Optimal. Leaf size=95 \[ \frac{3 C \sin (c+d x) (b \cos (c+d x))^{13/3}}{16 b^3 d}-\frac{3 (16 A+13 C) \sin (c+d x) (b \cos (c+d x))^{13/3} \, _2F_1\left (\frac{1}{2},\frac{13}{6};\frac{19}{6};\cos ^2(c+d x)\right )}{208 b^3 d \sqrt{\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.0701794, 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} \[ \frac{3 C \sin (c+d x) (b \cos (c+d x))^{13/3}}{16 b^3 d}-\frac{3 (16 A+13 C) \sin (c+d x) (b \cos (c+d x))^{13/3} \, _2F_1\left (\frac{1}{2},\frac{13}{6};\frac{19}{6};\cos ^2(c+d x)\right )}{208 b^3 d \sqrt{\sin ^2(c+d x)}} \]
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))^{4/3} \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac{\int (b \cos (c+d x))^{10/3} \left (A+C \cos ^2(c+d x)\right ) \, dx}{b^2}\\ &=\frac{3 C (b \cos (c+d x))^{13/3} \sin (c+d x)}{16 b^3 d}+\frac{(16 A+13 C) \int (b \cos (c+d x))^{10/3} \, dx}{16 b^2}\\ &=\frac{3 C (b \cos (c+d x))^{13/3} \sin (c+d x)}{16 b^3 d}-\frac{3 (16 A+13 C) (b \cos (c+d x))^{13/3} \, _2F_1\left (\frac{1}{2},\frac{13}{6};\frac{19}{6};\cos ^2(c+d x)\right ) \sin (c+d x)}{208 b^3 d \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.261575, size = 96, normalized size = 1.01 \[ -\frac{3 \sqrt{\sin ^2(c+d x)} \cos ^2(c+d x) \cot (c+d x) (b \cos (c+d x))^{4/3} \left (19 A \, _2F_1\left (\frac{1}{2},\frac{13}{6};\frac{19}{6};\cos ^2(c+d x)\right )+13 C \cos ^2(c+d x) \, _2F_1\left (\frac{1}{2},\frac{19}{6};\frac{25}{6};\cos ^2(c+d x)\right )\right )}{247 d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.349, 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{4}{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{4}{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 b \cos \left (d x + c\right )^{5} + A b \cos \left (d x + c\right )^{3}\right )} \left (b \cos \left (d x + c\right )\right )^{\frac{1}{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{4}{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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