Optimal. Leaf size=173 \[ -\frac{3 B \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m+2);\frac{1}{6} (3 m+8);\cos ^2(c+d x)\right )}{b d (3 m+2) \sqrt{\sin ^2(c+d x)} \sqrt [3]{b \cos (c+d x)}}-\frac{3 C \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m+5);\frac{1}{6} (3 m+11);\cos ^2(c+d x)\right )}{b d (3 m+5) \sqrt{\sin ^2(c+d x)} \sqrt [3]{b \cos (c+d x)}} \]
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Rubi [A] time = 0.131254, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {20, 3010, 2748, 2643} \[ -\frac{3 B \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m+2);\frac{1}{6} (3 m+8);\cos ^2(c+d x)\right )}{b d (3 m+2) \sqrt{\sin ^2(c+d x)} \sqrt [3]{b \cos (c+d x)}}-\frac{3 C \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m+5);\frac{1}{6} (3 m+11);\cos ^2(c+d x)\right )}{b d (3 m+5) \sqrt{\sin ^2(c+d x)} \sqrt [3]{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 20
Rule 3010
Rule 2748
Rule 2643
Rubi steps
\begin{align*} \int \frac{\cos ^m(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{4/3}} \, dx &=\frac{\sqrt [3]{\cos (c+d x)} \int \cos ^{-\frac{4}{3}+m}(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx}{b \sqrt [3]{b \cos (c+d x)}}\\ &=\frac{\sqrt [3]{\cos (c+d x)} \int \cos ^{-\frac{1}{3}+m}(c+d x) (B+C \cos (c+d x)) \, dx}{b \sqrt [3]{b \cos (c+d x)}}\\ &=\frac{\left (B \sqrt [3]{\cos (c+d x)}\right ) \int \cos ^{-\frac{1}{3}+m}(c+d x) \, dx}{b \sqrt [3]{b \cos (c+d x)}}+\frac{\left (C \sqrt [3]{\cos (c+d x)}\right ) \int \cos ^{\frac{2}{3}+m}(c+d x) \, dx}{b \sqrt [3]{b \cos (c+d x)}}\\ &=-\frac{3 B \cos ^{1+m}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (2+3 m);\frac{1}{6} (8+3 m);\cos ^2(c+d x)\right ) \sin (c+d x)}{b d (2+3 m) \sqrt [3]{b \cos (c+d x)} \sqrt{\sin ^2(c+d x)}}-\frac{3 C \cos ^{2+m}(c+d x) \, _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)}{b d (5+3 m) \sqrt [3]{b \cos (c+d x)} \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.350667, size = 140, normalized size = 0.81 \[ -\frac{3 \sqrt{\sin ^2(c+d x)} \csc (c+d x) \cos ^{m+2}(c+d x) \left (B (3 m+5) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m+2);\frac{1}{6} (3 m+8);\cos ^2(c+d x)\right )+C (3 m+2) \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m+5);\frac{1}{6} (3 m+11);\cos ^2(c+d x)\right )\right )}{d (3 m+2) (3 m+5) (b \cos (c+d x))^{4/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.382, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \cos \left ( dx+c \right ) \right ) ^{m} \left ( B\cos \left ( dx+c \right ) +C \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \left ( b\cos \left ( dx+c \right ) \right ) ^{-{\frac{4}{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 (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} \cos \left (d x + c\right )^{m}}{\left (b \cos \left (d x + c\right )\right )^{\frac{4}{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 (C \cos \left (d x + c\right ) + B\right )} \left (b \cos \left (d x + c\right )\right )^{\frac{2}{3}} \cos \left (d x + c\right )^{m}}{b^{2} \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} + B \cos \left (d x + c\right )\right )} \cos \left (d x + c\right )^{m}}{\left (b \cos \left (d x + c\right )\right )^{\frac{4}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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