Optimal. Leaf size=119 \[ -\frac{3 A \sin (c+d x) (b \cos (c+d x))^{5/3} \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right )}{5 b^3 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{8/3} \, _2F_1\left (\frac{1}{2},\frac{4}{3};\frac{7}{3};\cos ^2(c+d x)\right )}{8 b^4 d \sqrt{\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.0734252, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {16, 2748, 2643} \[ -\frac{3 A \sin (c+d x) (b \cos (c+d x))^{5/3} \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right )}{5 b^3 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{8/3} \, _2F_1\left (\frac{1}{2},\frac{4}{3};\frac{7}{3};\cos ^2(c+d x)\right )}{8 b^4 d \sqrt{\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2748
Rule 2643
Rubi steps
\begin{align*} \int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{4/3}} \, dx &=\frac{\int (b \cos (c+d x))^{2/3} (A+B \cos (c+d x)) \, dx}{b^2}\\ &=\frac{A \int (b \cos (c+d x))^{2/3} \, dx}{b^2}+\frac{B \int (b \cos (c+d x))^{5/3} \, dx}{b^3}\\ &=-\frac{3 A (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)}}-\frac{3 B (b \cos (c+d x))^{8/3} \, _2F_1\left (\frac{1}{2},\frac{4}{3};\frac{7}{3};\cos ^2(c+d x)\right ) \sin (c+d x)}{8 b^4 d \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.166257, size = 94, normalized size = 0.79 \[ -\frac{3 \sqrt{\sin ^2(c+d x)} \cos ^2(c+d x) \cot (c+d x) \left (8 A \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right )+5 B \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{4}{3};\frac{7}{3};\cos ^2(c+d x)\right )\right )}{40 d (b \cos (c+d x))^{4/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.327, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \left ( A+B\cos \left ( dx+c \right ) \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 (B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{2}}{\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 ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac{2}{3}}}{b^{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 (B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{2}}{\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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