Optimal. Leaf size=92 \[ \frac {3 A b^2 \sin (c+d x)}{8 d (b \cos (c+d x))^{8/3}}+\frac {3 (5 A+8 C) \sin (c+d x) \, _2F_1\left (-\frac {1}{3},\frac {1}{2};\frac {2}{3};\cos ^2(c+d x)\right )}{16 d \sqrt {\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}} \]
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Rubi [A] time = 0.10, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {16, 3012, 2643} \[ \frac {3 A b^2 \sin (c+d x)}{8 d (b \cos (c+d x))^{8/3}}+\frac {3 (5 A+8 C) \sin (c+d x) \, _2F_1\left (-\frac {1}{3},\frac {1}{2};\frac {2}{3};\cos ^2(c+d x)\right )}{16 d \sqrt {\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2643
Rule 3012
Rubi steps
\begin {align*} \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx &=b^3 \int \frac {A+C \cos ^2(c+d x)}{(b \cos (c+d x))^{11/3}} \, dx\\ &=\frac {3 A b^2 \sin (c+d x)}{8 d (b \cos (c+d x))^{8/3}}+\frac {1}{8} (b (5 A+8 C)) \int \frac {1}{(b \cos (c+d x))^{5/3}} \, dx\\ &=\frac {3 A b^2 \sin (c+d x)}{8 d (b \cos (c+d x))^{8/3}}+\frac {3 (5 A+8 C) \, _2F_1\left (-\frac {1}{3},\frac {1}{2};\frac {2}{3};\cos ^2(c+d x)\right ) \sin (c+d x)}{16 d (b \cos (c+d x))^{2/3} \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [C] time = 6.29, size = 473, normalized size = 5.14 \[ b \left (\frac {\cos ^4(c+d x) \left (A \sec ^2(c+d x)+C\right ) \left (\frac {3 \sec (c) \sec (c+d x) (5 A \sin (d x)+8 C \sin (d x))}{8 d}+\frac {3 (5 A+8 C) \csc (c) \sec (c)}{8 d}+\frac {3 A \sec (c) \sin (d x) \sec ^3(c+d x)}{4 d}+\frac {3 A \tan (c) \sec ^2(c+d x)}{4 d}\right )}{(b \cos (c+d x))^{5/3} (2 A+C \cos (2 c+2 d x)+C)}-\frac {i (5 A+8 C) \csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right ) \cos ^{\frac {11}{3}}(c+d x) \left (A \sec ^2(c+d x)+C\right ) \left (-\frac {3 i e^{-i d x} \left (2 i \sin (2 c) e^{2 i d x}+2 \cos (2 c) e^{2 i d x}+2\right )^{2/3} \, _2F_1\left (-\frac {1}{6},\frac {2}{3};\frac {5}{6};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right )}{d \left (e^{-i d x} \left (i \sin (c) \left (-1+e^{2 i d x}\right )+\cos (c) \left (1+e^{2 i d x}\right )\right )\right )^{2/3}}-\frac {3 i e^{i d x} \left (2 i \sin (2 c) e^{2 i d x}+2 \cos (2 c) e^{2 i d x}+2\right )^{2/3} \, _2F_1\left (\frac {2}{3},\frac {5}{6};\frac {11}{6};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right )}{5 d \left (e^{-i d x} \left (i \sin (c) \left (-1+e^{2 i d x}\right )+\cos (c) \left (1+e^{2 i d x}\right )\right )\right )^{2/3}}\right )}{32 (b \cos (c+d x))^{5/3} (2 A+C \cos (2 c+2 d x)+C)}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {1}{3}} \sec \left (d x + c\right )^{3}}{b \cos \left (d x + c\right )}, x\right ) \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \sec \left (d x + c\right )^{3}}{\left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.44, size = 0, normalized size = 0.00 \[ \int \frac {\left (A +C \left (\cos ^{2}\left (d x +c \right )\right )\right ) \left (\sec ^{3}\left (d x +c \right )\right )}{\left (b \cos \left (d x +c \right )\right )^{\frac {2}{3}}}\, dx \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \sec \left (d x + c\right )^{3}}{\left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {C\,{\cos \left (c+d\,x\right )}^2+A}{{\cos \left (c+d\,x\right )}^3\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{2/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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