Optimal. Leaf size=88 \[ \frac{3 A b \tan (c+d x)}{4 d (b \sec (c+d x))^{4/3}}-\frac{3 (A+4 C) \sin (c+d x) \text{Hypergeometric2F1}\left (\frac{1}{6},\frac{1}{2},\frac{7}{6},\cos ^2(c+d x)\right )}{4 d \sqrt{\sin ^2(c+d x)} \sqrt [3]{b \sec (c+d x)}} \]
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Rubi [A] time = 0.0968436, antiderivative size = 88, 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} \[ \frac{3 A b \tan (c+d x)}{4 d (b \sec (c+d x))^{4/3}}-\frac{3 (A+4 C) \sin (c+d x) \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right )}{4 d \sqrt{\sin ^2(c+d x)} \sqrt [3]{b \sec (c+d x)}} \]
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 )}{\sqrt [3]{b \sec (c+d x)}} \, dx &=b \int \frac{A+C \sec ^2(c+d x)}{(b \sec (c+d x))^{4/3}} \, dx\\ &=\frac{3 A b \tan (c+d x)}{4 d (b \sec (c+d x))^{4/3}}+\frac{(A+4 C) \int (b \sec (c+d x))^{2/3} \, dx}{4 b}\\ &=\frac{3 A b \tan (c+d x)}{4 d (b \sec (c+d x))^{4/3}}+\frac{\left ((A+4 C) \left (\frac{\cos (c+d x)}{b}\right )^{2/3} (b \sec (c+d x))^{2/3}\right ) \int \frac{1}{\left (\frac{\cos (c+d x)}{b}\right )^{2/3}} \, dx}{4 b}\\ &=-\frac{3 (A+4 C) \cos (c+d x) \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right ) (b \sec (c+d x))^{2/3} \sin (c+d x)}{4 b d \sqrt{\sin ^2(c+d x)}}+\frac{3 A b \tan (c+d x)}{4 d (b \sec (c+d x))^{4/3}}\\ \end{align*}
Mathematica [C] time = 0.551716, size = 121, normalized size = 1.38 \[ -\frac{3 i e^{-i (c+d x)} \left (2 (A+4 C) e^{2 i (c+d x)} \left (1+e^{2 i (c+d x)}\right )^{2/3} \text{Hypergeometric2F1}\left (\frac{1}{3},\frac{2}{3},\frac{4}{3},-e^{2 i (c+d x)}\right )+A \left (-1+e^{4 i (c+d x)}\right )\right )}{8 d \left (1+e^{2 i (c+d x)}\right ) \sqrt [3]{b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.184, 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 ){\frac{1}{\sqrt [3]{b\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 \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{1}{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 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \cos{\left (c + d x \right )}}{\sqrt [3]{b \sec{\left (c + d x \right )}}}\, dx \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{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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