Optimal. Leaf size=227 \[ -\frac {3 (A (3 m+4)+3 C m+C) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (3 m+1);\frac {1}{6} (3 m+7);\cos ^2(c+d x)\right )}{d (3 m+1) (3 m+4) \sqrt {\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}-\frac {3 B \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (3 m+4);\frac {1}{6} (3 m+10);\cos ^2(c+d x)\right )}{d (3 m+4) \sqrt {\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}+\frac {3 C \sin (c+d x) \cos ^{m+1}(c+d x)}{d (3 m+4) (b \cos (c+d x))^{2/3}} \]
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Rubi [A] time = 0.23, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {20, 3023, 2748, 2643} \[ -\frac {3 (A (3 m+4)+3 C m+C) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (3 m+1);\frac {1}{6} (3 m+7);\cos ^2(c+d x)\right )}{d (3 m+1) (3 m+4) \sqrt {\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}-\frac {3 B \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (3 m+4);\frac {1}{6} (3 m+10);\cos ^2(c+d x)\right )}{d (3 m+4) \sqrt {\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}+\frac {3 C \sin (c+d x) \cos ^{m+1}(c+d x)}{d (3 m+4) (b \cos (c+d x))^{2/3}} \]
Antiderivative was successfully verified.
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Rule 20
Rule 2643
Rule 2748
Rule 3023
Rubi steps
\begin {align*} \int \frac {\cos ^m(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{2/3}} \, dx &=\frac {\cos ^{\frac {2}{3}}(c+d x) \int \cos ^{-\frac {2}{3}+m}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx}{(b \cos (c+d x))^{2/3}}\\ &=\frac {3 C \cos ^{1+m}(c+d x) \sin (c+d x)}{d (4+3 m) (b \cos (c+d x))^{2/3}}+\frac {\left (3 \cos ^{\frac {2}{3}}(c+d x)\right ) \int \cos ^{-\frac {2}{3}+m}(c+d x) \left (\frac {1}{3} \left (3 C \left (\frac {1}{3}+m\right )+3 A \left (\frac {4}{3}+m\right )\right )+\frac {1}{3} B (4+3 m) \cos (c+d x)\right ) \, dx}{(4+3 m) (b \cos (c+d x))^{2/3}}\\ &=\frac {3 C \cos ^{1+m}(c+d x) \sin (c+d x)}{d (4+3 m) (b \cos (c+d x))^{2/3}}+\frac {\left (B \cos ^{\frac {2}{3}}(c+d x)\right ) \int \cos ^{\frac {1}{3}+m}(c+d x) \, dx}{(b \cos (c+d x))^{2/3}}+\frac {\left ((C+3 C m+A (4+3 m)) \cos ^{\frac {2}{3}}(c+d x)\right ) \int \cos ^{-\frac {2}{3}+m}(c+d x) \, dx}{(4+3 m) (b \cos (c+d x))^{2/3}}\\ &=\frac {3 C \cos ^{1+m}(c+d x) \sin (c+d x)}{d (4+3 m) (b \cos (c+d x))^{2/3}}-\frac {3 (C+3 C m+A (4+3 m)) \cos ^{1+m}(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 ) \sin (c+d x)}{d (1+3 m) (4+3 m) (b \cos (c+d x))^{2/3} \sqrt {\sin ^2(c+d x)}}-\frac {3 B \cos ^{2+m}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (4+3 m);\frac {1}{6} (10+3 m);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (4+3 m) (b \cos (c+d x))^{2/3} \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.41, size = 164, normalized size = 0.72 \[ -\frac {3 \sin (c+d x) \cos ^{m+1}(c+d x) \left ((A (3 m+4)+3 C m+C) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (3 m+1);\frac {1}{6} (3 m+7);\cos ^2(c+d x)\right )+(3 m+1) \left (B \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (3 m+4);\frac {m}{2}+\frac {5}{3};\cos ^2(c+d x)\right )-C \sqrt {\sin ^2(c+d x)}\right )\right )}{d (3 m+1) (3 m+4) \sqrt {\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.36, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {1}{3}} \cos \left (d x + c\right )^{m}}{b \cos \left (d x + c\right )}, x\right ) \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{m}}{\left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.49, size = 0, normalized size = 0.00 \[ \int \frac {\left (\cos ^{m}\left (d x +c \right )\right ) \left (A +B \cos \left (d x +c \right )+C \left (\cos ^{2}\left (d x +c \right )\right )\right )}{\left (b \cos \left (d x +c \right )\right )^{\frac {2}{3}}}\, dx \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{m}}{\left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\cos \left (c+d\,x\right )}^m\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\left (b\,\cos \left (c+d\,x\right )\right )}^{2/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (A + B \cos {\left (c + d x \right )} + C \cos ^{2}{\left (c + d x \right )}\right ) \cos ^{m}{\left (c + d x \right )}}{\left (b \cos {\left (c + d x \right )}\right )^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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