Optimal. Leaf size=146 \[ \frac {3 C \sin (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+1}(c+d x)}{d (3 m+8)}-\frac {3 (A (3 m+8)+C (3 m+5)) \sin (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+1}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (3 m+5);\frac {1}{6} (3 m+11);\cos ^2(c+d x)\right )}{d (3 m+5) (3 m+8) \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.12, antiderivative size = 136, normalized size of antiderivative = 0.93, 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 = {20, 3014, 2643} \[ \frac {3 C \sin (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+1}(c+d x)}{d (3 m+8)}-\frac {3 \left (\frac {A}{3 m+5}+\frac {C}{3 m+8}\right ) \sin (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+1}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (3 m+5);\frac {1}{6} (3 m+11);\cos ^2(c+d x)\right )}{d \sqrt {\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 20
Rule 2643
Rule 3014
Rubi steps
\begin {align*} \int \cos ^m(c+d x) (b \cos (c+d x))^{2/3} \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac {(b \cos (c+d x))^{2/3} \int \cos ^{\frac {2}{3}+m}(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx}{\cos ^{\frac {2}{3}}(c+d x)}\\ &=\frac {3 C \cos ^{1+m}(c+d x) (b \cos (c+d x))^{2/3} \sin (c+d x)}{d (8+3 m)}+\frac {\left (\left (C \left (\frac {5}{3}+m\right )+A \left (\frac {8}{3}+m\right )\right ) (b \cos (c+d x))^{2/3}\right ) \int \cos ^{\frac {2}{3}+m}(c+d x) \, dx}{\left (\frac {8}{3}+m\right ) \cos ^{\frac {2}{3}}(c+d x)}\\ &=\frac {3 C \cos ^{1+m}(c+d x) (b \cos (c+d x))^{2/3} \sin (c+d x)}{d (8+3 m)}-\frac {3 (C (5+3 m)+A (8+3 m)) \cos ^{1+m}(c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left (\frac {1}{2},\frac {1}{6} (5+3 m);\frac {1}{6} (11+3 m);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (5+3 m) (8+3 m) \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 142, normalized size = 0.97 \[ -\frac {3 \sqrt {\sin ^2(c+d x)} \csc (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+1}(c+d x) \left (A (3 m+11) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (3 m+5);\frac {1}{6} (3 m+11);\cos ^2(c+d x)\right )+C (3 m+5) \cos ^2(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (3 m+11);\frac {1}{6} (3 m+17);\cos ^2(c+d x)\right )\right )}{d (3 m+5) (3 m+11)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}} \cos \left (d x + c\right )^{m}, x\right ) \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}} \cos \left (d x + c\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.35, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{m}\left (d x +c \right )\right ) \left (b \cos \left (d x +c \right )\right )^{\frac {2}{3}} \left (A +C \left (\cos ^{2}\left (d x +c \right )\right )\right )\, dx \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}} \cos \left (d x + c\right )^{m}\,{d x} \]
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 \[ \int {\cos \left (c+d\,x\right )}^m\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{2/3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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