Optimal. Leaf size=85 \[ -\frac {a^3 \sin ^3(c+d x)}{d}+\frac {4 a^3 \sin (c+d x)}{d}+\frac {a^3 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {15 a^3 \sin (c+d x) \cos (c+d x)}{8 d}+\frac {15 a^3 x}{8} \]
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Rubi [A] time = 0.08, antiderivative size = 88, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2751, 2645, 2637, 2635, 8, 2633} \[ -\frac {a^3 \sin ^3(c+d x)}{4 d}+\frac {3 a^3 \sin (c+d x)}{d}+\frac {9 a^3 \sin (c+d x) \cos (c+d x)}{8 d}+\frac {15 a^3 x}{8}+\frac {\sin (c+d x) (a \cos (c+d x)+a)^3}{4 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 2637
Rule 2645
Rule 2751
Rubi steps
\begin {align*} \int \cos (c+d x) (a+a \cos (c+d x))^3 \, dx &=\frac {(a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {3}{4} \int (a+a \cos (c+d x))^3 \, dx\\ &=\frac {(a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {3}{4} \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\\ &=\frac {3 a^3 x}{4}+\frac {(a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {1}{4} \left (3 a^3\right ) \int \cos ^3(c+d x) \, dx+\frac {1}{4} \left (9 a^3\right ) \int \cos (c+d x) \, dx+\frac {1}{4} \left (9 a^3\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac {3 a^3 x}{4}+\frac {9 a^3 \sin (c+d x)}{4 d}+\frac {9 a^3 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {(a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {1}{8} \left (9 a^3\right ) \int 1 \, dx-\frac {\left (3 a^3\right ) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{4 d}\\ &=\frac {15 a^3 x}{8}+\frac {3 a^3 \sin (c+d x)}{d}+\frac {9 a^3 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {(a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}-\frac {a^3 \sin ^3(c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 51, normalized size = 0.60 \[ \frac {a^3 (104 \sin (c+d x)+32 \sin (2 (c+d x))+8 \sin (3 (c+d x))+\sin (4 (c+d x))+60 d x)}{32 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.53, size = 63, normalized size = 0.74 \[ \frac {15 \, a^{3} d x + {\left (2 \, a^{3} \cos \left (d x + c\right )^{3} + 8 \, a^{3} \cos \left (d x + c\right )^{2} + 15 \, a^{3} \cos \left (d x + c\right ) + 24 \, a^{3}\right )} \sin \left (d x + c\right )}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 71, normalized size = 0.84 \[ \frac {15}{8} \, a^{3} x + \frac {a^{3} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {a^{3} \sin \left (3 \, d x + 3 \, c\right )}{4 \, d} + \frac {a^{3} \sin \left (2 \, d x + 2 \, c\right )}{d} + \frac {13 \, a^{3} \sin \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 100, normalized size = 1.18 \[ \frac {a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+3 a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{3} \sin \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.84, size = 94, normalized size = 1.11 \[ -\frac {32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{3} - {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} - 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} - 32 \, a^{3} \sin \left (d x + c\right )}{32 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.49, size = 89, normalized size = 1.05 \[ \frac {15\,a^3\,x}{8}+\frac {\frac {15\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+\frac {55\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}+\frac {73\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{4}+\frac {49\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.03, size = 224, normalized size = 2.64 \[ \begin {cases} \frac {3 a^{3} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 a^{3} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {3 a^{3} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 a^{3} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {3 a^{3} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {2 a^{3} \sin ^{3}{\left (c + d x \right )}}{d} + \frac {5 a^{3} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {3 a^{3} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 a^{3} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {a^{3} \sin {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a \cos {\relax (c )} + a\right )^{3} \cos {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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