Optimal. Leaf size=226 \[ -\frac {3 (3 A m+A-C (2-3 m)) \sin (c+d x) \sec ^{m-1}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (5-3 m);\frac {1}{6} (11-3 m);\cos ^2(c+d x)\right )}{d (5-3 m) (3 m+1) \sqrt {\sin ^2(c+d x)} (b \sec (c+d x))^{2/3}}-\frac {3 B \sin (c+d x) \sec ^m(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (2-3 m);\frac {1}{6} (8-3 m);\cos ^2(c+d x)\right )}{d (2-3 m) \sqrt {\sin ^2(c+d x)} (b \sec (c+d x))^{2/3}}+\frac {3 C \sin (c+d x) \sec ^{m+1}(c+d x)}{d (3 m+1) (b \sec (c+d x))^{2/3}} \]
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Rubi [A] time = 0.19, antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {20, 4047, 3772, 2643, 4046} \[ -\frac {3 (3 A m+A-C (2-3 m)) \sin (c+d x) \sec ^{m-1}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (5-3 m);\frac {1}{6} (11-3 m);\cos ^2(c+d x)\right )}{d (5-3 m) (3 m+1) \sqrt {\sin ^2(c+d x)} (b \sec (c+d x))^{2/3}}-\frac {3 B \sin (c+d x) \sec ^m(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (2-3 m);\frac {1}{6} (8-3 m);\cos ^2(c+d x)\right )}{d (2-3 m) \sqrt {\sin ^2(c+d x)} (b \sec (c+d x))^{2/3}}+\frac {3 C \sin (c+d x) \sec ^{m+1}(c+d x)}{d (3 m+1) (b \sec (c+d x))^{2/3}} \]
Antiderivative was successfully verified.
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Rule 20
Rule 2643
Rule 3772
Rule 4046
Rule 4047
Rubi steps
\begin {align*} \int \frac {\sec ^m(c+d x) \left (A+B \sec (c+d x)+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+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx}{(b \sec (c+d x))^{2/3}}\\ &=\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 {\left (B \sec ^{\frac {2}{3}}(c+d x)\right ) \int \sec ^{\frac {1}{3}+m}(c+d x) \, 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 {\left (B \cos ^{\frac {1}{3}+m}(c+d x) \sec ^{1+m}(c+d x)\right ) \int \cos ^{-\frac {1}{3}-m}(c+d x) \, 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 {3 B \, _2F_1\left (\frac {1}{2},\frac {1}{6} (2-3 m);\frac {1}{6} (8-3 m);\cos ^2(c+d x)\right ) \sec ^m(c+d x) \sin (c+d x)}{d (2-3 m) (b \sec (c+d x))^{2/3} \sqrt {\sin ^2(c+d x)}}+\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)}}-\frac {3 B \, _2F_1\left (\frac {1}{2},\frac {1}{6} (2-3 m);\frac {1}{6} (8-3 m);\cos ^2(c+d x)\right ) \sec ^m(c+d x) \sin (c+d x)}{d (2-3 m) (b \sec (c+d x))^{2/3} \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [C] time = 10.81, size = 545, normalized size = 2.41 \[ -\frac {3 i 2^{m+\frac {1}{3}} e^{-\frac {1}{3} i (3 c+d (3 m+1) x)} \left (\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{m+\frac {1}{3}} \left (1+e^{2 i (c+d x)}\right )^{m+\frac {1}{3}} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left ((3 m+10) \left (2 (3 m-2) e^{\frac {1}{3} i (3 c+d (3 m+1) x)} \left ((3 m+1) e^{i (c+d x)} \left ((3 m+7) (A+2 C) \, _2F_1\left (m+\frac {4}{3},\frac {1}{6} (3 m+4);\frac {m}{2}+\frac {5}{3};-e^{2 i (c+d x)}\right )+B (3 m+4) e^{i (c+d x)} \, _2F_1\left (m+\frac {4}{3},\frac {1}{6} (3 m+7);\frac {1}{6} (3 m+13);-e^{2 i (c+d x)}\right )\right )+B \left (9 m^2+33 m+28\right ) \, _2F_1\left (m+\frac {4}{3},\frac {1}{6} (3 m+1);\frac {1}{6} (3 m+7);-e^{2 i (c+d x)}\right )\right )+A \left (27 m^3+108 m^2+117 m+28\right ) e^{\frac {1}{3} i d (3 m-2) x} \, _2F_1\left (m+\frac {4}{3},\frac {1}{6} (3 m-2);\frac {1}{6} (3 m+4);-e^{2 i (c+d x)}\right )\right )+A \left (81 m^4+270 m^3+135 m^2-150 m-56\right ) e^{4 i c+\frac {1}{3} i d (3 m+10) x} \, _2F_1\left (\frac {m}{2}+\frac {5}{3},m+\frac {4}{3};\frac {m}{2}+\frac {8}{3};-e^{2 i (c+d x)}\right )\right )}{d (3 m-2) (3 m+1) (3 m+4) (3 m+7) (3 m+10) \sec ^{\frac {4}{3}}(c+d x) (b \sec (c+d x))^{2/3} (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac {1}{3}} \sec \left (d x + c\right )^{m}}{b \sec \left (d x + c\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{m}}{\left (b \sec \left (d x + c\right )\right )^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.18, size = 0, normalized size = 0.00 \[ \int \frac {\left (\sec ^{m}\left (d x +c \right )\right ) \left (A +B \sec \left (d x +c \right )+C \left (\sec ^{2}\left (d x +c \right )\right )\right )}{\left (b \sec \left (d x +c \right )\right )^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{m}}{\left (b \sec \left (d x + c\right )\right )^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^m\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{2/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{m}{\left (c + d x \right )}}{\left (b \sec {\left (c + d x \right )}\right )^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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