Optimal. Leaf size=83 \[ \frac{3 \sin (c+d x) \cos ^m(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m-1);\frac{1}{6} (3 m+5);\cos ^2(c+d x)\right )}{b d (1-3 m) \sqrt{\sin ^2(c+d x)} \sqrt [3]{b \cos (c+d x)}} \]
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Rubi [A] time = 0.0311037, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {20, 2643} \[ \frac{3 \sin (c+d x) \cos ^m(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (3 m-1);\frac{1}{6} (3 m+5);\cos ^2(c+d x)\right )}{b d (1-3 m) \sqrt{\sin ^2(c+d x)} \sqrt [3]{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 20
Rule 2643
Rubi steps
\begin{align*} \int \frac{\cos ^m(c+d x)}{(b \cos (c+d x))^{4/3}} \, dx &=\frac{\sqrt [3]{\cos (c+d x)} \int \cos ^{-\frac{4}{3}+m}(c+d x) \, dx}{b \sqrt [3]{b \cos (c+d x)}}\\ &=\frac{3 \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 (1-3 m) \sqrt [3]{b \cos (c+d x)} \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.109982, size = 82, normalized size = 0.99 \[ -\frac{\sqrt{\sin ^2(c+d x)} \csc (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{2} \left (m-\frac{1}{3}\right );\frac{1}{2} \left (m+\frac{5}{3}\right );\cos ^2(c+d x)\right )}{d \left (m-\frac{1}{3}\right ) (b \cos (c+d x))^{4/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.129, 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{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{\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 (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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos ^{m}{\left (c + d x \right )}}{\left (b \cos{\left (c + d x \right )}\right )^{\frac{4}{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{\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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