Optimal. Leaf size=95 \[ \frac{3 C \sin (c+d x) (b \cos (c+d x))^{2/3}}{5 b^2 d}-\frac{3 (5 A+2 C) \sin (c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right )}{10 b^2 d \sqrt{\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.0658111, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {16, 3014, 2643} \[ \frac{3 C \sin (c+d x) (b \cos (c+d x))^{2/3}}{5 b^2 d}-\frac{3 (5 A+2 C) \sin (c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right )}{10 b^2 d \sqrt{\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3014
Rule 2643
Rubi steps
\begin{align*} \int \frac{\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{4/3}} \, dx &=\frac{\int \frac{A+C \cos ^2(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx}{b}\\ &=\frac{3 C (b \cos (c+d x))^{2/3} \sin (c+d x)}{5 b^2 d}+\frac{(5 A+2 C) \int \frac{1}{\sqrt [3]{b \cos (c+d x)}} \, dx}{5 b}\\ &=\frac{3 C (b \cos (c+d x))^{2/3} \sin (c+d x)}{5 b^2 d}-\frac{3 (5 A+2 C) (b \cos (c+d x))^{2/3} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right ) \sin (c+d x)}{10 b^2 d \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0918609, size = 90, normalized size = 0.95 \[ -\frac{3 \sqrt{\sin ^2(c+d x)} \cot (c+d x) \left (4 A \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right )+C \cos ^2(c+d x) \, _2F_1\left (\frac{1}{2},\frac{4}{3};\frac{7}{3};\cos ^2(c+d x)\right )\right )}{8 b d \sqrt [3]{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.326, size = 0, normalized size = 0. \begin{align*} \int{\cos \left ( dx+c \right ) \left ( A+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} + A\right )} \cos \left (d x + c\right )}{\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} + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac{2}{3}}}{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} + A\right )} \cos \left (d x + c\right )}{\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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