Optimal. Leaf size=132 \[ -\frac{B \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(c+d x)\right )}{d n \sqrt{\sin ^2(c+d x)}}-\frac{C \sin (c+d x) (b \cos (c+d x))^{n+1} \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(c+d x)\right )}{b d (n+1) \sqrt{\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.166147, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {16, 3010, 2748, 2643} \[ -\frac{B \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(c+d x)\right )}{d n \sqrt{\sin ^2(c+d x)}}-\frac{C \sin (c+d x) (b \cos (c+d x))^{n+1} \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(c+d x)\right )}{b d (n+1) \sqrt{\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3010
Rule 2748
Rule 2643
Rubi steps
\begin{align*} \int (b \cos (c+d x))^n \left (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 (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\\ &=b \int (b \cos (c+d x))^{-1+n} (B+C \cos (c+d x)) \, dx\\ &=(b B) \int (b \cos (c+d x))^{-1+n} \, dx+C \int (b \cos (c+d x))^n \, dx\\ &=-\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)}}-\frac{C (b \cos (c+d x))^{1+n} \, _2F_1\left (\frac{1}{2},\frac{1+n}{2};\frac{3+n}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{b d (1+n) \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.18466, size = 109, normalized size = 0.83 \[ -\frac{b \sqrt{\sin ^2(c+d x)} \cot (c+d x) (b \cos (c+d x))^{n-1} \left (B (n+1) \, _2F_1\left (\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(c+d x)\right )+C n \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(c+d x)\right )\right )}{d n (n+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.261, size = 0, normalized size = 0. \begin{align*} \int \left ( b\cos \left ( dx+c \right ) \right ) ^{n} \left ( 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 )\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 )\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 )\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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