Optimal. Leaf size=144 \[ \frac{C \sin (c+d x) (a \cos (c+d x))^{m+1} (b \cos (c+d x))^n}{a d (m+n+2)}-\frac{(A (m+n+2)+C (m+n+1)) \sin (c+d x) (a \cos (c+d x))^{m+1} (b \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{2} (m+n+1);\frac{1}{2} (m+n+3);\cos ^2(c+d x)\right )}{a d (m+n+1) (m+n+2) \sqrt{\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.111634, antiderivative size = 144, 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} \[ \frac{C \sin (c+d x) (a \cos (c+d x))^{m+1} (b \cos (c+d x))^n}{a d (m+n+2)}-\frac{(A (m+n+2)+C (m+n+1)) \sin (c+d x) (a \cos (c+d x))^{m+1} (b \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{2} (m+n+1);\frac{1}{2} (m+n+3);\cos ^2(c+d x)\right )}{a d (m+n+1) (m+n+2) \sqrt{\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 20
Rule 3014
Rule 2643
Rubi steps
\begin{align*} \int (a \cos (c+d x))^m (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx &=\left ((a \cos (c+d x))^{-n} (b \cos (c+d x))^n\right ) \int (a \cos (c+d x))^{m+n} \left (A+C \cos ^2(c+d x)\right ) \, dx\\ &=\frac{C (a \cos (c+d x))^{1+m} (b \cos (c+d x))^n \sin (c+d x)}{a d (2+m+n)}+\left (\left (A+\frac{C (1+m+n)}{2+m+n}\right ) (a \cos (c+d x))^{-n} (b \cos (c+d x))^n\right ) \int (a \cos (c+d x))^{m+n} \, dx\\ &=\frac{C (a \cos (c+d x))^{1+m} (b \cos (c+d x))^n \sin (c+d x)}{a d (2+m+n)}-\frac{\left (A+\frac{C (1+m+n)}{2+m+n}\right ) (a \cos (c+d x))^{1+m} (b \cos (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{1}{2} (1+m+n);\frac{1}{2} (3+m+n);\cos ^2(c+d x)\right ) \sin (c+d x)}{a d (1+m+n) \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.239387, size = 132, normalized size = 0.92 \[ -\frac{\sqrt{\sin ^2(c+d x)} \cot (c+d x) (a \cos (c+d x))^m (b \cos (c+d x))^n \left (A (m+n+3) \, _2F_1\left (\frac{1}{2},\frac{1}{2} (m+n+1);\frac{1}{2} (m+n+3);\cos ^2(c+d x)\right )+C (m+n+1) \cos ^2(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{2} (m+n+3);\frac{1}{2} (m+n+5);\cos ^2(c+d x)\right )\right )}{d (m+n+1) (m+n+3)} \]
Antiderivative was successfully verified.
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Maple [F] time = 2.474, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( dx+c \right ) a \right ) ^{m} \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 (a \cos \left (d x + c\right )\right )^{m} \left (b \cos \left (d x + c\right )\right )^{n}\,{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} + A\right )} \left (a \cos \left (d x + c\right )\right )^{m} \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 (a \cos \left (d x + c\right )\right )^{m} \left (b \cos \left (d x + c\right )\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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