Optimal. Leaf size=167 \[ -\frac{3 A \sin (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m+5);\frac{1}{6} (3 m+11);\cos ^2(c+d x)\right )}{d (3 m+5) \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+2}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m+8);\frac{1}{6} (3 m+14);\cos ^2(c+d x)\right )}{d (3 m+8) \sqrt{\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.0945796, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {20, 2748, 2643} \[ -\frac{3 A \sin (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m+5);\frac{1}{6} (3 m+11);\cos ^2(c+d x)\right )}{d (3 m+5) \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+2}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m+8);\frac{1}{6} (3 m+14);\cos ^2(c+d x)\right )}{d (3 m+8) \sqrt{\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 20
Rule 2748
Rule 2643
Rubi steps
\begin{align*} \int \cos ^m(c+d x) (b \cos (c+d x))^{2/3} (A+B \cos (c+d x)) \, dx &=\frac{(b \cos (c+d x))^{2/3} \int \cos ^{\frac{2}{3}+m}(c+d x) (A+B \cos (c+d x)) \, dx}{\cos ^{\frac{2}{3}}(c+d x)}\\ &=\frac{\left (A (b \cos (c+d x))^{2/3}\right ) \int \cos ^{\frac{2}{3}+m}(c+d x) \, dx}{\cos ^{\frac{2}{3}}(c+d x)}+\frac{\left (B (b \cos (c+d x))^{2/3}\right ) \int \cos ^{\frac{5}{3}+m}(c+d x) \, dx}{\cos ^{\frac{2}{3}}(c+d x)}\\ &=-\frac{3 A \cos ^{1+m}(c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left (\frac{1}{2},\frac{1}{6} (5+3 m);\frac{1}{6} (11+3 m);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (5+3 m) \sqrt{\sin ^2(c+d x)}}-\frac{3 B \cos ^{2+m}(c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left (\frac{1}{2},\frac{1}{6} (8+3 m);\frac{1}{6} (14+3 m);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (8+3 m) \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.28948, size = 140, normalized size = 0.84 \[ -\frac{3 \sqrt{\sin ^2(c+d x)} \csc (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+1}(c+d x) \left (A (3 m+8) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m+5);\frac{1}{6} (3 m+11);\cos ^2(c+d x)\right )+B (3 m+5) \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m+8);\frac{m}{2}+\frac{7}{3};\cos ^2(c+d x)\right )\right )}{d (3 m+5) (3 m+8)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.231, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( dx+c \right ) \right ) ^{m} \left ( b\cos \left ( dx+c \right ) \right ) ^{{\frac{2}{3}}} \left ( A+B\cos \left ( dx+c \right ) \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 (B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac{2}{3}} \cos \left (d x + c\right )^{m}\,{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 (B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac{2}{3}} \cos \left (d x + c\right )^{m}, 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 (B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac{2}{3}} \cos \left (d x + c\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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