Optimal. Leaf size=217 \[ \frac{2 (2 A n+A-C (1-2 n)) \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{4} (2 n-1);\frac{1}{4} (2 n+3);\cos ^2(c+d x)\right )}{d \left (1-4 n^2\right ) \sqrt{\sin ^2(c+d x)} \sqrt{\cos (c+d x)}}-\frac{2 B \sin (c+d x) \sqrt{\cos (c+d x)} (b \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{4} (2 n+1);\frac{1}{4} (2 n+5);\cos ^2(c+d x)\right )}{d (2 n+1) \sqrt{\sin ^2(c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^n}{d (2 n+1) \sqrt{\cos (c+d x)}} \]
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Rubi [A] time = 0.201545, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.098, Rules used = {20, 3023, 2748, 2643} \[ \frac{2 (2 A n+A-C (1-2 n)) \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{4} (2 n-1);\frac{1}{4} (2 n+3);\cos ^2(c+d x)\right )}{d \left (1-4 n^2\right ) \sqrt{\sin ^2(c+d x)} \sqrt{\cos (c+d x)}}-\frac{2 B \sin (c+d x) \sqrt{\cos (c+d x)} (b \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{4} (2 n+1);\frac{1}{4} (2 n+5);\cos ^2(c+d x)\right )}{d (2 n+1) \sqrt{\sin ^2(c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^n}{d (2 n+1) \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 20
Rule 3023
Rule 2748
Rule 2643
Rubi steps
\begin{align*} \int \frac{(b \cos (c+d x))^n \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac{3}{2}}(c+d x)} \, dx &=\left (\cos ^{-n}(c+d x) (b \cos (c+d x))^n\right ) \int \cos ^{-\frac{3}{2}+n}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 C (b \cos (c+d x))^n \sin (c+d x)}{d (1+2 n) \sqrt{\cos (c+d x)}}+\frac{\left (2 \cos ^{-n}(c+d x) (b \cos (c+d x))^n\right ) \int \cos ^{-\frac{3}{2}+n}(c+d x) \left (\frac{1}{2} \left (-2 C \left (\frac{1}{2}-n\right )+2 A \left (\frac{1}{2}+n\right )\right )+\frac{1}{2} B (1+2 n) \cos (c+d x)\right ) \, dx}{1+2 n}\\ &=\frac{2 C (b \cos (c+d x))^n \sin (c+d x)}{d (1+2 n) \sqrt{\cos (c+d x)}}+\left (B \cos ^{-n}(c+d x) (b \cos (c+d x))^n\right ) \int \cos ^{-\frac{1}{2}+n}(c+d x) \, dx+\frac{\left ((A-C (1-2 n)+2 A n) \cos ^{-n}(c+d x) (b \cos (c+d x))^n\right ) \int \cos ^{-\frac{3}{2}+n}(c+d x) \, dx}{1+2 n}\\ &=\frac{2 C (b \cos (c+d x))^n \sin (c+d x)}{d (1+2 n) \sqrt{\cos (c+d x)}}+\frac{2 (A-C (1-2 n)+2 A n) (b \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{4} (-1+2 n);\frac{1}{4} (3+2 n);\cos ^2(c+d x)\right ) \sin (c+d x)}{d \left (1-4 n^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sin ^2(c+d x)}}-\frac{2 B \sqrt{\cos (c+d x)} (b \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{4} (1+2 n);\frac{1}{4} (5+2 n);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (1+2 n) \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.445906, size = 157, normalized size = 0.72 \[ -\frac{2 \sin (c+d x) (b \cos (c+d x))^n \left ((2 A n+A+C (2 n-1)) \, _2F_1\left (\frac{1}{2},\frac{1}{4} (2 n-1);\frac{1}{4} (2 n+3);\cos ^2(c+d x)\right )+(2 n-1) \left (B \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{4} (2 n+1);\frac{1}{4} (2 n+5);\cos ^2(c+d x)\right )-C \sqrt{\sin ^2(c+d x)}\right )\right )}{d \left (4 n^2-1\right ) \sqrt{\sin ^2(c+d x)} \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.76, 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 ( \cos \left ( dx+c \right ) \right ) ^{-{\frac{3}{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 \frac{{\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 )^{\frac{3}{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 (\frac{{\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 )^{\frac{3}{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 \frac{{\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 )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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