Optimal. Leaf size=140 \[ \frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)} (b \cos (c+d x))^n}{d (2 n+3)}-\frac{2 (A (2 n+3)+2 C n+C) \sin (c+d x) \sqrt{\cos (c+d x)} (b \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{4} (2 n+1);\frac{1}{4} (2 n+5);\cos ^2(c+d x)\right )}{d (2 n+1) (2 n+3) \sqrt{\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.101933, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {20, 3014, 2643} \[ \frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)} (b \cos (c+d x))^n}{d (2 n+3)}-\frac{2 (A (2 n+3)+2 C n+C) \sin (c+d x) \sqrt{\cos (c+d x)} (b \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{4} (2 n+1);\frac{1}{4} (2 n+5);\cos ^2(c+d x)\right )}{d (2 n+1) (2 n+3) \sqrt{\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 20
Rule 3014
Rule 2643
Rubi steps
\begin{align*} \int \frac{(b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx &=\left (\cos ^{-n}(c+d x) (b \cos (c+d x))^n\right ) \int \cos ^{-\frac{1}{2}+n}(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 C \sqrt{\cos (c+d x)} (b \cos (c+d x))^n \sin (c+d x)}{d (3+2 n)}+\frac{\left (\left (C \left (\frac{1}{2}+n\right )+A \left (\frac{3}{2}+n\right )\right ) \cos ^{-n}(c+d x) (b \cos (c+d x))^n\right ) \int \cos ^{-\frac{1}{2}+n}(c+d x) \, dx}{\frac{3}{2}+n}\\ &=\frac{2 C \sqrt{\cos (c+d x)} (b \cos (c+d x))^n \sin (c+d x)}{d (3+2 n)}-\frac{2 (C+2 C n+A (3+2 n)) \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) (3+2 n) \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.170331, size = 140, normalized size = 1. \[ -\frac{2 \sqrt{\sin ^2(c+d x)} \sqrt{\cos (c+d x)} \csc (c+d x) (b \cos (c+d x))^n \left (A (2 n+5) \, _2F_1\left (\frac{1}{2},\frac{1}{4} (2 n+1);\frac{1}{4} (2 n+5);\cos ^2(c+d x)\right )+C (2 n+1) \cos ^2(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{4} (2 n+5);\frac{1}{4} (2 n+9);\cos ^2(c+d x)\right )\right )}{d (2 n+1) (2 n+5)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.74, size = 0, normalized size = 0. \begin{align*} \int{ \left ( b\cos \left ( dx+c \right ) \right ) ^{n} \left ( A+C \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ){\frac{1}{\sqrt{\cos \left ( dx+c \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 \frac{{\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{n}}{\sqrt{\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 (\frac{{\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{n}}{\sqrt{\cos \left (d x + c\right )}}, 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} + A\right )} \left (b \cos \left (d x + c\right )\right )^{n}}{\sqrt{\cos \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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