Optimal. Leaf size=117 \[ \frac{C \sin (c+d x) (b \cos (c+d x))^{n+2}}{b^2 d (n+3)}-\frac{(A (n+3)+C (n+2)) \sin (c+d x) (b \cos (c+d x))^{n+2} \, _2F_1\left (\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\cos ^2(c+d x)\right )}{b^2 d (n+2) (n+3) \sqrt{\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.10299, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {16, 3014, 2643} \[ \frac{C \sin (c+d x) (b \cos (c+d x))^{n+2}}{b^2 d (n+3)}-\frac{(A (n+3)+C (n+2)) \sin (c+d x) (b \cos (c+d x))^{n+2} \, _2F_1\left (\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\cos ^2(c+d x)\right )}{b^2 d (n+2) (n+3) \sqrt{\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3014
Rule 2643
Rubi steps
\begin{align*} \int \cos (c+d x) (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac{\int (b \cos (c+d x))^{1+n} \left (A+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{\left (A+\frac{C (2+n)}{3+n}\right ) \int (b \cos (c+d x))^{1+n} \, dx}{b}\\ &=\frac{C (b \cos (c+d x))^{2+n} \sin (c+d x)}{b^2 d (3+n)}-\frac{\left (A+\frac{C (2+n)}{3+n}\right ) (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) \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.175528, size = 120, normalized size = 1.03 \[ -\frac{\sqrt{\sin ^2(c+d x)} \cos (c+d x) \cot (c+d x) (b \cos (c+d x))^n \left (A (n+4) \, _2F_1\left (\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\cos ^2(c+d x)\right )+C (n+2) \cos ^2(c+d x) \, _2F_1\left (\frac{1}{2},\frac{n+4}{2};\frac{n+6}{2};\cos ^2(c+d x)\right )\right )}{d (n+2) (n+4)} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.463, size = 0, normalized size = 0. \begin{align*} \int \cos \left ( dx+c \right ) \left ( b\cos \left ( dx+c \right ) \right ) ^{n} \left ( A+C \left ( \cos \left ( dx+c \right ) \right ) ^{2} \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{\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \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 ({\left (C \cos \left (d x + c\right )^{3} + A \cos \left (d x + c\right )\right )} \left (b \cos \left (d x + c\right )\right )^{n}, 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} + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \cos \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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