \frac {{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{3}}{3 \, d} + \frac {3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3}}{4 \, d} + \frac {3 \, a^{3} \sin \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.40, size = 63, normalized size = 1.00 \[ \frac {5\,a^3\,x}{2}+\frac {11\,a^3\,\sin \left (c+d\,x\right )}{3\,d}+\frac {a^3\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{3\,d}+\frac {3\,a^3\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.50, size = 121, normalized size = 1.92 \[ \begin {cases} \frac {3 a^{3} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {3 a^{3} x \cos ^{2}{\left (c + d x \right )}}{2} + a^{3} x + \frac {2 a^{3} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {a^{3} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 a^{3} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {3 a^{3} \sin {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a \cos {\relax (c )} + a\right )^{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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