Optimal. Leaf size=67 \[ -\frac{a^2 \sin ^3(c+d x)}{3 d}+\frac{a^2 \sin (c+d x)}{d}-\frac{2 a b \cos ^3(c+d x)}{3 d}+\frac{b^2 \sin ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.0908333, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {3090, 2633, 2565, 30, 2564} \[ -\frac{a^2 \sin ^3(c+d x)}{3 d}+\frac{a^2 \sin (c+d x)}{d}-\frac{2 a b \cos ^3(c+d x)}{3 d}+\frac{b^2 \sin ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 3090
Rule 2633
Rule 2565
Rule 30
Rule 2564
Rubi steps
\begin{align*} \int \cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx &=\int \left (a^2 \cos ^3(c+d x)+2 a b \cos ^2(c+d x) \sin (c+d x)+b^2 \cos (c+d x) \sin ^2(c+d x)\right ) \, dx\\ &=a^2 \int \cos ^3(c+d x) \, dx+(2 a b) \int \cos ^2(c+d x) \sin (c+d x) \, dx+b^2 \int \cos (c+d x) \sin ^2(c+d x) \, dx\\ &=-\frac{a^2 \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d}-\frac{(2 a b) \operatorname{Subst}\left (\int x^2 \, dx,x,\cos (c+d x)\right )}{d}+\frac{b^2 \operatorname{Subst}\left (\int x^2 \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac{2 a b \cos ^3(c+d x)}{3 d}+\frac{a^2 \sin (c+d x)}{d}-\frac{a^2 \sin ^3(c+d x)}{3 d}+\frac{b^2 \sin ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.388145, size = 64, normalized size = 0.96 \[ \frac{\sin (c+d x) \left (\left (a^2-b^2\right ) \cos (2 (c+d x))+5 a^2+b^2\right )-3 a b \cos (c+d x)-a b \cos (3 (c+d x))}{6 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 52, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ({\frac{{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3}}-{\frac{2\,ab \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{3}}+{\frac{{a}^{2} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.124, size = 70, normalized size = 1.04 \begin{align*} -\frac{2 \, a b \cos \left (d x + c\right )^{3} - b^{2} \sin \left (d x + c\right )^{3} +{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{2}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.472204, size = 120, normalized size = 1.79 \begin{align*} -\frac{2 \, a b \cos \left (d x + c\right )^{3} -{\left ({\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, a^{2} + b^{2}\right )} \sin \left (d x + c\right )}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.545085, size = 85, normalized size = 1.27 \begin{align*} \begin{cases} \frac{2 a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{a^{2} \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} - \frac{2 a b \cos ^{3}{\left (c + d x \right )}}{3 d} + \frac{b^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} & \text{for}\: d \neq 0 \\x \left (a \cos{\left (c \right )} + b \sin{\left (c \right )}\right )^{2} \cos{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15405, size = 99, normalized size = 1.48 \begin{align*} -\frac{a b \cos \left (3 \, d x + 3 \, c\right )}{6 \, d} - \frac{a b \cos \left (d x + c\right )}{2 \, d} + \frac{{\left (a^{2} - b^{2}\right )} \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac{{\left (3 \, a^{2} + b^{2}\right )} \sin \left (d x + c\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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