Optimal. Leaf size=92 \[ \frac {2 b \left (2 a^2-b^2\right ) \sin (c+d x)}{3 d}+\frac {1}{2} a x \left (2 a^2-b^2\right )+\frac {a b^2 \sin (c+d x) \cos (c+d x)}{6 d}-\frac {b \sin (c+d x) (a+b \cos (c+d x))^2}{3 d} \]
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Rubi [A] time = 0.11, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {3016, 2753, 2734} \[ \frac {2 b \left (2 a^2-b^2\right ) \sin (c+d x)}{3 d}+\frac {1}{2} a x \left (2 a^2-b^2\right )+\frac {a b^2 \sin (c+d x) \cos (c+d x)}{6 d}-\frac {b \sin (c+d x) (a+b \cos (c+d x))^2}{3 d} \]
Antiderivative was successfully verified.
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Rule 2734
Rule 2753
Rule 3016
Rubi steps
\begin {align*} \int (a+b \cos (c+d x)) \left (a^2-b^2 \cos ^2(c+d x)\right ) \, dx &=-\int (-a+b \cos (c+d x)) (a+b \cos (c+d x))^2 \, 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-a b \cos (c+d x)\right ) \, dx\\ &=\frac {1}{2} a \left (2 a^2-b^2\right ) x+\frac {2 b \left (2 a^2-b^2\right ) \sin (c+d x)}{3 d}+\frac {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 = 0.17, size = 75, normalized size = 0.82 \[ -\frac {-12 a^3 d x+\left (9 b^3-12 a^2 b\right ) \sin (c+d x)+3 a b^2 \sin (2 (c+d x))+6 a b^2 c+6 a b^2 d x+b^3 \sin (3 (c+d x))}{12 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 67, normalized size = 0.73 \[ \frac {3 \, {\left (2 \, a^{3} - a b^{2}\right )} d x - {\left (2 \, b^{3} \cos \left (d x + c\right )^{2} + 3 \, a b^{2} \cos \left (d x + c\right ) - 6 \, a^{2} b + 4 \, b^{3}\right )} \sin \left (d x + c\right )}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.94, size = 74, normalized size = 0.80 \[ -\frac {b^{3} \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} - \frac {a b^{2} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {1}{2} \, {\left (2 \, a^{3} - a b^{2}\right )} x + \frac {{\left (4 \, a^{2} b - 3 \, b^{3}\right )} \sin \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 75, normalized size = 0.82 \[ \frac {-\frac {b^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}-b^{2} a \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{2} b \sin \left (d x +c \right )+a^{3} \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 73, normalized size = 0.79 \[ \frac {12 \, {\left (d x + c\right )} a^{3} - 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a b^{2} + 4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} b^{3} + 12 \, a^{2} b \sin \left (d x + c\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.33, size = 76, normalized size = 0.83 \[ a^3\,x-\frac {3\,b^3\,\sin \left (c+d\,x\right )}{4\,d}-\frac {b^3\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}-\frac {a\,b^2\,x}{2}-\frac {a\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {a^2\,b\,\sin \left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.54, size = 131, normalized size = 1.42 \[ \begin {cases} a^{3} x + \frac {a^{2} b \sin {\left (c + d x \right )}}{d} - \frac {a b^{2} x \sin ^{2}{\left (c + d x \right )}}{2} - \frac {a b^{2} x \cos ^{2}{\left (c + d x \right )}}{2} - \frac {a b^{2} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} - \frac {2 b^{3} \sin ^{3}{\left (c + d x \right )}}{3 d} - \frac {b^{3} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \cos {\relax (c )}\right ) \left (a^{2} - b^{2} \cos ^{2}{\relax (c )}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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