Optimal. Leaf size=95 \[ \frac{3 C \sin (c+d x) (b \cos (c+d x))^{11/3}}{14 b^3 d}-\frac{3 (14 A+11 C) \sin (c+d x) (b \cos (c+d x))^{11/3} \, _2F_1\left (\frac{1}{2},\frac{11}{6};\frac{17}{6};\cos ^2(c+d x)\right )}{154 b^3 d \sqrt{\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.0698885, 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} \[ \frac{3 C \sin (c+d x) (b \cos (c+d x))^{11/3}}{14 b^3 d}-\frac{3 (14 A+11 C) \sin (c+d x) (b \cos (c+d x))^{11/3} \, _2F_1\left (\frac{1}{2},\frac{11}{6};\frac{17}{6};\cos ^2(c+d x)\right )}{154 b^3 d \sqrt{\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3014
Rule 2643
Rubi steps
\begin{align*} \int \cos ^2(c+d x) (b \cos (c+d x))^{2/3} \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac{\int (b \cos (c+d x))^{8/3} \left (A+C \cos ^2(c+d x)\right ) \, dx}{b^2}\\ &=\frac{3 C (b \cos (c+d x))^{11/3} \sin (c+d x)}{14 b^3 d}+\frac{(14 A+11 C) \int (b \cos (c+d x))^{8/3} \, dx}{14 b^2}\\ &=\frac{3 C (b \cos (c+d x))^{11/3} \sin (c+d x)}{14 b^3 d}-\frac{3 (14 A+11 C) (b \cos (c+d x))^{11/3} \, _2F_1\left (\frac{1}{2},\frac{11}{6};\frac{17}{6};\cos ^2(c+d x)\right ) \sin (c+d x)}{154 b^3 d \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.109981, size = 96, normalized size = 1.01 \[ -\frac{3 \sqrt{\sin ^2(c+d x)} \cot (c+d x) (b \cos (c+d x))^{2/3} \left (17 A \cos ^2(c+d x) \, _2F_1\left (\frac{1}{2},\frac{11}{6};\frac{17}{6};\cos ^2(c+d x)\right )+11 C \cos ^4(c+d x) \, _2F_1\left (\frac{1}{2},\frac{17}{6};\frac{23}{6};\cos ^2(c+d x)\right )\right )}{187 d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.354, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( dx+c \right ) \right ) ^{2} \left ( b\cos \left ( dx+c \right ) \right ) ^{{\frac{2}{3}}} \left ( A+C \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \, 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{\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac{2}{3}} \cos \left (d x + c\right )^{2}\,{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 ({\left (C \cos \left (d x + c\right )^{4} + A \cos \left (d x + c\right )^{2}\right )} \left (b \cos \left (d x + c\right )\right )^{\frac{2}{3}}, 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{\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac{2}{3}} \cos \left (d x + c\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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