Optimal. Leaf size=146 \[ \frac {3 C \sin (c+d x) (b \sec (c+d x))^{2/3} \sec ^{m+1}(c+d x)}{d (3 m+5)}-\frac {3 (A (3 m+5)+C (3 m+2)) \sin (c+d x) (b \sec (c+d x))^{2/3} \sec ^{m-1}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (1-3 m);\frac {1}{6} (7-3 m);\cos ^2(c+d x)\right )}{d (1-3 m) (3 m+5) \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.12, antiderivative size = 146, normalized size of antiderivative = 1.00, 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} \[ \frac {3 C \sin (c+d x) (b \sec (c+d x))^{2/3} \sec ^{m+1}(c+d x)}{d (3 m+5)}-\frac {3 (A (3 m+5)+C (3 m+2)) \sin (c+d x) (b \sec (c+d x))^{2/3} \sec ^{m-1}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (1-3 m);\frac {1}{6} (7-3 m);\cos ^2(c+d x)\right )}{d (1-3 m) (3 m+5) \sqrt {\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 20
Rule 2643
Rule 3772
Rule 4046
Rubi steps
\begin {align*} \int \sec ^m(c+d x) (b \sec (c+d x))^{2/3} \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {(b \sec (c+d x))^{2/3} \int \sec ^{\frac {2}{3}+m}(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx}{\sec ^{\frac {2}{3}}(c+d x)}\\ &=\frac {3 C \sec ^{1+m}(c+d x) (b \sec (c+d x))^{2/3} \sin (c+d x)}{d (5+3 m)}+\frac {\left (\left (C \left (\frac {2}{3}+m\right )+A \left (\frac {5}{3}+m\right )\right ) (b \sec (c+d x))^{2/3}\right ) \int \sec ^{\frac {2}{3}+m}(c+d x) \, dx}{\left (\frac {5}{3}+m\right ) \sec ^{\frac {2}{3}}(c+d x)}\\ &=\frac {3 C \sec ^{1+m}(c+d x) (b \sec (c+d x))^{2/3} \sin (c+d x)}{d (5+3 m)}+\frac {\left (\left (C \left (\frac {2}{3}+m\right )+A \left (\frac {5}{3}+m\right )\right ) \cos ^{\frac {2}{3}+m}(c+d x) \sec ^m(c+d x) (b \sec (c+d x))^{2/3}\right ) \int \cos ^{-\frac {2}{3}-m}(c+d x) \, dx}{\frac {5}{3}+m}\\ &=\frac {3 C \sec ^{1+m}(c+d x) (b \sec (c+d x))^{2/3} \sin (c+d x)}{d (5+3 m)}-\frac {3 (C (2+3 m)+A (5+3 m)) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (1-3 m);\frac {1}{6} (7-3 m);\cos ^2(c+d x)\right ) \sec ^{-1+m}(c+d x) (b \sec (c+d x))^{2/3} \sin (c+d x)}{d (1-3 m) (5+3 m) \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [C] time = 3.23, size = 303, normalized size = 2.08 \[ -\frac {3 i 2^{m+\frac {5}{3}} e^{-\frac {1}{3} i (3 m+5) (c+d x)} \left (\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{m+\frac {5}{3}} (b \sec (c+d x))^{2/3} \left (A+C \sec ^2(c+d x)\right ) \left (\frac {2 (A+2 C) e^{\frac {1}{3} i (3 m+8) (c+d x)} \, _2F_1\left (1,\frac {1}{6} (-3 m-2);\frac {m}{2}+\frac {7}{3};-e^{2 i (c+d x)}\right )}{3 m+8}+\frac {A e^{\frac {1}{3} i (3 m+2) (c+d x)} \, _2F_1\left (1,\frac {1}{6} (-3 m-8);\frac {1}{6} (3 m+8);-e^{2 i (c+d x)}\right )}{3 m+2}+\frac {A e^{\frac {1}{3} i (3 m+14) (c+d x)} \, _2F_1\left (1,\frac {1}{6} (4-3 m);\frac {1}{6} (3 m+20);-e^{2 i (c+d x)}\right )}{3 m+14}\right )}{d \sec ^{\frac {8}{3}}(c+d x) (A \cos (2 c+2 d x)+A+2 C)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac {2}{3}} \sec \left (d x + c\right )^{m}, x\right ) \]
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 \[ \int {\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac {2}{3}} \sec \left (d x + c\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.44, size = 0, normalized size = 0.00 \[ \int \left (\sec ^{m}\left (d x +c \right )\right ) \left (b \sec \left (d x +c \right )\right )^{\frac {2}{3}} \left (A +C \left (\sec ^{2}\left (d x +c \right )\right )\right )\, dx \]
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 \[ \int {\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac {2}{3}} \sec \left (d x + c\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{2/3}\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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