Optimal. Leaf size=187 \[ -\frac{(A (n+3)+C (n+2)) \sin (c+d x) (b \cos (c+d x))^{n+2} \, _2F_1\left (\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\cos ^2(c+d x)\right )}{b^2 d (n+2) (n+3) \sqrt{\sin ^2(c+d x)}}-\frac{B \sin (c+d x) (b \cos (c+d x))^{n+3} \, _2F_1\left (\frac{1}{2},\frac{n+3}{2};\frac{n+5}{2};\cos ^2(c+d x)\right )}{b^3 d (n+3) \sqrt{\sin ^2(c+d x)}}+\frac{C \sin (c+d x) (b \cos (c+d x))^{n+2}}{b^2 d (n+3)} \]
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Rubi [A] time = 0.213301, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.108, Rules used = {16, 3023, 2748, 2643} \[ -\frac{(A (n+3)+C (n+2)) \sin (c+d x) (b \cos (c+d x))^{n+2} \, _2F_1\left (\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\cos ^2(c+d x)\right )}{b^2 d (n+2) (n+3) \sqrt{\sin ^2(c+d x)}}-\frac{B \sin (c+d x) (b \cos (c+d x))^{n+3} \, _2F_1\left (\frac{1}{2},\frac{n+3}{2};\frac{n+5}{2};\cos ^2(c+d x)\right )}{b^3 d (n+3) \sqrt{\sin ^2(c+d x)}}+\frac{C \sin (c+d x) (b \cos (c+d x))^{n+2}}{b^2 d (n+3)} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3023
Rule 2748
Rule 2643
Rubi steps
\begin{align*} \int \cos (c+d x) (b \cos (c+d x))^n \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac{\int (b \cos (c+d x))^{1+n} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx}{b}\\ &=\frac{C (b \cos (c+d x))^{2+n} \sin (c+d x)}{b^2 d (3+n)}+\frac{\int (b \cos (c+d x))^{1+n} (b (C (2+n)+A (3+n))+b B (3+n) \cos (c+d x)) \, dx}{b^2 (3+n)}\\ &=\frac{C (b \cos (c+d x))^{2+n} \sin (c+d x)}{b^2 d (3+n)}+\frac{B \int (b \cos (c+d x))^{2+n} \, dx}{b^2}+\frac{(C (2+n)+A (3+n)) \int (b \cos (c+d x))^{1+n} \, dx}{b (3+n)}\\ &=\frac{C (b \cos (c+d x))^{2+n} \sin (c+d x)}{b^2 d (3+n)}-\frac{(C (2+n)+A (3+n)) (b \cos (c+d x))^{2+n} \, _2F_1\left (\frac{1}{2},\frac{2+n}{2};\frac{4+n}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{b^2 d (2+n) (3+n) \sqrt{\sin ^2(c+d x)}}-\frac{B (b \cos (c+d x))^{3+n} \, _2F_1\left (\frac{1}{2},\frac{3+n}{2};\frac{5+n}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{b^3 d (3+n) \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.330168, size = 144, normalized size = 0.77 \[ -\frac{\sin (c+d x) \cos ^2(c+d x) (b \cos (c+d x))^n \left ((A (n+3)+C (n+2)) \, _2F_1\left (\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\cos ^2(c+d x)\right )+(n+2) \left (B \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{n+3}{2};\frac{n+5}{2};\cos ^2(c+d x)\right )-C \sqrt{\sin ^2(c+d x)}\right )\right )}{d (n+2) (n+3) \sqrt{\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.522, size = 0, normalized size = 0. \begin{align*} \int \cos \left ( dx+c \right ) \left ( b\cos \left ( dx+c \right ) \right ) ^{n} \left ( A+B\cos \left ( dx+c \right ) +C \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \cos \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (C \cos \left (d x + c\right )^{3} + B \cos \left (d x + c\right )^{2} + A \cos \left (d x + c\right )\right )} \left (b \cos \left (d x + c\right )\right )^{n}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \cos \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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