Optimal. Leaf size=235 \[ -\frac{3 (C (1-3 m)-A (3 m+2)) \sin (c+d x) \cos ^m(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m-1);\frac{1}{6} (3 m+5);\cos ^2(c+d x)\right )}{b d (1-3 m) (3 m+2) \sqrt{\sin ^2(c+d x)} \sqrt [3]{b \cos (c+d x)}}-\frac{3 B \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m+2);\frac{1}{6} (3 m+8);\cos ^2(c+d x)\right )}{b d (3 m+2) \sqrt{\sin ^2(c+d x)} \sqrt [3]{b \cos (c+d x)}}+\frac{3 C \sin (c+d x) \cos ^m(c+d x)}{b d (3 m+2) \sqrt [3]{b \cos (c+d x)}} \]
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Rubi [A] time = 0.232412, antiderivative size = 225, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 4, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.098, Rules used = {20, 3023, 2748, 2643} \[ \frac{3 \left (\frac{A}{1-3 m}-\frac{C}{3 m+2}\right ) \sin (c+d x) \cos ^m(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m-1);\frac{1}{6} (3 m+5);\cos ^2(c+d x)\right )}{b d \sqrt{\sin ^2(c+d x)} \sqrt [3]{b \cos (c+d x)}}-\frac{3 B \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m+2);\frac{1}{6} (3 m+8);\cos ^2(c+d x)\right )}{b d (3 m+2) \sqrt{\sin ^2(c+d x)} \sqrt [3]{b \cos (c+d x)}}+\frac{3 C \sin (c+d x) \cos ^m(c+d x)}{b d (3 m+2) \sqrt [3]{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 20
Rule 3023
Rule 2748
Rule 2643
Rubi steps
\begin{align*} \int \frac{\cos ^m(c+d x) \left (A+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 (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx}{b \sqrt [3]{b \cos (c+d x)}}\\ &=\frac{3 C \cos ^m(c+d x) \sin (c+d x)}{b d (2+3 m) \sqrt [3]{b \cos (c+d x)}}+\frac{\left (3 \sqrt [3]{\cos (c+d x)}\right ) \int \cos ^{-\frac{4}{3}+m}(c+d x) \left (\frac{1}{3} \left (-3 C \left (\frac{1}{3}-m\right )+3 A \left (\frac{2}{3}+m\right )\right )+\frac{1}{3} B (2+3 m) \cos (c+d x)\right ) \, dx}{b (2+3 m) \sqrt [3]{b \cos (c+d x)}}\\ &=\frac{3 C \cos ^m(c+d x) \sin (c+d x)}{b d (2+3 m) \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 (1-3 m)+A (2+3 m)) \sqrt [3]{\cos (c+d x)}\right ) \int \cos ^{-\frac{4}{3}+m}(c+d x) \, dx}{b (2+3 m) \sqrt [3]{b \cos (c+d x)}}\\ &=\frac{3 C \cos ^m(c+d x) \sin (c+d x)}{b d (2+3 m) \sqrt [3]{b \cos (c+d x)}}+\frac{3 \left (\frac{A}{1-3 m}-\frac{C}{2+3 m}\right ) \cos ^m(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (-1+3 m);\frac{1}{6} (5+3 m);\cos ^2(c+d x)\right ) \sin (c+d x)}{b d \sqrt [3]{b \cos (c+d x)} \sqrt{\sin ^2(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)}}\\ \end{align*}
Mathematica [A] time = 0.536624, size = 166, normalized size = 0.71 \[ -\frac{3 \sin (c+d x) \cos ^{m+1}(c+d x) \left ((A (3 m+2)+C (3 m-1)) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m-1);\frac{1}{6} (3 m+5);\cos ^2(c+d x)\right )+(3 m-1) \left (B \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m+2);\frac{1}{6} (3 m+8);\cos ^2(c+d x)\right )-C \sqrt{\sin ^2(c+d x)}\right )\right )}{d (3 m-1) (3 m+2) \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{4/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.307, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \cos \left ( dx+c \right ) \right ) ^{m} \left ( A+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 ) + A\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 )^{2} + 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}}{b^{2} \cos \left (d x + c\right )^{2}}, 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 ) + A\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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