Optimal. Leaf size=148 \[ -\frac{3 (7 A+4 C) \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right )}{28 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{7/3} \, _2F_1\left (\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right )}{7 b d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \sin (c+d x) (b \cos (c+d x))^{4/3}}{7 d} \]
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Rubi [A] time = 0.148493, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {16, 3023, 2748, 2643} \[ -\frac{3 (7 A+4 C) \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right )}{28 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{7/3} \, _2F_1\left (\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right )}{7 b d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \sin (c+d x) (b \cos (c+d x))^{4/3}}{7 d} \]
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 (c+d x) \, dx &=b \int \sqrt [3]{b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\\ &=\frac{3 C (b \cos (c+d x))^{4/3} \sin (c+d x)}{7 d}+\frac{3}{7} \int \sqrt [3]{b \cos (c+d x)} \left (\frac{1}{3} b (7 A+4 C)+\frac{7}{3} b B \cos (c+d x)\right ) \, dx\\ &=\frac{3 C (b \cos (c+d x))^{4/3} \sin (c+d x)}{7 d}+B \int (b \cos (c+d x))^{4/3} \, dx+\frac{1}{7} (b (7 A+4 C)) \int \sqrt [3]{b \cos (c+d x)} \, dx\\ &=\frac{3 C (b \cos (c+d x))^{4/3} \sin (c+d x)}{7 d}-\frac{3 (7 A+4 C) (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)}{28 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B (b \cos (c+d x))^{7/3} \, _2F_1\left (\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right ) \sin (c+d x)}{7 b d \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.224451, size = 109, normalized size = 0.74 \[ -\frac{3 b \sin (2 (c+d x)) \sqrt [3]{b \cos (c+d x)} \left ((7 A+4 C) \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right )+4 B \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right )-4 C \sqrt{\sin ^2(c+d x)}\right )}{56 d \sqrt{\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.362, 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 ) \sec \left ( dx+c \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} + 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 )\,{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 ), 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 )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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