Optimal. Leaf size=167 \[ \frac{3 A b \sin (c+d x) \sqrt [3]{b \sec (c+d x)} \sec ^m(c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{6} (-3 m-1),\frac{1}{6} (5-3 m),\cos ^2(c+d x)\right )}{d (3 m+1) \sqrt{\sin ^2(c+d x)}}+\frac{3 b B \sin (c+d x) \sqrt [3]{b \sec (c+d x)} \sec ^{m+1}(c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{6} (-3 m-4),\frac{1}{6} (2-3 m),\cos ^2(c+d x)\right )}{d (3 m+4) \sqrt{\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.120175, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {20, 3787, 3772, 2643} \[ \frac{3 A b \sin (c+d x) \sqrt [3]{b \sec (c+d x)} \sec ^m(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (-3 m-1);\frac{1}{6} (5-3 m);\cos ^2(c+d x)\right )}{d (3 m+1) \sqrt{\sin ^2(c+d x)}}+\frac{3 b B \sin (c+d x) \sqrt [3]{b \sec (c+d x)} \sec ^{m+1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (-3 m-4);\frac{1}{6} (2-3 m);\cos ^2(c+d x)\right )}{d (3 m+4) \sqrt{\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 20
Rule 3787
Rule 3772
Rule 2643
Rubi steps
\begin{align*} \int \sec ^m(c+d x) (b \sec (c+d x))^{4/3} (A+B \sec (c+d x)) \, dx &=\frac{\left (b \sqrt [3]{b \sec (c+d x)}\right ) \int \sec ^{\frac{4}{3}+m}(c+d x) (A+B \sec (c+d x)) \, dx}{\sqrt [3]{\sec (c+d x)}}\\ &=\frac{\left (A b \sqrt [3]{b \sec (c+d x)}\right ) \int \sec ^{\frac{4}{3}+m}(c+d x) \, dx}{\sqrt [3]{\sec (c+d x)}}+\frac{\left (b B \sqrt [3]{b \sec (c+d x)}\right ) \int \sec ^{\frac{7}{3}+m}(c+d x) \, dx}{\sqrt [3]{\sec (c+d x)}}\\ &=\left (A b \cos ^{\frac{1}{3}+m}(c+d x) \sec ^m(c+d x) \sqrt [3]{b \sec (c+d x)}\right ) \int \cos ^{-\frac{4}{3}-m}(c+d x) \, dx+\left (b B \cos ^{\frac{1}{3}+m}(c+d x) \sec ^m(c+d x) \sqrt [3]{b \sec (c+d x)}\right ) \int \cos ^{-\frac{7}{3}-m}(c+d x) \, dx\\ &=\frac{3 A b \, _2F_1\left (\frac{1}{2},\frac{1}{6} (-1-3 m);\frac{1}{6} (5-3 m);\cos ^2(c+d x)\right ) \sec ^m(c+d x) \sqrt [3]{b \sec (c+d x)} \sin (c+d x)}{d (1+3 m) \sqrt{\sin ^2(c+d x)}}+\frac{3 b B \, _2F_1\left (\frac{1}{2},\frac{1}{6} (-4-3 m);\frac{1}{6} (2-3 m);\cos ^2(c+d x)\right ) \sec ^{1+m}(c+d x) \sqrt [3]{b \sec (c+d x)} \sin (c+d x)}{d (4+3 m) \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.381677, size = 140, normalized size = 0.84 \[ \frac{3 \sqrt{-\tan ^2(c+d x)} \csc (c+d x) (b \sec (c+d x))^{4/3} \sec ^m(c+d x) \left (A (3 m+7) \cos (c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{6} (3 m+4),\frac{m}{2}+\frac{5}{3},\sec ^2(c+d x)\right )+B (3 m+4) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{6} (3 m+7),\frac{1}{6} (3 m+13),\sec ^2(c+d x)\right )\right )}{d (3 m+4) (3 m+7)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.15, size = 0, normalized size = 0. \begin{align*} \int \left ( \sec \left ( dx+c \right ) \right ) ^{m} \left ( b\sec \left ( dx+c \right ) \right ) ^{{\frac{4}{3}}} \left ( A+B\sec \left ( dx+c \right ) \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 (B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac{4}{3}} \sec \left (d x + c\right )^{m}\,{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 \sec \left (d x + c\right )^{2} + A b \sec \left (d x + c\right )\right )} \left (b \sec \left (d x + c\right )\right )^{\frac{1}{3}} \sec \left (d x + c\right )^{m}, 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 \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac{4}{3}} \sec \left (d x + c\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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