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 = 0.21, antiderivative size = 187, normalized size of antiderivative = 1.00, 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 2643
Rule 2748
Rule 3023
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*}
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Mathematica [A] time = 0.33, 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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fricas [F] time = 1.35, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C \cos \left (d x + c\right )^{3} + B \cos \left (d x + c\right )^{2} + A \cos \left (d x + c\right )\right )} \left (b \cos \left (d x + c\right )\right )^{n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \cos \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 3.64, size = 0, normalized size = 0.00 \[ \int \cos \left (d x +c \right ) \left (b \cos \left (d x +c \right )\right )^{n} \left (A +B \cos \left (d x +c \right )+C \left (\cos ^{2}\left (d x +c \right )\right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \cos \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \cos \left (c+d\,x\right )\,{\left (b\,\cos \left (c+d\,x\right )\right )}^n\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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