Optimal. Leaf size=90 \[ \frac{3 A b \tan (c+d x)}{7 d (b \sec (c+d x))^{7/3}}-\frac{3 (4 A+7 C) \sin (c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{2}{3},\frac{5}{3},\cos ^2(c+d x)\right )}{28 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{4/3}} \]
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Rubi [A] time = 0.0901464, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {16, 4045, 3772, 2643} \[ \frac{3 A b \tan (c+d x)}{7 d (b \sec (c+d x))^{7/3}}-\frac{3 (4 A+7 C) \sin (c+d x) \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right )}{28 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{4/3}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 4045
Rule 3772
Rule 2643
Rubi steps
\begin{align*} \int \frac{\cos (c+d x) \left (A+C \sec ^2(c+d x)\right )}{(b \sec (c+d x))^{4/3}} \, dx &=b \int \frac{A+C \sec ^2(c+d x)}{(b \sec (c+d x))^{7/3}} \, dx\\ &=\frac{3 A b \tan (c+d x)}{7 d (b \sec (c+d x))^{7/3}}+\frac{(4 A+7 C) \int \frac{1}{\sqrt [3]{b \sec (c+d x)}} \, dx}{7 b}\\ &=\frac{3 A b \tan (c+d x)}{7 d (b \sec (c+d x))^{7/3}}+\frac{\left ((4 A+7 C) \left (\frac{\cos (c+d x)}{b}\right )^{2/3} (b \sec (c+d x))^{2/3}\right ) \int \sqrt [3]{\frac{\cos (c+d x)}{b}} \, dx}{7 b}\\ &=-\frac{3 (4 A+7 C) \cos ^2(c+d x) \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right ) (b \sec (c+d x))^{2/3} \sin (c+d x)}{28 b^2 d \sqrt{\sin ^2(c+d x)}}+\frac{3 A b \tan (c+d x)}{7 d (b \sec (c+d x))^{7/3}}\\ \end{align*}
Mathematica [A] time = 0.109125, size = 92, normalized size = 1.02 \[ -\frac{3 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) \left (A \cos ^2(c+d x) \text{Hypergeometric2F1}\left (-\frac{7}{6},\frac{1}{2},-\frac{1}{6},\sec ^2(c+d x)\right )+7 C \text{Hypergeometric2F1}\left (-\frac{1}{6},\frac{1}{2},\frac{5}{6},\sec ^2(c+d x)\right )\right )}{7 b d \sqrt [3]{b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.197, size = 0, normalized size = 0. \begin{align*} \int{\cos \left ( dx+c \right ) \left ( A+C \left ( \sec \left ( dx+c \right ) \right ) ^{2} \right ) \left ( b\sec \left ( dx+c \right ) \right ) ^{-{\frac{4}{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 \frac{{\left (C \sec \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )}{\left (b \sec \left (d x + c\right )\right )^{\frac{4}{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 (\frac{{\left (C \cos \left (d x + c\right ) \sec \left (d x + c\right )^{2} + A \cos \left (d x + c\right )\right )} \left (b \sec \left (d x + c\right )\right )^{\frac{2}{3}}}{b^{2} \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 \frac{{\left (C \sec \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )}{\left (b \sec \left (d x + c\right )\right )^{\frac{4}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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