Optimal. Leaf size=76 \[ a^3 x+\frac {b \left (3 a^2+b^2\right ) \sin (c+d x)}{d}+\frac {3 a b^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {3}{2} a b^2 x-\frac {b^3 \sin ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.07, antiderivative size = 90, normalized size of antiderivative = 1.18, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2656, 2734} \[ \frac {2 b \left (4 a^2+b^2\right ) \sin (c+d x)}{3 d}+\frac {1}{2} a x \left (2 a^2+3 b^2\right )+\frac {5 a b^2 \sin (c+d x) \cos (c+d x)}{6 d}+\frac {b \sin (c+d x) (a+b \cos (c+d x))^2}{3 d} \]
Antiderivative was successfully verified.
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Rule 2656
Rule 2734
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^3 \, dx &=\frac {b (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac {1}{3} \int (a+b \cos (c+d x)) \left (3 a^2+2 b^2+5 a b \cos (c+d x)\right ) \, dx\\ &=\frac {1}{2} a \left (2 a^2+3 b^2\right ) x+\frac {2 b \left (4 a^2+b^2\right ) \sin (c+d x)}{3 d}+\frac {5 a b^2 \cos (c+d x) \sin (c+d x)}{6 d}+\frac {b (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 80, normalized size = 1.05 \[ \frac {12 a^3 c+12 a^3 d x+9 b \left (4 a^2+b^2\right ) \sin (c+d x)+9 a b^2 \sin (2 (c+d x))+18 a b^2 c+18 a b^2 d x+b^3 \sin (3 (c+d x))}{12 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 66, normalized size = 0.87 \[ \frac {3 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} d x + {\left (2 \, b^{3} \cos \left (d x + c\right )^{2} + 9 \, a b^{2} \cos \left (d x + c\right ) + 18 \, a^{2} b + 4 \, b^{3}\right )} \sin \left (d x + c\right )}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 72, normalized size = 0.95 \[ \frac {b^{3} \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac {3 \, a b^{2} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {1}{2} \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} x + \frac {3 \, {\left (4 \, a^{2} b + b^{3}\right )} \sin \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 76, normalized size = 1.00 \[ \frac {\frac {b^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+3 b^{2} a \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 a^{2} b \sin \left (d x +c \right )+a^{3} \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 72, normalized size = 0.95 \[ a^{3} x + \frac {3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a b^{2}}{4 \, d} - \frac {{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} b^{3}}{3 \, d} + \frac {3 \, a^{2} b \sin \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.60, size = 77, normalized size = 1.01 \[ a^3\,x+\frac {3\,b^3\,\sin \left (c+d\,x\right )}{4\,d}+\frac {b^3\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {3\,a\,b^2\,x}{2}+\frac {3\,a\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {3\,a^2\,b\,\sin \left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.52, size = 128, normalized size = 1.68 \[ \begin {cases} a^{3} x + \frac {3 a^{2} b \sin {\left (c + d x \right )}}{d} + \frac {3 a b^{2} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {3 a b^{2} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {3 a b^{2} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {2 b^{3} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {b^{3} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \cos {\relax (c )}\right )^{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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