Optimal. Leaf size=154 \[ -\frac{3 (5 A+2 C) \sin (c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right )}{10 b^2 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{5/3} \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right )}{5 b^3 d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \sin (c+d x) (b \cos (c+d x))^{2/3}}{5 b^2 d} \]
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Rubi [A] time = 0.140053, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {16, 3023, 2748, 2643} \[ -\frac{3 (5 A+2 C) \sin (c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right )}{10 b^2 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{5/3} \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right )}{5 b^3 d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \sin (c+d x) (b \cos (c+d x))^{2/3}}{5 b^2 d} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3023
Rule 2748
Rule 2643
Rubi steps
\begin{align*} \int \frac{\cos (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{\int \frac{A+B \cos (c+d x)+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{3 \int \frac{\frac{1}{3} b (5 A+2 C)+\frac{5}{3} b B \cos (c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx}{5 b^2}\\ &=\frac{3 C (b \cos (c+d x))^{2/3} \sin (c+d x)}{5 b^2 d}+\frac{B \int (b \cos (c+d x))^{2/3} \, dx}{b^2}+\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)}}-\frac{3 B (b \cos (c+d x))^{5/3} \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right ) \sin (c+d x)}{5 b^3 d \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.205271, size = 111, normalized size = 0.72 \[ -\frac{3 \sin (2 (c+d x)) \left ((5 A+2 C) \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right )+2 B \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right )-2 C \sqrt{\sin ^2(c+d x)}\right )}{20 b d \sqrt{\sin ^2(c+d x)} \sqrt [3]{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.297, size = 0, normalized size = 0. \begin{align*} \int{\cos \left ( dx+c \right ) \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 )}{\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}}}{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 ) + 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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