Optimal. Leaf size=115 \[ \frac{3 A b^2 \sin (c+d x) \, _2F_1\left (-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right )}{2 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}-\frac{3 b B \sin (c+d x) \sqrt [3]{b \cos (c+d x)} \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right )}{d \sqrt{\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.0982533, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {16, 2748, 2643} \[ \frac{3 A b^2 \sin (c+d x) \, _2F_1\left (-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right )}{2 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}-\frac{3 b B \sin (c+d x) \sqrt [3]{b \cos (c+d x)} \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right )}{d \sqrt{\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2748
Rule 2643
Rubi steps
\begin{align*} \int (b \cos (c+d x))^{4/3} (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx &=b^3 \int \frac{A+B \cos (c+d x)}{(b \cos (c+d x))^{5/3}} \, dx\\ &=\left (A b^3\right ) \int \frac{1}{(b \cos (c+d x))^{5/3}} \, dx+\left (b^2 B\right ) \int \frac{1}{(b \cos (c+d x))^{2/3}} \, dx\\ &=\frac{3 A b^2 \, _2F_1\left (-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right ) \sin (c+d x)}{2 d (b \cos (c+d x))^{2/3} \sqrt{\sin ^2(c+d x)}}-\frac{3 b B \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)}{d \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.103253, size = 88, normalized size = 0.77 \[ \frac{3 b^2 \sqrt{\sin ^2(c+d x)} \csc (c+d x) \left (A \, _2F_1\left (-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right )-2 B \cos (c+d x) \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right )\right )}{2 d (b \cos (c+d x))^{2/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.335, 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 ) \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{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{\left (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 )^{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 ({\left (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 )^{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 (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 )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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