Optimal. Leaf size=145 \[ \frac{3 C \sin (c+d x) \sec ^{m+1}(c+d x)}{d (3 m+1) (b \sec (c+d x))^{2/3}}-\frac{3 (3 A m+A-C (2-3 m)) \sin (c+d x) \sec ^{m-1}(c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{6} (5-3 m),\frac{1}{6} (11-3 m),\cos ^2(c+d x)\right )}{d (5-3 m) (3 m+1) \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{2/3}} \]
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Rubi [A] time = 0.126757, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {20, 4046, 3772, 2643} \[ \frac{3 C \sin (c+d x) \sec ^{m+1}(c+d x)}{d (3 m+1) (b \sec (c+d x))^{2/3}}-\frac{3 (3 A m+A-C (2-3 m)) \sin (c+d x) \sec ^{m-1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (5-3 m);\frac{1}{6} (11-3 m);\cos ^2(c+d x)\right )}{d (5-3 m) (3 m+1) \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{2/3}} \]
Antiderivative was successfully verified.
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Rule 20
Rule 4046
Rule 3772
Rule 2643
Rubi steps
\begin{align*} \int \frac{\sec ^m(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(b \sec (c+d x))^{2/3}} \, dx &=\frac{\sec ^{\frac{2}{3}}(c+d x) \int \sec ^{-\frac{2}{3}+m}(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx}{(b \sec (c+d x))^{2/3}}\\ &=\frac{3 C \sec ^{1+m}(c+d x) \sin (c+d x)}{d (1+3 m) (b \sec (c+d x))^{2/3}}+\frac{\left (\left (C \left (-\frac{2}{3}+m\right )+A \left (\frac{1}{3}+m\right )\right ) \sec ^{\frac{2}{3}}(c+d x)\right ) \int \sec ^{-\frac{2}{3}+m}(c+d x) \, dx}{\left (\frac{1}{3}+m\right ) (b \sec (c+d x))^{2/3}}\\ &=\frac{3 C \sec ^{1+m}(c+d x) \sin (c+d x)}{d (1+3 m) (b \sec (c+d x))^{2/3}}+\frac{\left (\left (C \left (-\frac{2}{3}+m\right )+A \left (\frac{1}{3}+m\right )\right ) \cos ^{\frac{1}{3}+m}(c+d x) \sec ^{1+m}(c+d x)\right ) \int \cos ^{\frac{2}{3}-m}(c+d x) \, dx}{\left (\frac{1}{3}+m\right ) (b \sec (c+d x))^{2/3}}\\ &=\frac{3 C \sec ^{1+m}(c+d x) \sin (c+d x)}{d (1+3 m) (b \sec (c+d x))^{2/3}}-\frac{3 (A-C (2-3 m)+3 A m) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (5-3 m);\frac{1}{6} (11-3 m);\cos ^2(c+d x)\right ) \sec ^{-1+m}(c+d x) \sin (c+d x)}{d (5-3 m) (1+3 m) (b \sec (c+d x))^{2/3} \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [C] time = 8.24401, size = 311, normalized size = 2.14 \[ -\frac{3 i 2^{m+\frac{1}{3}} \left (\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{m-\frac{2}{3}} \left (1+e^{2 i (c+d x)}\right )^{m-\frac{2}{3}} \left (A+C \sec ^2(c+d x)\right ) \left ((3 m+10) \left (2 (3 m-2) (A+2 C) e^{2 i (c+d x)} \text{Hypergeometric2F1}\left (m+\frac{4}{3},\frac{1}{6} (3 m+4),\frac{m}{2}+\frac{5}{3},-e^{2 i (c+d x)}\right )+A (3 m+4) \text{Hypergeometric2F1}\left (m+\frac{4}{3},\frac{1}{6} (3 m-2),\frac{1}{6} (3 m+4),-e^{2 i (c+d x)}\right )\right )+A \left (9 m^2+6 m-8\right ) e^{4 i (c+d x)} \text{Hypergeometric2F1}\left (\frac{m}{2}+\frac{5}{3},m+\frac{4}{3},\frac{m}{2}+\frac{8}{3},-e^{2 i (c+d x)}\right )\right )}{d (3 m-2) (3 m+4) (3 m+10) \sec ^{\frac{4}{3}}(c+d x) (b \sec (c+d x))^{2/3} (A \cos (2 c+2 d x)+A+2 C)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.149, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \sec \left ( dx+c \right ) \right ) ^{m} \left ( A+C \left ( \sec \left ( dx+c \right ) \right ) ^{2} \right ) \left ( b\sec \left ( dx+c \right ) \right ) ^{-{\frac{2}{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 )} \sec \left (d x + c\right )^{m}}{\left (b \sec \left (d x + c\right )\right )^{\frac{2}{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 \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac{1}{3}} \sec \left (d x + c\right )^{m}}{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 ) \sec ^{m}{\left (c + d x \right )}}{\left (b \sec{\left (c + d x \right )}\right )^{\frac{2}{3}}}\, 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 )} \sec \left (d x + c\right )^{m}}{\left (b \sec \left (d x + c\right )\right )^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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