Optimal. Leaf size=145 \[ -\frac{3 b (4 A+C) \sin (c+d x) \sqrt [3]{b \cos (c+d x)} \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right )}{4 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right )}{4 d \sqrt{\sin ^2(c+d x)}}+\frac{3 b C \sin (c+d x) \sqrt [3]{b \cos (c+d x)}}{4 d} \]
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Rubi [A] time = 0.173694, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.098, Rules used = {16, 3023, 2748, 2643} \[ -\frac{3 b (4 A+C) \sin (c+d x) \sqrt [3]{b \cos (c+d x)} \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right )}{4 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right )}{4 d \sqrt{\sin ^2(c+d x)}}+\frac{3 b C \sin (c+d x) \sqrt [3]{b \cos (c+d x)}}{4 d} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3023
Rule 2748
Rule 2643
Rubi steps
\begin{align*} \int (b \cos (c+d x))^{4/3} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx &=b^2 \int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx\\ &=\frac{3 b C \sqrt [3]{b \cos (c+d x)} \sin (c+d x)}{4 d}+\frac{1}{4} (3 b) \int \frac{\frac{1}{3} b (4 A+C)+\frac{4}{3} b B \cos (c+d x)}{(b \cos (c+d x))^{2/3}} \, dx\\ &=\frac{3 b C \sqrt [3]{b \cos (c+d x)} \sin (c+d x)}{4 d}+(b B) \int \sqrt [3]{b \cos (c+d x)} \, dx+\frac{1}{4} \left (b^2 (4 A+C)\right ) \int \frac{1}{(b \cos (c+d x))^{2/3}} \, dx\\ &=\frac{3 b C \sqrt [3]{b \cos (c+d x)} \sin (c+d x)}{4 d}-\frac{3 b (4 A+C) \sqrt [3]{b \cos (c+d x)} \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right ) \sin (c+d x)}{4 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B (b \cos (c+d x))^{4/3} \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right ) \sin (c+d x)}{4 d \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.204434, size = 108, normalized size = 0.74 \[ -\frac{3 b^2 \sin (2 (c+d x)) \left ((4 A+C) \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right )+B \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right )+C \left (-\sqrt{\sin ^2(c+d x)}\right )\right )}{8 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.376, size = 0, normalized size = 0. \begin{align*} \int \left ( b\cos \left ( dx+c \right ) \right ) ^{{\frac{4}{3}}} \left ( A+B\cos \left ( dx+c \right ) +C \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}\, 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} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac{4}{3}} \sec \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 )^{3} + B b \cos \left (d x + c\right )^{2} + A b \cos \left (d x + c\right )\right )} \left (b \cos \left (d x + c\right )\right )^{\frac{1}{3}} \sec \left (d x + c\right )^{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{\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{4}{3}} \sec \left (d x + c\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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