Optimal. Leaf size=95 \[ \frac{3 C \sin (c+d x) (b \cos (c+d x))^{5/3}}{8 b^3 d}-\frac{3 (8 A+5 C) \sin (c+d x) (b \cos (c+d x))^{5/3} \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right )}{40 b^3 d \sqrt{\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.0668854, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {16, 3014, 2643} \[ \frac{3 C \sin (c+d x) (b \cos (c+d x))^{5/3}}{8 b^3 d}-\frac{3 (8 A+5 C) \sin (c+d x) (b \cos (c+d x))^{5/3} \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right )}{40 b^3 d \sqrt{\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3014
Rule 2643
Rubi steps
\begin{align*} \int \frac{\cos ^2(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{4/3}} \, dx &=\frac{\int (b \cos (c+d x))^{2/3} \left (A+C \cos ^2(c+d x)\right ) \, dx}{b^2}\\ &=\frac{3 C (b \cos (c+d x))^{5/3} \sin (c+d x)}{8 b^3 d}+\frac{(8 A+5 C) \int (b \cos (c+d x))^{2/3} \, dx}{8 b^2}\\ &=\frac{3 C (b \cos (c+d x))^{5/3} \sin (c+d x)}{8 b^3 d}-\frac{3 (8 A+5 C) (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)}{40 b^3 d \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.192695, size = 96, normalized size = 1.01 \[ -\frac{3 \sqrt{\sin ^2(c+d x)} \cos ^2(c+d x) \cot (c+d x) \left (11 A \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right )+5 C \cos ^2(c+d x) \, _2F_1\left (\frac{1}{2},\frac{11}{6};\frac{17}{6};\cos ^2(c+d x)\right )\right )}{55 d (b \cos (c+d x))^{4/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.348, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \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 )^{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 (C \cos \left (d x + c\right )^{2} + 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 (C \cos \left (d x + c\right )^{2} + 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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