Optimal. Leaf size=31 \[ \frac {b \cos ^3(c+d x)}{3 d}-\frac {(a+b) \cos (c+d x)}{d} \]
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Rubi [A] time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {3013} \[ \frac {b \cos ^3(c+d x)}{3 d}-\frac {(a+b) \cos (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3013
Rubi steps
\begin {align*} \int \sin (c+d x) \left (a+b \sin ^2(c+d x)\right ) \, dx &=-\frac {\operatorname {Subst}\left (\int \left (a+b-b x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {(a+b) \cos (c+d x)}{d}+\frac {b \cos ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 54, normalized size = 1.74 \[ \frac {a \sin (c) \sin (d x)}{d}-\frac {a \cos (c) \cos (d x)}{d}-\frac {3 b \cos (c+d x)}{4 d}+\frac {b \cos (3 (c+d x))}{12 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 27, normalized size = 0.87 \[ \frac {b \cos \left (d x + c\right )^{3} - 3 \, {\left (a + b\right )} \cos \left (d x + c\right )}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 40, normalized size = 1.29 \[ \frac {1}{3} \, {\left (\frac {\cos \left (d x + c\right )^{3}}{d} - \frac {3 \, \cos \left (d x + c\right )}{d}\right )} b - \frac {a \cos \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 34, normalized size = 1.10 \[ \frac {-\frac {b \left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}-a \cos \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 34, normalized size = 1.10 \[ \frac {{\left (\cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} b - 3 \, a \cos \left (d x + c\right )}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.31, size = 27, normalized size = 0.87 \[ \frac {\frac {b\,{\cos \left (c+d\,x\right )}^3}{3}-\cos \left (c+d\,x\right )\,\left (a+b\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.00, size = 58, normalized size = 1.87 \[ \begin {cases} - \frac {a \cos {\left (c + d x \right )}}{d} - \frac {b \sin ^{2}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} - \frac {2 b \cos ^{3}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin ^{2}{\relax (c )}\right ) \sin {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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