Optimal. Leaf size=167 \[ -\frac{3 A \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m+1);\frac{1}{6} (3 m+7);\cos ^2(c+d x)\right )}{d (3 m+1) \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}-\frac{3 B \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m+4);\frac{1}{6} (3 m+10);\cos ^2(c+d x)\right )}{d (3 m+4) \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}} \]
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Rubi [A] time = 0.0880054, 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} \[ -\frac{3 A \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m+1);\frac{1}{6} (3 m+7);\cos ^2(c+d x)\right )}{d (3 m+1) \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}-\frac{3 B \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m+4);\frac{1}{6} (3 m+10);\cos ^2(c+d x)\right )}{d (3 m+4) \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}} \]
Antiderivative was successfully verified.
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Rule 20
Rule 2748
Rule 2643
Rubi steps
\begin{align*} \int \frac{\cos ^m(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{2/3}} \, dx &=\frac{\cos ^{\frac{2}{3}}(c+d x) \int \cos ^{-\frac{2}{3}+m}(c+d x) (A+B \cos (c+d x)) \, dx}{(b \cos (c+d x))^{2/3}}\\ &=\frac{\left (A \cos ^{\frac{2}{3}}(c+d x)\right ) \int \cos ^{-\frac{2}{3}+m}(c+d x) \, dx}{(b \cos (c+d x))^{2/3}}+\frac{\left (B \cos ^{\frac{2}{3}}(c+d x)\right ) \int \cos ^{\frac{1}{3}+m}(c+d x) \, dx}{(b \cos (c+d x))^{2/3}}\\ &=-\frac{3 A \cos ^{1+m}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (1+3 m);\frac{1}{6} (7+3 m);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (1+3 m) (b \cos (c+d x))^{2/3} \sqrt{\sin ^2(c+d x)}}-\frac{3 B \cos ^{2+m}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (4+3 m);\frac{1}{6} (10+3 m);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (4+3 m) (b \cos (c+d x))^{2/3} \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.285737, size = 140, normalized size = 0.84 \[ -\frac{3 \sqrt{\sin ^2(c+d x)} \csc (c+d x) \cos ^{m+1}(c+d x) \left (A (3 m+4) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m+1);\frac{1}{6} (3 m+7);\cos ^2(c+d x)\right )+B (3 m+1) \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m+4);\frac{m}{2}+\frac{5}{3};\cos ^2(c+d x)\right )\right )}{d (3 m+1) (3 m+4) (b \cos (c+d x))^{2/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.207, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \cos \left ( dx+c \right ) \right ) ^{m} \left ( A+B\cos \left ( dx+c \right ) \right ) \left ( b\cos \left ( dx+c \right ) \right ) ^{-{\frac{2}{3}}}}\, 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 (B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{m}}{\left (b \cos \left (d x + c\right )\right )^{\frac{2}{3}}}\,{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 (B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac{1}{3}} \cos \left (d x + c\right )^{m}}{b \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (A + B \cos{\left (c + d x \right )}\right ) \cos ^{m}{\left (c + d x \right )}}{\left (b \cos{\left (c + d x \right )}\right )^{\frac{2}{3}}}\, dx \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 (B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{m}}{\left (b \cos \left (d x + c\right )\right )^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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