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 = 0.09, antiderivative size = 67, normalized size of antiderivative = 1.00, 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 30
Rule 2564
Rule 2565
Rule 2633
Rule 3090
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*}
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Mathematica [A] time = 0.39, 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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fricas [A] time = 0.50, size = 53, normalized size = 0.79 \[ -\frac {2 \, a b \cos \left (d x + c\right )^{3} - {\left ({\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, a^{2} + b^{2}\right )} \sin \left (d x + c\right )}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 73, normalized size = 1.09 \[ -\frac {a b \cos \left (3 \, d x + 3 \, c\right )}{6 \, d} - \frac {a b \cos \left (d x + c\right )}{2 \, d} + \frac {{\left (a^{2} - b^{2}\right )} \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac {{\left (3 \, a^{2} + b^{2}\right )} \sin \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.43, size = 52, normalized size = 0.78 \[ \frac {\frac {b^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{3}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right ) a b}{3}+\frac {a^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 52, normalized size = 0.78 \[ -\frac {2 \, a b \cos \left (d x + c\right )^{3} - b^{2} \sin \left (d x + c\right )^{3} + {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{2}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.51, size = 77, normalized size = 1.15 \[ \frac {2\,\left (\frac {\sin \left (c+d\,x\right )\,a^2\,{\cos \left (c+d\,x\right )}^2}{2}+\sin \left (c+d\,x\right )\,a^2-a\,b\,{\cos \left (c+d\,x\right )}^3-\frac {\sin \left (c+d\,x\right )\,b^2\,{\cos \left (c+d\,x\right )}^2}{2}+\frac {\sin \left (c+d\,x\right )\,b^2}{2}\right )}{3\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.48, size = 85, normalized size = 1.27 \[ \begin {cases} \frac {2 a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {a^{2} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} - \frac {2 a b \cos ^{3}{\left (c + d x \right )}}{3 d} + \frac {b^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (a \cos {\relax (c )} + b \sin {\relax (c )}\right )^{2} \cos {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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