Optimal. Leaf size=173 \[ -\frac{b (C (1-n)-A n) \sin (c+d x) (b \cos (c+d x))^{n-1} \, _2F_1\left (\frac{1}{2},\frac{n-1}{2};\frac{n+1}{2};\cos ^2(c+d x)\right )}{d (1-n) n \sqrt{\sin ^2(c+d x)}}-\frac{B \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(c+d x)\right )}{d n \sqrt{\sin ^2(c+d x)}}+\frac{b C \sin (c+d x) (b \cos (c+d x))^{n-1}}{d n} \]
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Rubi [A] time = 0.230114, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {16, 3023, 2748, 2643} \[ -\frac{b (C (1-n)-A n) \sin (c+d x) (b \cos (c+d x))^{n-1} \, _2F_1\left (\frac{1}{2},\frac{n-1}{2};\frac{n+1}{2};\cos ^2(c+d x)\right )}{d (1-n) n \sqrt{\sin ^2(c+d x)}}-\frac{B \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(c+d x)\right )}{d n \sqrt{\sin ^2(c+d x)}}+\frac{b C \sin (c+d x) (b \cos (c+d x))^{n-1}}{d n} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3023
Rule 2748
Rule 2643
Rubi steps
\begin{align*} \int (b \cos (c+d x))^n \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx &=b^2 \int (b \cos (c+d x))^{-2+n} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\\ &=\frac{b C (b \cos (c+d x))^{-1+n} \sin (c+d x)}{d n}+\frac{b \int (b \cos (c+d x))^{-2+n} (-b (C (1-n)-A n)+b B n \cos (c+d x)) \, dx}{n}\\ &=\frac{b C (b \cos (c+d x))^{-1+n} \sin (c+d x)}{d n}+(b B) \int (b \cos (c+d x))^{-1+n} \, dx-\frac{\left (b^2 (C (1-n)-A n)\right ) \int (b \cos (c+d x))^{-2+n} \, dx}{n}\\ &=\frac{b C (b \cos (c+d x))^{-1+n} \sin (c+d x)}{d n}-\frac{b (C (1-n)-A n) (b \cos (c+d x))^{-1+n} \, _2F_1\left (\frac{1}{2},\frac{1}{2} (-1+n);\frac{1+n}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{d (1-n) n \sqrt{\sin ^2(c+d x)}}-\frac{B (b \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{n}{2};\frac{2+n}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{d n \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.287624, size = 131, normalized size = 0.76 \[ \frac{\tan (c+d x) (b \cos (c+d x))^n \left ((-A n+C (-n)+C) \, _2F_1\left (\frac{1}{2},\frac{n-1}{2};\frac{n+1}{2};\cos ^2(c+d x)\right )-(n-1) \left (B \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(c+d x)\right )-C \sqrt{\sin ^2(c+d x)}\right )\right )}{d (n-1) n \sqrt{\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.304, size = 0, normalized size = 0. \begin{align*} \int \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 ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}\, 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} \sec \left (d x + c\right )^{2}\,{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 )^{2} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{2}, 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} \sec \left (d x + c\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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